Equations in HNN-extensions.

M. Lohrey and Geraud Senizergues

1 Inst. fur Informatik, Universitat Leipzig, Germany2 LABRI, Universite Bordeaux I, France

lohrey@informatik.uni-leipzig.de, ges@labri.fr

Abstract. Let H be a cancellative monoid and let G be an HNN-extension of H with finite associated subgroups A,B ≤ H. We showthat, if equations are algorithmically solvable in H, then they are alsoalgorithmically solvable in G. The result also holds for equations withrational constraints and for the existential first order theory. Analogousresults are derived for amalgamated product with finite amalgamatedsubgroups. Finally, a transfer theorem is shown for the fundamentalgroup of a finite graph of groups with finite edge groups and vertexgroups where the existential first-order theory is decidable.

Keywords:Equations; groups and monoids; HNN-extensions; amalgamated product.Date:June 20, 2008

1 Introduction

General context Theories of equations over groups are a classical research topicat the borderline between algebra, mathematical logic, and theoretical computerscience. This line of research was initiated by the work of Lyndon, Tarski, andothers in the first half of the 20th century. A major driving force for the de-velopment of this field was a question that was posed by Tarski around 1945:Is the first-order theory of a free group F of rank two, i.e, the set of all state-ments of first-order logic with equations as atomic propositions that are true inF , decidable. Decidability results for fragments of this theory were obtained byMakanin (for the existential theory of a free group) [Mak83] and Merzlyakov andMakanin (for the positive theory of a free group) [Mak84,Mer66]. A complete(positive) solution of Tarski’s problem was finally announced in [KM98]; thecomplete solution is spread over a series of papers. The complexity of Makanin’salgorithm for deciding the existential theory of a free group was shown to benot primitive recursive in [KP98]. Based on [Pla99], a new PSPACE algorithmfor the existential theory of a free group, which also allows to include rationalconstraints for variables, was presented in [DHG05]. Beside these results for freegroups, also extensions to larger classes of groups were obtained in the past:[DL04,DM06,KMS05,RS95]. In [DL03], a general transfer theorem for existen-tial theories was shown: the decidability of the existential theory is preservedby graph products over groups — a construction that generalizes both free anddirect products, see e.g. [Gre90]. Moreover, it is shown in [DL03] that for a largeclass of graph products, the positive theory can be reduced to the existentialtheory.

The aim of this paper is to prove similar transfer theorems for HNN-extensions(this kind of extension was introduced in [HNN49], its definition is recalled byeq. (1) of section 2) and amalgamated free products (which is a classical toolin algebraic topology, its definition is recalled by eq. (10) of section 2). Thesetwo operations are of fundamental importance in combinatorial group theory[LS77]. One of the first important applications of HNN-extensions was a moretransparent proof of the celebrated result of Novikov and Boone on the existenceof a finitely presented group with an undecidable word problem, see e.g. [LS77].Such a group can be constructed by a series of HNN-extensions starting froma free group. This shows that there is no hope to prove a transfer theorem forHNN-extensions, similar to the one for graph products from [DL03]. Thereforewe mainly consider HNN-extensions and amalgamated free products, where thesubgroup A in (1) and (10), respectively, is finite. These restrictions appear alsoin other contexts in combinatorial group theory: A seminal result of Stallings[Sta71] states that every group G with more than one end can be either writtenin the form (10) with A finite or in the form (1) with A finite. Those groupswhich can be built up from finite groups using the operations of amalgamatedfree products and HNN-extensions, both subject to the finiteness restrictionsabove, are precisely the virtually-free groups [DD90] (i.e., those groups with afree subgroup of finite index). Virtually-free groups also have strong connectionsto formal language theory and infinite graph theory [MS83].

2

Main results of the paper This paper is part of a sequence of three articles dealingwith transfer theorems for HNN-extensions and free products with amalgama-tion where the associated subgroups are finite (see in [LS06] a survey of the fullsequence). In [LS08] we studied the membership problem and other related al-gorithmic problems about rational subsets of monoids and subgroups of groups.Here we study the satisfiability problem for systems of equations (possibly to-gether with disequations and rational constraints) in monoids. In a forthcomingpaper ([LS05]) we study the validity of positive first-order formulas in groups.

The main result of this paper is Theorem 2: it states that the satisfiabil-ity problem for equations with rational constraints in a monoid G, which is anHNN-extension of a cancellative monoid H , with finite associated subgroups, isTuring-reducible to the same problem over the base monoid H.Several variations and corollaries of this result are also derived:- the same transfer property is shown for systems of equations with constants(Theorem 5) and for systems of equations and disequations with rational con-straints (Theorem 6);- a similar theorem is proved for systems of equations and disequations withrational constraints in an amalgamated product (Theorem 10).As a corollary, the satisfiability problem for equations and disequations withrational constraints in the fundamental group of a finite graph of groups withfinite edge groups is Turing-reducible to the join of the same problems in thevertex groups (Theorem 11).

ContentsSection 2 consists of recalls and notation on the subject of monoids and groups,rational subsets of monoids, equations in monoids and algorithmic problems ingeneral. In section 3 we introduce the main technical tool allowing us to dealwith equations in HNN-extensions: the notion of AB-algebra. This notion is de-fined in general. We then study two particular AB-algebras, denoted by Ht andW, which are crucial in our reductions. The domain of the AB-algebra Ht isessentially the set of reduced sequences of the HNN-extension; the domain ofW is a free product of the finite associated subgroups A,B with a free monoidgenerated by all the possible “types” of sequences that might occur in Ht, quo-tiented by relations expressing the pseudo-commutations between each type ofsequence and the elements of A ∪ B. In section 4 we make precise our notionof system of equations with rational constraints in a monoid and give normalforms for such systems. In section 5 we provide a reduction of equations overan HNN-extension G of a monoid H to equations over the AB-algebra Ht andequations over the base monoid H. In section 6 we reduce equations over theAB-algebra Ht to equations over the AB-algebra W. In section 7 we reduceequations over the the AB-algebra W to equations over some finitely generatedgroup U which turns out to be virtually-free. In section 8 we show, by induc-tion over the size of the associated subgroups A,B, that equations in U arealgorithmically solvable. We then prove a transfer theorem concerning groupsonly and prove finally our main Theorem 2 concerning cancellative monoids. Insection 9 we adapt the proof-techniques developed in the previous sections in

3

order to prove some variants of Theorem 2 where the constraints can be pos-itive only, in particular to the case of equations with constants. In section 10we extend to equations and disequations the reduction exposed in section 5. Wethus prove some transfer theorems for systems of equations and disequationsin HNN-extensions of cancellative monoids. In section 11 we deduce from ourprevious results on HNN-extensions a transfer theorem for systems of equationsand disequations with rational constraints in free products with amalgamationof cancellative monoids (with finite amalgamated subgroups). In section 12 wesynthesize two transfer theorems obtained previously, one for HNN-extensionsand the other for free products with amalgamation, into a single theorem aboutfinite graphs of groups.

4

Table of Contents

Equations in HNN-extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Partial semi-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Monoids and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7HNN-extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Amalgamated products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Rational subsets of a monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Equations and disequations over a monoid . . . . . . . . . . . . . . . . . . . . . 10

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Disequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Special constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Definable subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Reductions among decision problems . . . . . . . . . . . . . . . . . . . . . . . . . . 123 AB-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 AB-algebra axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 AB-homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Finite t-automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 The AB-algebra Ht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

H ∗ t, t∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Ht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 The AB-algebra W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26W∗ ∗A ∗B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Wt,WH, W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Equations with rational constraints over G . . . . . . . . . . . . . . . . . . . . . . . . . 404.1 Equations with rational constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Quadratic normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Rational constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Equations and disequations with rational constraints . . . . . . . . . . . . 425 Equations over Ht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1 t-equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2 From G-equations to t-equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

From G-solutions to t-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47From t-solutions to G-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Equations over W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.1 W-equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 From t-equations to W-equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

From t-solutions to W-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5

From W-solutions to t-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 Equations over U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.1 The group U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.2 From W-equations to U-equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

From W-solutions to U-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66From U-solutions to W-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Bijection Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8 Transfer of solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708.1 The structure of U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708.2 Transfer results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Transfer for groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Transfer for cancellative monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

9 Equations with positive rational constraints over G . . . . . . . . . . . . . . . . . . 819.1 Positive rational constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819.2 Positive subgroup constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819.3 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

10 Equations and disequations with rational constraints over G . . . . . . . . . . 8410.1 Rational constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

From G-solutions to t-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86From t-solutions to G-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10.2 Positive rational constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.3 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

11 Equations and disequations over an amalgamated product . . . . . . . . . . . . 9311.1 Free product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9311.2 Free product with amalgamation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9311.3 Equations over an amalgamated product . . . . . . . . . . . . . . . . . . . . . . 94

12 Equations and disequations over a graph of groups . . . . . . . . . . . . . . . . . . 95

6

2 Preliminaries

We recall in this section all the needed definitions and classical results concerningpartial semi-groups, semi-groups, monoids and groups.

2.1 Partial semi-groups

Let (P, ·) be a set endowed with a function · : P ×P → P . By D(·) ⊆ P ×P wedenote the domain of the law ·. The structure

〈P, ·〉

is a partial semi-group iff, for every p, q, r ∈ P ,

(p, q) ∈ D(·) ∧ ((p · q), r) ∈ D(·) ⇔ (q, r) ∈ D(·) ∧ (p, (q · r)) ∈ D(·)

and, in the case where ((p · q), r) ∈ D(·)

(p · q) · r = p · (q · r).

Let us notice that, when P is a partial semi-group, the following structure is asemi-group:

〈P(P ), ·〉

where the product is defined by , for every R,S ∈ P(P )

R · S = r · s | (r, s) ∈ R× S ∩D(·).

Let A,B be two groups. For every K,K ′ ∈ A,B, we denote by PIs(K,K ′) theset of all group isomorphisms ϕ with dom(ϕ) ⊆ K, im(ϕ) ⊆ K ′.We then define

PGI(A,B) := PIs(A,A) ∪ PIs(A,B) ∪ PIs(B,A) ∪ PIs(B,B).

The pair 〈PGI(A,B), 〉 is a partial semi-group.

2.2 Monoids and groups

Monoids A semi-group is a pair (M, ·) such that · is a total mapping M×M →M which is associative i.e. ∀x, y, z ∈ M, ((x · y) · z)) = (x · (y · z)). A monoid isa triple (M, ·, e) such that (M, ·) is a semi-group and e is a neutral element forthe law · which means that, for every m ∈ M , m · e = e · m = m. A monoid(M, ·, e) is called cancellative iff, for every x, y, z ∈M , x · y = x · z ⇒ y = z andy · x = z · x ⇒ y = z. A submonoid of a monoid (M, ·, e) is a monoid (N, , f)such that N ⊆M, ∀n, n′ ∈ N,n n′ = n ·n′ and e = f . A subgroup of a monoid(M, ·, e) is a submonoid (N, , e) of (M, ·, e) which is a group i.e. such that,∀x ∈ N, ∃x′ ∈ N, x x′ = x′ x = e. Let (M, ·, 1) be a monoid. Let us considersome elements s ∈M and define D(s) := (m,m′) ∈M×M | m·s = s·m′. Thissubset D(s) is always a sub-monoid of the product monoid M ×M . Moreover,when M is cancellative, for every s ∈M , D(s) is a partial isomorphism.

7

HNN-extensions Let us fix throughout this section, a monoid H (the basemonoid), two finite subgroups A ≤ H, B ≤ H and an isomorphism ϕ : A → B.We then consider the HNN-extension

G = 〈H, t; t−1at = ϕ(a)(a ∈ A)〉 (1)

We denote byπG : H ∗ t, t∗ → G

the homomorphism sending every h ∈ H on itself in G and mapping t to t (resp.t to t−1).

The kernel of πG coincides with the congruence ≈ over H ∗ t, t∗ generatedby the set of rules

tt ≈ tt ≈ 1, (2)

at ≈ tϕ(a), for all a ∈ A and (3)

bt ≈ tϕ−1(b), for all b ∈ B. (4)

An element of s ∈ H ∗ t, t∗ can be viewed as a word over the alphabetH ∪ t, t which has the form:

s = h0tα1h1 · · · t

αihi · · · tαnhn, (5)

where n ∈ IN, αi ∈ +1,−1, t−1 means the letter t and hi ∈ H.We name t-sequence every such s ∈ H ∗ t, t∗. The t-sequence s is said to be areduced sequence iff it does not contain any factor of the form tat (with a ∈ A)nor tbt (with b ∈ B). We denote by Red(H, t) the subset of H ∗ t, t∗ consistingof all reduced sequences. Let us denote by ∼ the congruence over H ∗ t, t∗

generated by all the rules of type (3),(4) above. The set Red(H, t) is saturatedby the congruence ∼. The following lemma is fundamental.

Lemma 1. Let s, s′ be some reduced sequences. Then s ≈ s′ if and only if s ∼ s′.

This lemma could be named “Britton’s lemma for monoids”. (See [?, AppendixA], for example, for a proof).

Lemma 2. If the monoid H is cancellative then the extension monoid G iscancellative too.

Proof. Let us use the properties of normal forms established in [?, Appendix A].We recall that N denotes the set of normal forms w.r.t. ≈ (which is also a set ofnormal forms for elements of H ∗ t, t∗ w.r.t. ∼) and ρ : H ∗ t, t∗ → N is thereduction map, associating to every t-sequence its normal form. Let us assumethat H is cancellative. We show in (1) that G is right-cancellative and in (2) thatit is left-cancellative.1- Let s, s′ ∈ N and h ∈ H:

s = h0tα1 · · · tαnhn, s′ = h′0t

α′

1 · · · tα′

mh′m

8

where h0 ∈ RA(α1), · · · , hn−1 ∈ RA(αn), hn ∈ H, h′0 ∈ RA(α′

1), · · · , h

′m−1 ∈

RA(α′m), h

′m ∈ H. For every h ∈ H:

ρ(s · h) = ρ(s′ · h)

⇔ ρ(h0tα1 · · · tαnhn · h) = ρ(h′0t

α′

1 · · · tα′

mh′m · h)

⇔ h0tα1 · · · tαn · (hn · h) = h′0t

α′

1 · · · tα′

m · (h′m · h)

⇔ n = m ∧ (∀i ∈ [1, n], αi = α′i) ∧ (∀i ∈ [1, n], hi−1 = h′i−1) ∧ hn · h = h′n · h ( prop. of the free product )

⇔ n = m ∧ (∀i ∈ [1, n], αi = α′i) ∧ (∀i ∈ [1, n], hi−1 = h′i−1) ∧ hn = h′n ( cancellativity of H).

We have thus established that, for every g, g′ ∈ G, h ∈ H

g · h = g′ · h⇒ g = g′ (6)

Since G is generated by H ∪ t, t−1, properties (6-??) imply that G is right-cancellative.2- Using a notion of left-normal form, dual to the notion of right-normal formused in (1), one can also show that G is left-cancellative.

We define the norm of a given sequence s ∈ H ∗ t, t∗ by:

‖s‖ = |s|t,t. (7)

One can easily check that, for every s, s′ ∈ H ∗ t, t∗

‖s · s′‖ = ‖s‖ + ‖s′‖; ‖s‖ = 0 ⇔ s ∈ H. (8)

The boolean norm of a t-sequence s is the boolean defined by

‖|s‖| = 1 ⇔ ‖s‖ ≥ 1. (9)

Amalgamated products Let us consider two monoids H1,H2, two finite sub-groups A1 ≤ H1, A2 ≤ H2, and an isomorphism ϕ : A1 → A2. The correspondingamalgamated product

G = 〈H1,H2; a = ϕ(a)(a ∈ A1)〉 (10)

is defined by G = (H1 ∗H2)/≈, where ≈ is the congruence on the free productH1 ∗H2 generated by the equations a = ϕ(a) for a ∈ A1. An (H1, H2)-sequenceis an element s ∈ H1 ∗H2; it has a unique decomposition of the form:

s = h0k1h1 · · · kihi · · · knhn, (11)

where n ≥ 0, h1, · · · , hi, · · · , hn−1 ∈ H2 −1, k1, · · · , ki, . . . , kn ∈ H1 −1 andh0, hn ∈ H2. We name reduced (H1, H2)-sequence every s ∈ H1 ∗H2 of the form:

s = h0k1h1 · · · kihi · · · knhn, (12)

9

where n ≥ 0, h1, · · · , hi, · · · , hn−1 ∈ H2 − A2, k1, · · · , ki, . . . , kn ∈ H1 − A1

and h0, hn ∈ H2. We denote by Red(H1 ∗ H2) the set of all reduced (H1, H2)-sequences.It is well-known that G is embedded into the HNN-extension

G = 〈H1 ∗ H2, t; t−1at = ϕ(a)(a ∈ A1)〉

by the map

h1 ∈ H1 7→ t−1h1t; h2 ∈ H2 7→ h2 (13)

(see [LS77, Theorem 2.6. p. 187] in the case where H1,H2 are groups and [LS08]for the adaptation to the free product of monoids with amalgamation over sub-groups). Further basic terminology and results on free products with amalgama-tion are given in [LS08], section 5.

2.3 Rational subsets of a monoid

Let M = (M, ·, 1M ) be some monoid. The set

Rat(M) ∈ P(P(M))

is the smallest element of P(P(M)) which posseses the finite subsets of M andwhich is closed under the operations ∪ (the union operation), · (the productoperation) and ∗ (the star operation, associating with a subset P the smallestsubmonoid of M containing P ).

We introduced in [LS08], in the particular case where M is an HNN-extension,a kind of finite automata called t-automata, which recognize exactly the rationalsubsets of M. We postpone to §3.4 a precise normal form for these automata.Given a normal finite t-automaton A and an element g ∈ G we set:

µA,G(g) = µA,1((1, H, ‖|s‖|, 1, 1), s), (14)

where s is any reduced t-sequence representing g. Since A is ∼-saturated, thevalue of µA,1(s) does not depend of the chosen representative s.

2.4 Equations and disequations over a monoid

Equations Let M = (M, ·, 1M) be some monoid and let

C ∈ P(P(M)).

A system of equations S , over M, with variables in a set U , is a family ((ui, u′i))i∈I

of elements of U∗ × U∗. This system is said to be quadratic if, for every i ∈ I,

|ui| = 1, |u′i| = 2.

10

A C-constraint over U is a map C : U → C. A M-solution of the system S withC-constraint C is any monoid homomorphism

σM : U∗ → M

fulfilling both conditions:

∀i ∈ I, σM(ui) = σM(u′i) (15)

∀U ∈ U , σM(U) ∈ C(U). (16)

Disequations A system of equations and disequations over M is a family ((ui, ci, u′i))i∈I

of elements of U∗ × =, 6= × U∗, where U is a set of variables.A M-solution of the system ((ui, ci, u

′i))i∈I with constraint C is any monoid ho-

momorphismσM : U∗ → M

fulfilling both conditions:

∀i ∈ I, σM(ui)ciσM(u′i) (17)

and (16) above. One sometimes also consider systems of equations and dise-quations with partial involution. This means that some variables u ∈ U have aformal inverse u−1 ∈ U and that a solution σM of the system is asked to respectthe partial involution i.e. to map the variable u−1 onto the inverse of σM(u). Sucha system “with partial involution” can be reduced to a system of the form aboveby adding the equations uu−1 = 1 and u−1u = 1 in the system and removingthe explicit constraint that σM respects some partial involution.

Special constraints In the case where C = m | m ∈M∪M, any systemof equations with constraints in C is called a system of equations with constants(since the variables U ∈ U such that C(U) is a singleton are seen as constants inM while the variables U ∈ U such that C(U) = M are seen as variables withoutany constraint).In the case where C = B(Rat(M)), (i.e. the boolean closure of the set of rationalsubsets of M), any system of equations with constraints in C is called a systemof equations with rational constraints.In the case where C = Rat(M), in order to emphasize the fact that we do not useconstraints that are complement of rational subsets, any system of equations withconstraints in C is called a system of equations with positive rational constraints.

Definable subsets Let M be a monoid and let C ∈ P(P(M)). We denote byEQ(M, C) the set of all subsets of M which can be defined by a system of equa-tions with constants and with constraints in C.We denote by Def∃+(M, C)) the set of all subsets of M which can be defined by

11

a logical formula of the form ∃w1 · ∃w2 · · · ∃wnΨ where Ψ is a positive booleancombination of statements which are either equations with constants or of theform v ∈ P (for some variable v and some element P ∈ C). The elements ofEQ(M, C) are called the equational subsets (with constraints in C); the elementsof Def∃+(M, C) are called the positively definable subsets (with constraints in C).

2.5 Reductions among decision problems

A a one tape Turing machine M with oracle is a Turing-machine, with threedifferent special state q?, qy, qn such that, when the current state of M is q?,M makes a transition that does not move the two tape heads, does not printanything, and enters the state qy or the state qn according to whether the wordon tape 2 belongs to L2 or not (see, for example [HU79, §8.9, p. 209-210] or[Rog87, p. 128] for a more precise definition). Such an oracle machine M is calleda (Turing)-reduction from L1 to L2 if the language recognized by M , using theoracle L2, is the language L1. When such a reduction exists we say that L1 isTuring-reducible to L2. The underlying idea is that, if somebody knows both Mand a machine M2 that recognizes L2, then he can combine them into a Turingmachine M1 that recognizes L1. Let us consider three languages L1, L2, L3. Wecall Turing reduction from L1 to (L2, L3) any Turing-reduction from L1 to theunion of a copy of L2 over an alphabet A2 with a copy of L3 over an alphabetA3 such that A2 ∩ A3 = ∅. Given decision problems P1, P2, P3, we say that P1

is Turing-reducible to P2 (or to (P2, P3)) when the formal languages L1, L2, L3

that encode the set of instances where the correct answer is “yes” are fulfilingthe above definition of the notion of reduction.

12

3 AB-algebras

We define here the notion of AB-algebra, which we devised for handling equa-tions with rational constraints in an HNN-extension.

3.1 Types

Let us consider the finite set of types T6 defined in [LS08]. We define a finitepartial semi-group 〈T , ·〉 as follows:

T = T6 × B × T6

where B = 〈0, 1, ·〉 is the monoid of booleans. The partial product is definedby: for every (p, b, q), (p′, b′, q′) ∈ T6 × B × T6, if q = p′ then

(p, b, q) · (p′, b′, q′) = (p, b+ b′, q′),

otherwise the product is undefined. As we noticed in §2.1, 〈P(T ), ·〉 is thus asemi-group.We define an involutory map IR : T6 → T6 by:

(A, T ) 7→ (A,H), (A,H) 7→ (A, T ), (B, T ) 7→ (B,H), (B,H) 7→ (B, T ), (1, H) 7→ (1, 1), (1, 1) 7→ (1, H).

We then define an involution IT : T → T by:

IT (p, b, q) = (IR(q), b, IR(p)).

One can check that IT is an involutory anti-automorphism of T . This involutioninduces an involutory semi-group anti-automorphism of 〈P(T ), ·〉 that will bedenoted by IT too. We associate to every element θ of T an “initial type”τ i(θ) ∈ T6, an “end type” τe(θ) ∈ T6, an “initial group” Gi(θ) ∈ 1, A,Band an “end group” Ge(θ) ∈ 1, A,B:

τ i(p, b, q) = p, Gi(p, b, q) = p1(p), τe(p, b, q) = q, Ge(p, b, q) = p1(q).

Each element of T is called a path-type while the elements of T6 are calledvertex-types. This terminology refers to the graph R exhibited on Figure 1.(It is a variant of the t-automaton R6 defined in [LS08], which recognizes theset of reduced t-sequences, Red(H, t)). Let us call atomic types the path-typescorresponding to the edges of R i.e. :

(A, T, 1, B,H), (B, T, 1, A,H) (18)

(A,H, 0, B, T ), (B,H, 0, A, T ), (A,H, 0, A, T ), (B,H, 0, B, T ), (19)

(1, H, 0, A, T ), (1, H, 0, B, T ), (B,H, 0, 1, 1), (A,H, 0, 1, 1), (1, H, 0, 1, 1) (20)

(1, H, 0, 1, H), (A, T, 0, A, T ), (B, T, 0, B, T ), (B,H, 0, B,H), (A,H, 0, A,H), (1, 1, 0, 1, 1).

(21)

13

0

(A,H)

(B,H)

(1, H) (1, 1)

0

0 0 0

0

0

0

0

1

0 0

0

0

1(A, T )

(B,T )

0 0

Fig. 1. graph R

This set is closed under IT . It is denoted by TA. The partial submonoid generatedby this set of atomic path-types will be denoted by TR. It is closed under theinvolution IT . The only path-types used in this work are those from TR. We callT -types all the atomic types listed in (18), H-types all the atomic types listed in(19)(20), A ∪ B-types all the atomic types listed in (21). Note that some typesare non-atomic: for example (A, T, 1, A,H) = (A, T, 1, B,H) · (B,H, 0, B, T ) ·(B, T, 1, A,H) is an element of TR − TA.

3.2 AB-algebra axioms

Let A,B be two groups (what we have in mind are the two subgroups A,B ofH leading to the HNN-extension G defined by (1)) and Q be some finite set (wehave in mind the set of states of some t-automaton A over H ∗ t, t∗). Let usdenote by B(Q) the monoid (P(Q×Q)) of binary relations over Q and by B2(Q)the direct product of the monoid B(Q) by itself. Given m ∈ B(Q),m−1 is thebinary relation m−1 = (p, q) ∈ Q× Q | (q, p) ∈ m. We consider the involutory

14

monoid anti-isomorphism IQ : B2(Q) → B2(Q) defined by

∀m,m′ ∈ B(Q), IQ(m,m′) = (m′−1,m−1).

We call AB-algebra a structure of the form

〈M, ·, 1M, I, ιA, ιB, γ, µ, δ〉 (22)

where ιA : A → M, ιB : B → M are total maps, I : M → M is a partialmap, γ : M → P(T ) is a total map, µ : T × M → B2(Q) is a total map,δ : T × M → PGI(A,B) is a total map ;fulfilling the eleven axioms (23-33) below.monoid:

(M, ·, 1M) is a monoid , (23)

embeddings:

ιA, ιB are injective monoid homomorphisms , (24)

involution I:

ιA(A) ∪ ιB(B) ⊆ dom(I) ⊆ M \ γ−1(∅) (25)

for every m,m′ ∈ M,

[γ(m) · γ(m′) 6= ∅] ⇒ [m ·m′ ∈ dom(I) ⇔ (m ∈ dom(I) ∧m′ ∈ dom(I))] (26)

I : (dom(I), ·, 1M) → (dom(I), ·, 1M) is a monoid anti-isomorphism, II = Iddom(I),(27)

almost homomorphisms:for every m,m′ ∈ M,

γ(m ·m′) ⊇ γ(m) · γ(m′), (28)

for every m,m′ ∈ M,θ ∈ γ(m),θ′ ∈ γ(m′), such that (θ,θ′) ∈ D(·),

µ(θ · θ′,m ·m′) = µ(θ,m) · µ(θ′,m′), (29)

dom(δ(θ,m)) ⊆ Gi(θ), im(δ(θ,m)) ⊆ Ge(θ) (30)

δ(θ · θ′,m ·m′) = δ(θ,m) δ(θ′,m′). (31)

commutation with I:for every a ∈ A, b ∈ B,m ∈ dom(I),θ ∈ γ(m),

I(ιA(a)) = ιA(a−1); I(ιB(b)) = ιB(b−1) (32)

γ(I(m)) = IT (γ(m)); µ(IT (θ), I(m)) = IQ(µ(θ,m)); δ(IT (θ), I(m)) = δ(θ,m)−1.(33)

Axiom (27) includes the assumption that dom(I) is a submonoid. From now on,

we denote by M this submonoid.

15

3.3 AB-homomorphisms

Let M1 = 〈M1, ·, 1M1, ιA,1, ιB,1, I1, γ1, µ1, δ1〉,M2 = 〈M2, ·, 1M2

, ιA,2, ιB,2, I2, γ2, µ2, δ2〉be two AB-algebras with the same underlying groups A,B and set Q. We callAB-homomorphism from M1 to M2 any map ψ : M1 → M2 fulfilling the sevenproperties (34-40) below:m-homomorphism:

ψ : (M1, ·, 1M1) → (M2, ·, 1M2

) is a monoid homomorphism (34)

ι-preservation:

∀a ∈ A, ∀b ∈ B,ψ(ιA,1(a)) = ιA,2(a), ψ(ιB,1(b)) = ιB,2(b) (35)

I-preservation:

∀m ∈ M1 − γ−11 (∅), m ∈ dom(I1) ⇔ ψ(m) ∈ dom(I2) (36)

∀m ∈ M1, I2(ψ(m)) = ψ(I1(m)) (37)

γ-compatibility:∀m ∈ M1, γ2(ψ(m)) ⊇ γ1(m) (38)

µ-preservation:

∀m ∈ M1, ∀θ ∈ γ1(m), µ2(θ, ψ(m)) = µ1(θ,m), (39)

δ-preservation:

∀m ∈ M1, ∀θ ∈ γ1(m), δ2(θ, ψ(m)) = δ1(θ,m). (40)

M1 is said to be a sub-AB-algebra of M2 if M1 ⊆ M2 and the inclusion mapι : M1 → M2 is an AB-homomorphism.Given a submonoid (M1, ·, 1) of (M2, ·, 1) which contains ι2(A) ∪ ι2(B) and isclosed under I2, there is a natural way to endow M1 with a structure of sub-AB-algebra of M2: it suffices to set ι1 := ι ι2 , I1 := ι I2, γ1 := ι γ2, µ1 := ι µ2

and δ1 := ι δ2 (where ι denotes the inclusion map from M1 into M2). Thisstructure is called the AB-structure induced by M2 on M1.

3.4 Finite t-automata

Given an HNN-extension of the form (1), we have defined in §4.1 of [LS08] anotion of finite automata that recognize subsets of H ∗ t, t−1∗. We recall herethe main definitions and refer to [LS08] for further details and proofs. A finitet-automaton over H ∗ t, t∗ with labeling set F ⊆ P(H) is a 5-tuple

A = 〈L,Q, ∆, I,T〉, (41)

where:(i) L is a finite subset of F ∪ P(A) ∪ P(B) ∪ t, t−1,

16

(ii) Q is a finite set of states,(iii) I ⊆ Q is the set of initial states,(iv) T ⊆ Q is the set of terminal states, and(v) ∆ ⊆ Q × L× Q is the set of transitions.Such an automaton induces a representation map µA : H∗t, t−1∗ → P(Q×Q)defined by: for every x ∈ H∪t, t−1 \ 1 and every s ∈ H∗ t, t−1∗of the form(5):

µA,0(1) = (q, q) | q ∈ Q ∪ (q, r) ∈ Q×Q | ∃(q, L, r) ∈ ∆ : 1 ∈ L

µA,0(x) = (q, r) ∈ Q×Q | ∃(q, L, r) ∈ ∆ : x ∈ L

µA(s) = µA,0(h0) µA,0(tα1) µA,0(h1) · · ·µA,0(t

αn) µA,0(hn).

A recognizes the set

L(A) = s ∈ H ∗ t, t−1∗ | (I × T) ∩ µA(s) 6= ∅.

We recall that a partitioned finite t-automaton over F is a 6-tuple A = 〈L,Q, τ,∆, I,T〉,where 〈L,Q, ∆, I,T〉 is as in (41) and τ : Q→ T6 maps each state to a vertex-typein such a way that

τ(I) ⊆ (1, H), τ(T) ⊆ (1, 1), ∀(q, L, r) ∈ ∆ : τ(q) × L× τ(r) ⊆ E6.

(here E6 is, roughly speaking, the set of transitions of the natural t-automatonR6 recognizing the set of all reduced sequences; see [LS08] §4.1).The representation map µA associated to such a partitioned t-automaton is justthe map µA associated to its underlying t-automaton (obtained by forgettingthe map τ). For every state q ∈ Q, we denote by γ(q) the subgroup

γ(q) = p1(τ(q)) ∈ 1, A,B

where p1 denotes the projection onto the first component of a vertex-type. Wedefine an additional function µA,1 : T × H ∗ t, t∗ → B(QA) by: for everys ∈ H ∗ t, t∗,θ ∈ T

µA,1(θ, s) = µA(s) ∩ (τ−1(τ i(θ)) × τ−1(τe(θ))).

Let us define a map γt associating to every sequence s ∈ H ∗ t, t∗ the set of allpath-types that can be realized by s in the automaton R6:

γt(s) = (p, b, q) ∈ T6 × B × T6 | (p, q) ∈ µR6(s) ∧ b = (‖s‖ 6= 0) (42)

where (‖s‖ 6= 0) is the boolean 1 if and only if it is true that (‖s‖ 6= 0). We nowdefine some properties of t-automata that play a role in the process of solvingequations with rational constraints in G.The automaton A is said ≈-compatible iff:

[L(A)]≈ = [L(A) ∩ Red(H, t)]≈, (43)

17

A is said ∼-saturated iff:

∀s, s′ ∈ H ∗ t, t−1∗, s ∼ s′ ⇒ µA(s) = µA(s′), (44)

A is said unitary iff:

∀θ ∈ T6, µA,1((θ, 0, θ), 1) = idτ−1(θ). (45)

A is said subgroups-compatible iff, for every subgroup C ∈ A,B, there exists aright-action ⊙ of the subgroup C over the set of states q ∈ Q | γ(q) = C suchthat: for every q, r ∈ Q, c ∈ C, h ∈ H ,

γ(q) = C ⇒ qc

→A q ⊙ c (46)

(γ(q) = C ∧ qh

→A r) ⇒ q ⊙ c−1 ch→A r (47)

(γ(r) = C ∧ qh

→A r) ⇒ qhc→A r ⊙ c (48)

A is said multiplicative iff for every θ ∈ γt(s),θ′ ∈ γt(s

′) such that θ · θ′ isdefined in T :

µA,1(θ · θ′, s · s′) = µA,1(θ, s) · µA,1(θ′, s′), (49)

A is said strict if,L(A) ⊆ Red(H, t). (50)

(note that every strict fta (50) is ≈-compatible (43)). Finally, a partitioned finitet-automaton A is said to be normal iff it is ≈-compatible, ∼-saturated, unitaryand multiplicative.

Proposition 1. Let R ∈ P(G).1- R ∈ RAT(G) iff R = πG(L(A)) for some normal partitioned finite t-automatonA with labeling set RAT(H).2- If R ∈ B(RAT(G)) then R = πG(L(A)) for some strict normal partitionedfinite t-automaton A with labeling set B(RAT(H)).

It is proved in [LS08, Proposition 33, points (1),(3) and (a)] that R is a rationalsubset of G iff π−1

G (R) ∩ Red(H, t) = L(A) ∩ Red(H, t) for some partitioned,≈-compatible, ∼-saturated and unitary fta A with labelling set Rat(H).

Moreover, the automaton A = 〈L,Q, τ, δ, I,T〉 can be chosen in such a waythat it is subgroups-compatible: this follows from equations (39-40) in the theproof of [LS08, Proposition 22]. In order to prove Proposition 1, point (1), it re-mains to show that this automaton A is also multiplicative. For ease of notation,we skip the index A in the maps µA, µA,0, µA,1, up to the end of this subsection.

Lemma 3. For every h ∈ H,hi ∈ H,αi ∈ −1,+1,1- µ0(t) = µ1((A, T, 1, B,H), t)2- µ0(t) = µ1((B, T, 1, A,H), t)3- µ0(h) = ∪θ∈γ(h)µ1(θ, h)

4- µ1((A, T, 1, B,H), (∏n−1i=1 t

αihi)tαn) =

∏n−1i=1 µ0(t

αi)µ0(hi)µ0(tαn).

18

Sketch of proof: Points (1)(2)(3) follow directly from the definition of µ. Point(4) follows from the fact that A is unitary. 2

Lemma 4. Let h, k ∈ H, θ, θ′, θ′′ ∈ T6, such that (θ, 0, θ′) ∈ γt(h), (θ′, 0, θ′′) ∈

γt(k) and (θ, 0, θ′′) ∈ T . Then µ1((θ, 0, θ′) · (θ′, 0, θ′′), h · k) = µ1((θ, 0, θ

′), h) ·µ1((θ

′, 0, θ′′), k).

Sketch of proof: If θ′ /∈ θ, θ′′, then the product (θ, 0, θ′′) /∈ T .Let us suppose that θ′ = θ. Hence h belongs to the subgroup p1(θ), and thus,the equality µ1((θ, 0, θ

′′), h ·k) = µ1((θ, 0, θ), h) ·µ1((θ, 0, θ′′), k) follows from the

fact that A is subgroups-compatible.The case where θ′ = θ′′ can be treated symmetrically. 2

Lemma 5.Let n ≥ 1 , θ, θ′ be vertex-types and α1, . . . , αn ∈ −1,+1, h0, h1, . . . , hn ∈ H.

Let s = (∏n−1i=1 t

αihi)tαn . Then

µ1((θ, 1, θ′), h0shn) = µ1((θ, 0, A(α1), T ), h0)·µ1((A(α1), T, 1, B(αn), H), s)·µ1((B(αn), H, 0, θ′), hn).

Sketch of proof: Follows immediately from the definitions of µA and µ1. 2

Lemma 6. For every path-types θ,θ′ and every t-sequences s, s′ ∈ H∗t, t∗, ifθ ∈ γ(s), θ′ ∈ γ(s′) and θ ·θ′ is defined, then µ1(θ ·θ′, s ·s′) = µ1(θ, s)·µ1(θ

′, s′).

Sketch of proof: Suppose that θ = (θ, 1, θ′),θ′ = (θ′, 1, θ′′), s = hsk, s′ = h′s′k′

where s = (∏n−1i=1 t

αihi)tαn , s′ = (

∏m−1j=1 tβjh′j)t

βm .We use below the notation: A(+1) := A,A(−1) := B,B(+1) := B,B(−1) := A.Case 1: θ′ = (B(αn), H).Let us determine

µ1((θ, 1, θ′′), s · s′). (51)

By Lemma 5 it is equal to

µ1((θ, 0, A(α1), T ), h0)·µ1((A(α1), T, 1, B(βm), H), h), s·kh·s′)·µ1((B(βm), H, 0, θ′′), k′).(52)

By claim 3, point (4), this can be rewritten:

µ1((θ, 0, A(α1), T ), h0)·n−1∏

i=1

µ0(tαi)µ0(hi)µ0(t

αn)·µ0(kh)·m−1∏

j=1

µ0(tβj )µ0(h

′j)µ0(t

βm)µ1((B(βm), H, 0, θ′′), k′).

(53)but since the image of µ0(t

αn) is included in τ−1(B(αn), H) and the domain ofµ0(t

β1) is included in τ−1(A(β1), T ), the factor µ0(kh) in formula (53) can bereplaced by

µ1((B(αn), H, 0, A(β1), T ), kh),

19

which, by Lemma 4 can be replaced by

µ1((B(αn), H, 0, B(αn), H), k) · µ1((B(αn), H, 0, A(β1), T ), h).

After this replacement in formula (53) we obtain:

µ1((θ, 0, A(α1), T ), h0)·∏n−1i=1 µ0(t

αi)µ0(hi)µ0(tαn)· µ1(B(αn), H, 0, B(αn), H), k)

·µ1((B(αn), H, 0, A(β1), T ), h)·∏m−1j=1 µ0(t

βj )µ0(h′j)µ0(t

βm)· µ1((B(βm), H, 0, θ′′), k′).

Using Lemma 5 backwards and twice, we obtain

µ1((θ, 0, B(αn), H), s) · µ1((B(αn), H, 0, θ′′), s′) (54)

which is exactlyµ1((θ, 1, θ

′), s) · µ1((θ′, 1, θ′′), s′) (55)

We have established that (51) and (55) have the same value, as required.Case 2: θ′ = (A(β1), T ).Symetric arguments can be applied.Since every other value of θ′ makes impossible that both (θ, 1, θ′) ∈ γ(s) and(θ′, 1, θ′′) ∈ γ(s′), we have treated all the possible cases.It remains to treat the case where s ∈ H or s′ ∈ H :Case 3: s ∈ H, s′ /∈ H .By Lemma 4,

µ1((θ, 0, A(β1), T )), s · h′) = µ1((θ, 0, θ′), s) · µ1((θ

′, 0, A(β1), T ), h′).

Applying Lemma 5 we get

µ1((θ, 0, θ′′), s · s′) = µ1((θ, 0, A(β1), T ), sh′) · µ1((A(β1), T, 0, θ

′′), s′k′)

= µ1((θ, 0, θ′), s) · µ1((θ

′, 0, θ′′), h′s′k′)

as required.Case 4: s /∈ H, s′ ∈ H .can be handled as case 3.Case 5: s ∈ H, s′ ∈ H .This case is treated by Lemma 4. 2

Proof. Let us prove Proposition 1:1- The initial properties of A together with the multiplicativity property givenby Lemma 6 show that, for every R ∈ RAT(G), there exists a normal partitionedfinite t-automaton A such that R = πG(L(A)). Conversely, by Proposition 33 of[LS08], if such an automaton A exists, then G is rational.2- Let K be a boolean combination of rational subsets K1, . . . ,Ki, . . . ,Kp ofG. By [LS08, Proposition 33, point (3)], for every i ∈ [1, p] there exists a par-titionned, ∼-saturated, deterministic, complete, finite t-automaton Ci, whoselabelling set is B(Rat(H)), and such that

L(Ci) = π−1G (K) ∩ Red(H, t).

20

Since every boolean operation can be translated over partitioned, ∼-saturated,deterministic, complete, finite t-automata (see [LS08, Lemma 17, Lemma 20]),there exists a partitioned, ∼-saturated, deterministic, complete, finite t-automatonA, whose labelling set is B(Rat(H)), and such that

L(A) = π−1G (K) ∩ Red(H, t). (56)

Applying [LS08, Proposition 22] to this automaton A and the set F = B(Rat(H)),we obtain a partitioned,≈-compatible, ∼-saturated, unitary fta B , with labellingset B(Rat(H)), such that

[L(B)]≈ ∩ Red(H, t) = π−1G (K) ∩ Red(H, t).

Moreover: by equations (39-40) of [LS08], B is subgroups-compatible, thus, byLemma 6, multiplicative; by [LS08, Claim 26], L(B) ⊆ L(A) while by (56) L(A) ⊆Red(H, t), hence B is also strict; finally, B is strict and normal.

3.5 The AB-algebra Ht

H ∗ t, t∗ Given an HNN-extension (1) and a normal finite t-automaton A, wedefine an AB-algebra with underlying monoid H ∗ t, t∗ and set of states QA

〈H ∗ t, t∗, ·, 1H, ιA, ιB, It, µt, γ, δt〉 (57)

as follows:ιA, ιB

are the natural injections from A (resp. B) into H ∗ t, t∗,

dom(It) = (I(H) ∪ t, t)∗

where I(H) is the set of invertible elements of H; It is the unique monoid anti-homomorphism dom(It) → dom(It) such that

∀h ∈ I(H), It(h) = h−1; It(t) = t; It(t) = t.

The map γt is the one previously defined in (42). The map µt : T ×H∗ t, t∗ →B2(Q) is defined by:

µt(θ, s) = (µ1(θ, s), (µ1(IT (θ), It(s))−1))) if s ∈ dom(It);

µt(θ, s) = (µ1(θ, s), ∅) if s /∈ dom(It). (58)

where µ1 is the representation map µA,1 associated with the partitioned t-automaton A fixed above. The map δt : T × H ∗ t, t∗ → PGI(A,B) is definedby:

δt(θ, s) = (g, g′) ∈ Gi(θ) × Ge(θ) | g · s ∼ s · g′.

It is noteworthy that, for every s ∈ H ∗ t, t∗

γt(s) 6= ∅ ⇔ s ∈ Red(H, t). (59)

21

Proposition 2. The above structure 〈H ∗ t, t∗, ·, 1H, ιA, ιB , It, µt, γ, δt〉 is anAB-algebra.

In order to prove this proposition we show the following lemma.

Lemma 7. For every s, s′ ∈ H ∗ t, t∗, γt(s · s′) ⊇ γt(s) · γt(s′).

Sketch of proof: Let Γ be the homomorphism from the partial semi-groupof paths in R into the partial semi-group T which associates to every edge ofR the same edge, viewed as an atomic path-type. Let θ ∈ T , s ∈ Red(H, t). ByPath(θ, s) we denote the unique path in the graph R such that Γ (Path(θ, s)) = θ

and Λ(Path(θ, s)) = s (where Λ is the labelling map). By definition, a path-type θ belongs to γt(s) iff Path(θ, s) exists. Suppose that s, s′ ∈ Red(H, t),θ ∈γt(s),θ

′ ∈ γt(s′), and (θ,θ′) ∈ D(·). Then the product Path(θ, s) · Path(θ′, s′)

is defined too, and

Γ (Path(θ, s) · Path(θ′, s′)) = θ · θ′, Λ(Path(θ, s) · Path(θ′, s′)) = s · s′,

showing thatγt(s · s

′) ⊇ γt(s) · γt(s′).

2

From the multiplicativity of µ1 we can immediately deduce the multiplicativityof µt. All the other checks are easy.

Positive AB-structure Another AB-structure over H ∗ t, t∗ can be defined bychoosing, in place of the map γt defined in (42), the map γ+ defined by: for everys ∈ H ∗ t, t∗,

γ+(s) = (p, b, q) ∈ T6 × B × T6 | (p, q) ∈ µG6(s) ∧ b = (‖s‖ 6= 0) (60)

Assertion (59) is now replaced by

∀s ∈ H ∗ t, t∗, γ+(s) 6= ∅. (61)

We call the resulting structure

〈H ∗ t, t∗, ·, 1H, ιA, ιB , I, µt, γ+, δt〉 (62)

the positive AB-algebra structure over H ∗ t, t∗. This variant will be used insection 9 where we deal with positive rational constraints.

Ht One can check that the monoid-congruence∼ is compatible with It, ιA, ιB, γt, µt, δtin the sense that: for every s, s′ ∈ H ∗ t, t∗, a ∈ A, b ∈ B,θ ∈ γt(s), if s ∼ s′

then,

It(s) ∼ It(s′), s = ιA(a) ⇔ s′ = ιA(a), s = ιB(b) ⇔ s′ = ιB(b),

γt(s) = γt(s′), µt(θ, s) = µt(θ, s

′), δt(θ, s) = δt(θ, s′).

22

Let us denote by Ht the quotient set H ∗ t, t∗/ ∼. We can naturally endow Ht

with a structure of AB-algebra:

Ht := 〈Ht, ·, 1H, ιA,∼, ιB,∼, I∼, µ∼, γ∼, δ∼〉 (63)

where all the required maps are just obtained from the corresponding map in theAB-structure of H ∗ t, t∗, by composition by π∼ : H ∗ t, t∗ → H ∗ t, t∗/ ∼.In addition

s ∼ s′ ⇒ ‖s‖ = ‖s′‖

so that the notion of norm remains well-defined in the quotient Ht. In the restof the paper we sometimes skip the subscript t when the context makes clearwhich AB-structure we are using.

Positive AB-structure Similarly, we can define the structure of AB-algebra:

Ht+ := 〈Ht, ·, 1H, ιA,∼, ιB,∼, I∼, µ∼, γ+∼, δ∼〉 (64)

where γ+ is defined via the automaton G6 instead of R6, see (60).

Algebraic properties The monoid (Ht, ·, 1) has some properties which resem-ble equidivisibility. We detail here these properties.

Lemma 8. Let P, P ′, S, S′ ∈ Ht,θ ∈ γ(P ) ∩ γ(P ′),ρ ∈ γ(S) ∩ γ(S′) such thatθ · ρ is defined , θ is a H-type and

P · S = P ′ · S′.

Then, there exists c ∈ Ge(θ) such that

P = P ′c, cS = S′.

1

P ′S′

1

SP

c

Fig. 2. Lemma 8

23

Proof. Let p, p′, s, s′ ∈ H ∗ t, t∗ such that P = [p]∼, P′ = [p′]∼, S = [s]∼, S

′ =[s′]∼. Since θ is a H-type, p, p′ ∈ H and since the product θ · ρ is defined,Gi(ρ) = Ge(θ). Moreover the second component p2(ρ) cannot be H, hence onlyone of the following two cases can occur.Case 1: ρ = (1, 1, 0, 1, 1).Thus s = s′ ∈ Ge(θ) = 1. Choosing c = 1, we obtain that

c ∈ Ge(θ), P = P ′c, cS = S′.

Case 2: ρ = (C, T, b,D,K) for some b ∈ 0, 1, D ∈ A,B,K ∈ T,H andC = Ge(θ).Suppose that

s = h0tα1h1 · · · t

αihi · · · tαnhn, s′ = h′0t

α′

1h′1 · · · tα′

jh′j · · · tα′

mhm,

where hi, h′j ∈ H, αi, α

′j ∈ −1,+1. Moreover, h0, h

′0 ∈ Gi(ρ). Since p ·s ∼ p′ ·s′,

we get that:n = m, ∀i ∈ [1, n], αi = α′

i,

and there exist connecting elements c1, . . . , c2n between p ·s and p′ ·s′ as recalledin [LS08, section 3]. Hence

c1 ∈ Ge(θ), p·h0·c1 = p′·h′0, tα1h1 · · · tαihi · · · t

αnhn ∼ c1·tα1h′1 · · · t

αih′i · · · tαnhn.

Let us choose c := h′0 · c−11 · h−1

0 ∈ Ge(θ). We obtain that

c ∈ Ge(θ), P = P ′c, cS = S′.

Lemma 9. Let P, P ′, S, S′ ∈ Ht,θ ∈ γ(P ),θ′ ∈ γ(P ′),ρ ∈ γ(S),ρ′ ∈ γ(S′)such that θ · ρ = θ′ · ρ′ is defined , θ,θ′ are T-types and

P · S = P ′ · S′.

One of the following cases must occur:1- ‖P‖ = ‖P ′‖ and there exists c ∈ Ge(θ) such that

P = P ′c, cS = S′

2- ‖P‖ < ‖P ′‖ and there exist c ∈ Ge(θ),P ′1,P

′2,P

′3 ∈ Ht,P

′1 has a T-type θ′

1,P′3

has a T-type θ′3,P

′2 has a H-type θ′

2 such that

P = P ′1c, P ′ = P ′

1P′2P

′3, cS = P ′

2P′3S

′, θ′ = θ′1θ

′2θ

′3.

3- ‖P‖ > ‖P ′‖ and there exists c ∈ Ge(P ′),P1,P2,P3 ∈ Ht,P1 has a T-type θ1,P3

has a T-type θ3,P2 has a H-type θ2 such that

P = P1P2P3, P1 = P ′c, cP2P3S = S′, θ = θ1θ2θ3.

24

1 1

P

c

S

S′P ′

1 P ′

3P ′

2

Fig. 3. Lemma 9

Proof. Let p, p′, s, s′ ∈ H ∗ t, t∗ such that P = [p]∼, P′ = [p′]∼, S = [s]∼, S

′ =[s′]∼. Since the products θ · ρ,θ′ · ρ′ are defined, Gi(ρ) = Ge(θ) and Gi(ρ′) =Ge(θ′). Suppose that

p = h0tα1h1 · · · t

αihi · · · tαnhn, s = k0t

β1k1 · · · tβjkj · · · t

βℓhℓ,

p′ = h′0tα′

1h′1 · · · tα′

ιh′ι · · · tα′

νh′ν , s′ = k′0tβ′

1k′1 · · · tβ′

κk′κ · · · tβλhλ,

where hi, kj , h′ι, k

′κ ∈ H, αi, βj, α

′ι, β

′κ ∈ −1,+1 and hn ∈ Ge(θ), h′ν ∈ Ge(θ′).

Since p · s ∼ p′ · s′, we get that:

n+ ν = ℓ+ λ, (α1, . . . , αn, β1, . . . , βℓ) = (α′1, . . . , α

′ν , β

′1, . . . , β

′λ)

and there exist connecting elements c1, . . . , c2n+2ν between p · s and p′ · s′ (see[LS08, section 3]).Case 1:‖P‖ = ‖P ′‖Hence n = ν. Let us choose c := h

′−1n · c−1

2n ·hn. The fact that hn, h′n, c2n ∈ Ge(θ)

ensures that c ∈ Ge(θ). Since c2n is a connecting element

h0tα1h1 · · · t

αihi · · · tαnc2n ∼ h′0t

α′

1h′1 · · · tα′

ih′i · · · tα′

n

andhnk0t

β1k1 · · · tβjkj · · · t

βℓhℓ ∼ c2nh′nk

′0tβ′

1k′1 · · · tβ′

jk′j · · · tβℓhℓ

so thatp ∼ p′c ∧ cs ∼ s′,

as required.Case 2:‖P‖ < ‖P ′‖Hence n < ν. Let us choose

p′1 := h′0tα1h′1 · · · t

αιhι · · · tαn , c := c−1

2n hn, p′2 := h′n, p′3 := tα′

n+1h′n+1 · · · tα′

νh′ν .

The fact that hn, c2n ∈ Ge(θ) ensures that c ∈ Ge(θ). A possible type for P ′1 is

θ1 := (A(α1), T, 1, B(αn), H) (where we use the notation A(+1) := A,A(−1) :=B,B(+1) := B,B(−1) := A), a possible type for P ′

2 is θ2 := (B(αn), H, 0, A(α′n+1), T )

and a possible type for P ′3 is θ3 := (A(α′

n+1), T, 1, B(α′nu), H). Thus θ′

1,θ′3 are

T-types and θ′2 is a H-type such that θ′ = θ′

1θ′2θ

′3. From the fact that c2n is a

connecting ellement one deduces that p ∼ p′1c and cs ∼ p′2p′3s

′ (see Figure 4).Finally the classes P ′

i = [p′i]∼ are fulfiling the conclusion of the lemma.Case 3:‖P‖ > ‖P ′‖

This case is similar to Case 2.

25

1 1

P S

hn tβ1

tαn h′

n

P ′

P ′

1 P ′

2 P ′

3

S′

k′0tα′

n+1 h′

νtα′

ν

k0tαn

cc2n

Fig. 4. Proof of Lemma 9, Case 2

3.6 The AB-algebra W

W∗ ∗A∗B Let S be a system of equations over Ht with involution and rationalconstraints. The rational constraints are expressed via the map µt defined by (58)in §3.5. We define an alphabet of “generic” symbols W with the underlying ideaof representing inside each symbol the values of the functions γt, µt, δt for the“concrete” value (i.e. in H ∗ t, t∗) of that variable that leads to a solution ofthe system of equations.Let V0 be some starting set. Let us denote by TA0 the set of all atomic typeswhich are either H-types or T-types. We then define

Ω := V0 × −1, 0, 1× TA0 × B2(Q) × PGI(A,B). (65)

W := (V, ǫ,θ,m, ϕ) ∈ Ω | ϕ ∈ PIs(Gi(θ),Ge(θ)), ∀(c, d) ∈ ϕ, µA(c)·m = m·µA(d).(66)

LetW = W ∈ W | p2(W ) = 0, W = W ∈ W | p2(W ) 6= 0

Let us consider the free product W∗ ∗A ∗B. We denote by ιA : A→ W∗ ∗A ∗Bthe natural embedding of A into W∗ ∗ A ∗ B ( and use similarly the notationιB). Note that ιA(A) ∩ ιB(B) = 1. We define an AB-algebra with underlyingmonoid W∗ ∗A ∗B and set of states Q

〈W∗ ∗A ∗B, ·, 1, ιA, ιB , I, µ, γ, δ〉 (67)

as follows:dom(I) = W∗ ∗A ∗B

(the submonoid generated by W ∪ ιA(A) ∪ ιB(B));I is the unique monoid anti-homomorphism W∗ ∗A ∗B → W∗ ∗A ∗B such that:

∀a ∈ A, I(ιA(a)) = ιA(a−1); ∀b ∈ B, I(ιB(b)) = ιB(b−1)

26

I(V, ǫ,θ,m, ϕ) = (V,−ǫ, Iθ(t), IQ(m), ϕ−1).

γ : W∗ ∗A ∗B → P(T )

is defined by:

γ(V, ǫ,θ,m, ϕ) = θ,

for every a ∈ A− 1, b ∈ B − 1

γ(ιA(a)) = (A, T, 0, A, T ), (A,H, 0, A,H), γ(ιB(b)) = (B, T, 0, B, T ), (B,H, 0, B,H),

γ(1) = (1, H, 0, 1, H), (1, 1, 0, 1, 1), (A, T, 0, A, T ), (A,H, 0, A,H), (B, T, 0, B, T ), (B,H, 0, B,H)

and finally, for every g1, . . . , gi, . . . , gn ∈ W ∪ ιA(A) ∪ ιB(B),

γ(

n∏

i=1

gi) =

n∏

i=1

γ(gi) (68)

µ is defined by

µ(θ, ιA(a)) = µt(θ, a), µ(θ, ιB(b)) = µt(θ, b),

for every θ′ ∈ T ,θ′ 6= θ,

µ(θ, (V, ǫ,θ,m, ϕ)) = m, µ(θ′, (V, ǫ,θ,m, ϕ)) = ∅.

The map δ : T × H ∗ t, t∗ → PGI(A,B) is defined by: for every a ∈ ιA(A), b ∈ιB(B),θ ∈ T ,

δ(θ, ιA(a)) = (ιA(c), ιA(d)) | (c, d) ∈ Gi(θ) × Ge(θ), ca = ad, if θ ∈ γ(ιA(a))

δ(θ, ιA(a)) = (1, 1), if θ /∈ γ(ιA(a))

δ(θ, ιB(b)) = (ιB(c), ιB(d)) | (c, d) ∈ Gi(θ) × Ge(θ), ca = ad, if θ ∈ γ(ιB(b))

δ(θ, ιB(b)) = (1, 1), if θ /∈ γ(ιB(b)).

For every θ′ ∈ T ,θ′ 6= θ,

δ(θ, (V, ǫ,θ,m, ϕ)) = ϕ, δ(θ′, (V, ǫ,θ,m, ϕ)) = (1, 1)

Since for every W ∈ W , γ(W ) is simply a singleton of the form θ, we also usethe (abusive) notation

τ i(W ), τe(W ),Gi(W ),Ge(W ), µ(W ), δ(W )

for what should be denoted, in full rigor, by

τ i(θ), τe(θ),Gi(θ),Ge(θ), µ(θ,W ), δ(θ,W ).

27

lengths For every w ∈ (ιA(A) ∪ ιB(B) ∪W)∗, we set

‖w‖ = |w|W .

One can check that

‖w · w′‖ = ‖w‖ + ‖w′‖; ‖w‖ = 0 ⇔ w ∈ A ∗B.

χAB(w) := 1 if w ∈ ιA(A) ∪ ιB(B);χAB(w) := 0 otherwise .

χH(w) := 1 if Card(γ(w)) = 1 ∧ p2(γ(w)) ∈ H;χH(w) := 0 otherwise .

Given P, S, P ′, S′ ∈ W and ψt ∈ HomAB(W,Ht) we define

∆(P, S, P ′, S′, ψt) = 1−1

2(χAB(P )+χAB(P ′))+χH(S)+χH(S′)+2‖ψt(S)‖+2‖ψt(S

′)‖

(69)(this notion will serve as a “progression measure” in the induction provingLemma 21, which is a key-point of this paper).

W Let us consider the monoid congruence ≡ over W∗ ∗A ∗B generated by theset of pairs

cW ≡Wd, (70)

for all W ∈ W , (c, d) ∈ δ(W ). We define the monoid 〈W, ·, 1W〉 as the quotient-monoid W∗ ∗ A ∗ B/ ≡. One can check that the monoid-congruence ≡ is com-patible with I, ιA, ιB , γ, µ, δ in the sense that: for every u, u′ ∈ W∗ ∗ A ∗ B, a ∈A, b ∈ B,θ ∈ γ(u), if u ≡ u′ then,

I(u) ≡ I(u′), u = ιA(a) ⇔ u′ = ιA(a), u = ιB(b) ⇔ u′ = ιB(b), (71)

γ(u) ≡ γ(u′), µ(θ, u) = µ(θ, u′), δ(θ, u) = δ(θ, u′). (72)

We can naturally endow W with a structure of AB-algebra:

〈W, ·, 1W, ιA,≡, ιB,≡, I≡, µ≡, γ≡, δ≡〉 (73)

where all the required maps are just obtained from the corresponding map in theAB-structure of W∗∗A∗B, by composition by π≡ : W∗∗A∗B → W∗∗A∗B/ ≡.In addition

u ≡ u′ ⇒ ‖u‖ = ‖u′‖

so that the notion of norm remains well-defined in the quotient W.

Wt, WH, W Let us consider the set Wt consisting of all the letters W ∈ Wfulfilling

∃s ∈ H ∗ t, t∗,W ∈ dom(IW) ⇔ s ∈ dom(It), γ(W ) ⊆ γ(s), and

∀θ ∈ γ(W ), µ(θ,W ) = µ(θ, s), δ(θ,W ) = δ(θ, s).

28

We define the subset

WH := W ∈ Wt | γ(W ) is a H-type .

We then define the quotients

Wt := W∗t ∗A ∗B/ ≡, WH := W∗

H ∗A ∗B/ ≡, W := W∗ ∗A ∗B/ ≡

and consider the AB-structures induced by W on Wt,WH, W. One can eas-ily check that the inclusions WH → Wt, Wt → W, W → W are AB-homomorphisms.

Involutions

Involutions over W Apart from the natural involution I : W → W there mightexist other involutory monoid anti-isomorphisms I′ : (W, ·, 1W) → (W, ·, 1W).Let us consider those I′ defined by a partition W = W0 ∪ W1 ∪ W1,W1 =W1, · · · ,Wp,W1 = W1, · · · , Wp, a tuple (a1, b1, · · · , ak, bk, · · · , ap, bp), with(ak, bk) ∈ (Gi(Wk),Ge(Wk)) and formulas of the form

I′(e) = e−1, for all e ∈ ι(A) ∪ ι(B), I

′(W ) = I(W ), for all W ∈ W0 (74)

I′(Wk) = akWkbk; I

′(Wk) = a−1k Wkb

−1k for all k ∈ [1, p]. (75)

Lemma 10. Let us consider a partition of W and a tuple of group elements(a1, b1, · · · , ak, bk, · · · , ap, bp) as above.

1- If there exists some anti-automorphism I′ : W → W satisfying (74,75) then,for every k ∈ [1, p]

δ(Wk)−1 = δ(ak) δ(Wk) δ(bk). (76)

2- If condition (76) is fulfiled, then there exists a unique anti-automorphism

I′ : W → W satisfying (74,75).3- The monoid anti-automorphism I′ given in point (2) is involutive iff, for everyk ∈ [1, p],

(b−1k ak, akb

−1k ) ∈ δ(Wk) (77)

4- The involutive monoid anti-automorphism I′ given in point (3) makes 〈W, ιA, ιB, I′, γ, µ, δ〉

into an AB-algebra iff, in addition, for every k ∈ [1, p]

γ(Wk) = IT (γ(Wk)), µ(akWkbk) = IQ(µ(Wk)). (78)

Proof: 1- Suppose that I′ is an anti-isomorphism of (W, ·) fulfiling (74,75). Letk ∈ [1, p] and (c, d) ∈ Gi(Wk) × Ge(Wk). On one hand:

(c, d) ∈ δ(I′(Wk)) ⇔ (c, d) ∈ δ(akWkbk)

⇔ (c, d) ∈ δ(ak) δ(Wk) δ(bk).

29

On the other hand:

(c, d) ∈ δ(I′(Wk)) ⇔ cI′(Wk) = I′(Wk)d

⇔ I′(Wkc

−1) = I′(d−1Wk)

⇔Wkc−1 = d−1Wk

⇔ dWk = Wkc

⇔ (d, c) ∈ I′(Wk)

⇔ (c, d) ∈ (I′(Wk))−1.

We have thus established that

(I′(Wk))−1 = δ(ak) δ(Wk) δ(bk).

2- Suppose that condition (76) is met. By the universal property of the free-product, there exists a monoid-anti-homomorphism I : (W∗ ∗A ∗B, ·) → (W∗ ∗A ∗B, ·) such that,

I(e) = e, for all e ∈ ι(A) ∪ ι(B), I(W ) = W, for all W ∈ W0

I(Wk) = akWkbk, I(Wk) = a−1k Wkb

−1k for all k ∈ [1, p].

Similarly, there exists a unique monoid-anti-homomorphism J : (W∗∗A∗B, ·) →(W∗ ∗A ∗B, ·) such that,

J(e) = e, for all e ∈ ι(A) ∪ ι(B), J(W ) = W, for all W ∈ W0

J(Wk) = bkWkak, J(Wk) = b−1k Wka

−1k for all k ∈ [1, p].

Let k ∈ [1, p], (c, d) ∈ δ(Wk).

I(cWk) = akWkbk · c−1, I(Wkd) = d−1 · akWkbk.

From (c, d) ∈ δ(Wk) we get that (d−1, c−1) ∈ δ(Wk) and by hypothesis (76) weobtain that (d−1, c−1) ∈ δ(aWkbk), so that

I(cWk) = I(Wkd) (79)

and by a similar argument

I(Wkc) = I(dWk). (80)

It follows from the equalities (79,80), that I induces a monoid anti-homomorphism

I′ : W → W.Let us remark that condition (76) also implies the similar condition

δ(Wk)−1 = δ(bk) δ(Wk) δ(ak). (81)

This implication can be detailed as

δ(Wk)−1 = δ(ak) δ(Wk) δ(bk) ⇔ δ(Wk) = δ(bk)

−1 δ(Wk)−1 δ(ak)

−1

⇔ δ(bk) δ(Wk) δ(ak) = δ(Wk)−1.

30

Hence, by same means as those used for I, one can prove that J induces a monoidanti-homomorphism J′ : W → W.Moreover the action of I′ J′ over the generators of W is given by:

I′ J

′(e) = e, for all e ∈ ι(A) ∪ ι(B), I′ J

′(W ) = W, for all W ∈ W0,

and, for every k ∈ [1, p],

I′ J

′(Wk) = J′(akWkbk) = b−1

k J′(Wk)a

−1k = Wk

I′ J

′(Wk) = J′(a−1

k Wkb−1k ) = bkJ

′(Wk)ak = Wk

which shows that I′ J′ = IdW

. By analogous calculations J′ I′ = IdW

. Finaly,I′ is a monoid anti-automorphism (with reciprocal J′). The unicity of such an

anti-automorphism follows from the fact that W is generated by the images ofι(A) ∪ ι(B) ∪ W by π≡.3- I′ I′ = Id iff, for every k ∈ [1, p], I′(I′(Wk)) = Wk i.e. I′(akWkbk) = Wk

i.e. b−1k akWkbka

−1k = Wk, which is equivalent to b−1

k akWk = Wkakb−1k i.e.

(b−1k ak, akb

−1k ) ∈ δ(Wk), which is exactly condition (77).

4- I′ is a suitable involution for making 〈W, ιA, ιB , I′, γ, µ, δ〉 into an AB-albebra,

iff I′ is compatible with γ, µ, δ (as imposed by axiom (33)). It is clear that I′

is compatible with γ and µ iff it fulfils γ(Wk) = IT (γ(Wk)), µ(akWkbk) =IQ(µ(Wk)) for every k ∈ [1, p]. The condition that I′ is compatible with δ can beexpressed as δ(akWkbk) = (δ(Wk))

−1, which is just the condition (76) that I′ isalready assumed to fulfil. 2

An involution I′ of the form (74,75), fulfiling conditions (76-78) is called H-realizable if, for every k ∈ [1, p]

∃hk ∈ I(H), γ(Wk) ⊆ γt(hk), µ(Wk) = µt(hk), δ(Wk) = δt(hk), akhkbk = h−1k

(82)

We denote by I the set of all partial involution I′ of the form (74,75) satisfyingthe four conditions (76)(77)(78) and (82). We call the elements of I the H-involutions of W.

Homomorphisms Let us show here some properties of AB-homomorphismsfrom W∗ ∗ A ∗ B or W to other AB-algebras. Let us denote by GW the set ofgenerators:

GW = W ∪ ιA(A) ∪ ιB(B).

Lemma 11. Let M2 = 〈M2, ·, 1M2, ιA,2, ιB,2, I2, γ2, µ2, δ2〉 be some AB-algebra.

Let ψ : W → M2 be some monoid-homomorphism. This map ψ is an AB-homomorphism if and only if,1- ιA ψ = ιA,2, ιB ψ = ιB,2and for every g ∈ GW,θ ∈ γ(g):2- g ∈ dom(I) ⇔ ψ(g) ∈ dom(I2)2’- I2(ψ(g)) = ψ(I(g))

31

3- γ2(ψ(g)) ⊇ γ(g)4- µ2(θ, ψ(g)) = µ(θ, g)5- δ2(θ, ψ(g)) = δ(θ, g).

Proof: Suppose that ψ is an AB-homomorphism. By definition it must fulfilllconditions (35-40). But for all g ∈ GW, γ(g) 6= ∅. Hence condition (36) impliescondition (2) of the lemma. The other five conditions translate immediately into(1)(2’)(3)(4)(5).Conversely, let suppose that ψ fulfills conditions (1-5) of the lemma.By (1), condition (35) is fulfilled.Extending (2) to W

Let w ∈ W with γ(w) 6= ∅. It must have a decomposition

w = g1g2 · · · gn

with 1 ≤ n, ∀i ∈ [1, n], gi ∈ GW.Let us suppose that

w ∈ dom(I). (83)

By definition of γW, γW(w) =∏ni=1 γ(gi), hence, (83) and axiom (25) imply

n∏

i=1

γ(gi) 6= ∅ (84)

and, in particular∀i ∈ [1, n], γ(gi) 6= ∅ (85)

Applying axiom (26) of AB-algebras, (83) and (84) give:

∀i ∈ [1, n], gi ∈ dom(I). (86)

By condition (2) of the lemma, (85) and (86) entail that

∀i ∈ [1, n], ψ(gi) ∈ dom(I2). (87)

By condition (3) of the lemma and (84), we know that

n∏

i=1

γ(ψ(gi)) 6= ∅. (88)

Using again axiom (26) and (87) we obtain that

n∏

i=1

ψ(gi) ∈ dom(I2). (89)

But ψ is a monoid homomorphism, hence this implies that

ψ(w) ∈ dom(I2). (90)

32

We have proved that, under the hypothesis that γ(w) 6= ∅, (83) implies (90).Let us establish the converse. Let us assume that

γ(w) 6= ∅ (91)

and (90). As ψ is a monoid-homomorphism, we obtain (89). From (91) we get(84) and, as above (88). From (89) and (88), by axiom (26) of AB-algebras wecan deduce (87). By condition (2) of the lemma, (87) implies (86). From (86)and (84), by axiom (26) we obtain (83), as required.Extending (2’) to W

By the point above, we know that im(I ψ) ⊆ dom(I2). Let us consider the map

θ = I ψ I2 : W → M2.

Condition (2’) shows that,

∀g ∈ GW, θ(g) = ψ(g).

As θ, ψ are monoid-homomorphism and GW is a set of monoid generators of W,it follows that θ = ψ, thus θ I2 = ψ I2 i.e.

I ψ = ψ I2,

as required.Extending (3) to W

Let us consider the pairs (W,=),(M2,=),(P(T ),⊆). Each of them is an orderedmonoid i.e., the second element of each pair is an ordering relation which iscompatible with right(resp. left) product. Let us consider the sequence of twomaps:

Wψ

−→ M2γ2−→ P(T )

These two maps are overmorphic in the sense that:

ψ(w · w′) = ψ(w) · ψ(w′), γ2(m ·m′) ⊇ γ2(m) · γ2(m′).

(see axiom (28) for the second inequality). It follows that their composition ψγ2

is overmorphic too, i.e. for every w,w′ ∈ W

γ2(ψ(w · w′)) ⊇ γ2(ψ(w)) · γ2(ψ(w′)) (92)

From this property (92), one can show, by induction over the integer n that, forevery g1, · · · , gi, · · · , gn

γ2(ψ(

n∏

i=1

gi)) ⊇n∏

i=1

γ2(ψ(gi)),

which, by condition (3) of the lemma , and the definition of γW implies that

γ2(ψ(

n∏

i=1

gi)) ⊇ γW(

n∏

i=1

gi),

33

as required.Extending (4) to W

Let w ∈ W such thatγW(w) 6= ∅. (93)

This w must have a decomposition

w =

n∏

i=1

gi (94)

where gi ∈ GW. As we saw in the extension of (3) to W , hypothesis (93) entailsthat

n∏

i=1

γW(gi) 6= ∅,n∏

i=1

γ2(ψ(gi)) 6= ∅. (95)

Let θ ∈ γW(w): by definition of γW it must have the form

θ =n∏

i=1

θi, ∀i ∈ [1, n],θi ∈ γW(gi). (96)

We thus have

µ2(θ, ψ(w)) = µ2(∏i=1 θi,

∏ni=1 ψ(gi)) (ψ is a monoid hom. )

=∏ni=1 µ2(ψ(θi, gi)) ( axiom (29))

=∏ni=1 µW(θi, gi) ( by condition (4))

= µW(θ, w) ( axiom (29)), (97)

as required.Extending (5) to W

We start again with some w ∈ W fulfilling (93), hence (95) and consider someθ ∈ γW(w), hence of the form (96). We thus have

δ2(θ, ψ(w)) = δ2(∏i=1 θi,

∏ni=1 ψ(gi)) (ψ is a monoid hom. )

=∏ni=1 δ2(θi, ψ(gi)) ( axiom (31))

=∏ni=1 δ(θi, gi) ( by condition (5))

= δW(θ, w) ( axiom (31)). (98)

2

Lemma 12. Let P, S, P ′, S′ ∈ W such γ(P ) = γ(P ′) and γ(PS) = γ(P ′S′) 6= ∅.Then one of the following occurs1- there exist θ, θ′ ∈ T6, γ(S), γ(S′) ∈ (θ, 0, θ′), (θ, 1, θ′).2- P, S, P ′, S′ ∈ A3- P, S, P ′, S′ ∈ B

34

Proof: Let P, S, P ′, S′ fulfill the hypothesis of the lemma.Case 1: ‖P‖ ≥ 1, ‖P ′‖ ≥ 1, ‖S‖ ≥ 1, ‖S′‖ ≥ 1.Since γ(PS) = γ(P ′S′) 6= ∅ we must have τ i(S) = τe(P ), τi(S

′) = τe(P ′).Similarly we obtain that τe(S) = τe(S′), so that conslusion 1 holds.Case 2: ‖P‖ ≥ 1, ‖P ′‖ ≥ 1, ‖S‖ = ‖S′‖ = 0.In this case, S, S′ ∈ A ∪ B. Moreover, γ(P ) = γ(P ′) ⇒ Ge(P ) = Ge(P ′) ⇒S, S′ ∈ Ge(P ) ⇒ γ(S) = γ(S′).Case 3: ‖P‖ ≥ 1, ‖P ′‖ ≥ 1, ‖S‖ ≥ 1, ‖S′‖ = 0.As ‖S‖ ≥ 1,Card(γ(S)) = 1 and τe(S) 6= τ i(S) = τe(P ) so that

τe(S) 6= τe(P ) (99)

Since ‖S′‖ = 0, for every θ ∈ γ(S′), τ i(θ) = τe(θ), hence

τe(P ′S′) = τe(P ′) (100)

We also have γ(PS) = γ(P ′S′) 6= ∅ which implies that τe(S) = τe(PS) =τe(P ′S′) hence, taking into account (100) we get

τe(S) = τe(P ′) (101)

But equations (99)(101) entail that τe(P ) 6= τe(P ′) contradicting the hypothesisthat γ(P ) = γ(P ′). This case is thus impossible.Case 4: ‖P‖ = ‖P ′‖ = 0, ‖S‖ ≥ 1, ‖S′‖ ≥ 1.In this case P, P ′ ∈ A ∪B. The fact that γ(P ) = γ(P ′) implies that

(P ∈ A− 1 ∧ P ′ ∈ A− 1) ∨ (P ∈ B − 1 ∧ P ′ ∈ B − 1) ∨ (P = P ′ = 1).

Here γ(S) = γ(PS) = γ(P ′S′) = γ(S′). Hence conclusion 1 holds.Case 5: ‖P‖ = ‖P ′‖ = 0, ‖S‖ ≥ 1, ‖S′‖ = 0.We should then have Card(γ(PS)) = 1 while Card(γ(P ′S′)) ∈ 0, 2, 6. Thiscontradicts the hypothesis that γ(PS) = γ(P ′S′). This case is thus impossible.Case 6: ‖P‖ = ‖P ′‖ = 0, ‖S‖ = ‖S′‖ = 0.Then P, P ′, S, S′ ∈ A ∪ B. But γ(PS) = γ(P ′S′) implies that conclusion 2 orconlusion 3 holds. 2

Lemma 13. Let ψ : W → M be some AB-homomorphism into some AB-algebra M. Let P, S, P ′, S′ ∈ W such that γ(P ) = γ(P ′), ψ(P ) = ψ(P ′), ψ(PS) =ψ(P ′S′) and γ(PS) = γ(P ′S′) 6= ∅. Then γ(S) = γ(S′).

Proof: One of conclusions 1,2,3 of Lemma 12 must hold. If conclusion 1 holds,‖ψ(P )‖ =‖ψ(P ′)‖, ‖ψ(PS)‖ = ‖ψ(P ′S′)‖ imply that ‖ψ(S)‖ = ‖ψ(S′)‖. The booleancomponent of γ(S), γ(S′) are thus equal and finally, γ(S) = γ(S′).If conclusion 2 (resp. 3) holds, since ψ restricted to the group A (resp. B) is agroup isomorphism onto its image, S = S′. 2

35

1

Q

P S

P ′S′

1

ψ

Fig. 5. Lemma 14, the statement.

Lemma 14. Let ψ : W → Ht be some AB-homomorphism. Let Q ∈ W, P ′, S′ ∈Ht,θ ∈ γ(P ′),ρ ∈ γ(S′) such that, θ is a H-type, θ · ρ is defined and ψ(Q) =P ′ · S′. Then, there exist P, S ∈ W such that:

Q = P · S, ψ(P ) = P ′, γ(P ) = θ, ψ(S) = S′, ρ ∈ γ(S).

Proof: Let us consider ψ,Q, P ′, S′,θ,ρ fulfilling the hypothesis of the lemma.Decomposing Q over the generators of GW gives

Q = e0W0 ·Q1 (102)

where W0 ∈ W , e0 ∈ ι(A) ∪ ι(B). Let us choose some reduced sequencess0, s1, p

′, s′ such that

ψ(e0W0) = [s0]∼, ψ(Q1) = [s1]∼, P ′ = [p′]∼, S′ = [s′]∼, (103)

see Figure 6. Since W0 ∈ W , W0 has either a H-type or a T-type. The equalityγ(W0) · γ(Q1) = θ · ρ shows that the following equality between vertex-typesholds: γ1(W0) = γ1(θ). Hence γ(W0) is a H-type. It follows that Q1 has a T-type or the type (1, 1, 0, 1, 1). Similarly, p′ has a H-type (namely θ) and, bythe same argument, s′ has a T-type or the type (1, 1, 0, 1, 1). Let us choose therepresentatives s1 (resp. s′) in such a way that it begins with a letter t or t or itis equal to 1. We know that ψ(Q) = P ′Q′ i.e.

s0 · s1 ∼ p′ · s′ (104)

and the choice of representatives s1, s′ ensures the sequences s0 · s1, p′ · s′ are

reduced. Thus there exist connecting elements c1, . . . between s0 · s1 and p′ · s′

([LS08, section 3]); in particular

c1 ∈ Ge(θ), s0 · c1 = p′, c1s′ ∼ s1 (105)

36

Q

e0W0

1

Q1

ψ

1

11

s0 s1

p′ s′c1

Fig. 6. Lemma 14, the proof.

Let us define

P := e0W0c1, S := c−11 Q1. (106)

We obtain from (102,106) that Q = P · S. We obtain from (105) that ψ(P ) =P ′, ψ(S) = S′. It remains to determine the types of P,Q. Let us note γ(P ) =θ1 and let ρ1 ∈ γ(S). The fact that ψ(P · S) = P ′ · S′ implies that

θ1 · ρ1 = θ · ρ. (107)

Since θ is a H-type and θ · ρ is defined, ρ must be either a T-type or the type(1, 1, 0, 1, 1).Case 1: ρ is a T-type.ρ,ρ1 are both types of s′. Since s′ starts with either a letter t or a letter t,this letter completely determines the first vertex-type p1(ρ) = p1(ρ1); it alsoimposes that, concerning the middle boolean component, p2(ρ) = p2(ρ1) = 1;and finally, by (107), p3(ρ) = p3(ρ1) = p3(θ · ρ). This establishes that ρ1 = ρ.Using (107) we know that θ1,θ have same initial and ending vertex-types and,since they are H-types, they also have the same boolean component, namely 0.Hence θ1 = θ. Finally, γ(P ) = θ, γ(S) = ρ.Case 2: ρ = (1, 1, 0, 1, 1).In this caseQ′ = 1 and the choice of the representatives s′, s1 implies s′ = s1 = 1,hence c1 = 1 and Q1 = S = 1 so that ρ ∈ γ(S). By hypothesis, γ(Q) = θ · ρ,hence γ(P ) = θ.In both cases the types of P, S fulfil the required property. 2

Lemma 15 (factorization). Let ψ : W∗∗A∗B → W be some AB-homomorphism.Then, there exists a unique AB-homomorphism ψ : W → W such that π≡ ψ =ψ.

37

Proof: Since ψ is anAB-homomorphism, for everyW ∈ W , δ(W ) = δ(ψ(W )).Henceker(π≡) ⊆ ker(ψ), which ensures the existence of ψ ∈ Hom(W,W) such thatπ≡ ψ = ψ. By hypothesis, ψ preserves ιA, ιB , I, γ, µ, δ, over the elements of GW.As π≡ does also preserves all these maps, it follows that ψ fulfills conditions(1-5) of Lemma 11, hence it is an AB-homomorphism. 2

Lemma 16 (quotient). Let σ : W∗ ∗ A ∗ B → W∗ ∗ A ∗ B be some AB-homomorphism. Then, there exists a unique AB-homomorphism σ : W → W

such that π≡ σ = σ π≡.

Proof: Applying Lemma 15 to the AB-homomorphism ψ = σ π≡ we obtainthis lemma. 2

Lemma 17 (lifting). Let ψ : W∗ ∗ A ∗ B → W be some AB-homomorphism.Then, there exists an AB-homomorphism ψ : W∗ ∗ A ∗ B → W∗ ∗ A ∗ B suchthat ψ = ψ π≡.

Proof: Let us consider some map ψ : GW → W∗ ∗ A ∗ B fulfilling, for everya ∈ A, b ∈ B,W ∈ W :

ψ(ιA(a)) = ιA(a), ψ(ιB(b)) = ιB(b)

ψ(W ) ∈ π−1≡ (ψ(W )).

Since π≡ preserves ιA, ιB , I, γ, µ, δ and ψ fulfills conditions (1-5) of Lemma 11,the map ψ does also fulfill conditions (1-5). By the universal property of the freeproduct,it can be extended into a monoid homomorphism from W∗ ∗ A ∗ B toW∗ ∗A ∗B. By Lemma 11, it is also an AB-homomorphism from W∗ ∗A ∗B toW∗ ∗A ∗B. Moreover, for every g ∈ GW, π≡(ψ(g)) = ψ(g), hence ψ π≡ = ψ, asrequired. 2

Lemma 18 (inverse image). Let σ : W → W be some AB-homomorphism.Then, there exists an AB-homomorphism σ : W∗ ∗ A ∗ B → W∗ ∗ A ∗ B suchthat π≡ σ = σ π≡.

Proof: Applying Lemma 17 to the AB-homomorphism ψ = π≡ σ, we obtainthis lemma. 2

38

WW

ψ

W

W∗ ∗A ∗BW∗ ∗A ∗B W∗ ∗A ∗B

ψ

W∗ ∗A ∗B W∗ ∗A ∗B

π≡

σ

σ

π≡

π≡

π≡π≡

σ

σ

WW

W

ψ

ψ

W∗ ∗ A ∗ BW∗ ∗ A ∗B

W

π≡

(factorization) (quotient)

(lifting) (inverse image)

Fig. 7. AB-homomorphisms

39

4 Equations with rational constraints over G

Let us consider a system of equations S over G:

((ui, u′i))i∈I

where I is a finite set, U is a finite set of variables and ui ∈ U∗, u′i ∈ U∗; let usconsider also a rational constraint C : U → B(Rat(G)).

4.1 Equations with rational constraints

Quadratic normal form Such a system can be effectively transformed into afinite system S′, with set of variables U ′ and with rational constraint C′ : U ′ →B(Rat(G)) such that(1) S′ is quadratic(2) U ⊆ U ′

(3) the solutions of (S,C) are exactly the restrictions to U∗ of the solutions of(S′,C′).Such a transformation consists of applying iteratively the following elementarytransformations Tk, for 1 ≤ k ≤ 4:T1 Suppose that |u′i| ≤ 1.Then set U ′ := U ∪ U ′, where U ′ is a new variable, and set

vj := uj for all j ∈ I, v′j := u′j for all j ∈ I − i, v′i := u′iU′,

C′(U) = C(U) for all U ∈ U , C′(U ′) = ε.

T2 Suppose that |ui| = 0.Then set U ′ := U ∪ U ′, where U ′ is a new variable, and set

vj := uj for all j ∈ I − i, v′j := u′j for all j ∈ I, vi := U ′,

C′(U) = C(U) for all U ∈ U , C′(U ′) = ε.

T3 Suppose that |ui| ≥ 2.Then set U ′ := U ∪ U ′, where U ′ is a new variable, I ′ := I ∪ ı where ı /∈ I

vj := uj for all j ∈ I − i, v′j := u′j for all j ∈ I − i,

vi := U ′, v′i := ui, vı := U ′, v′ı := u′i,

C′(U) = C(U) for all U ∈ U , C′(U ′) = G.

T4 Suppose that |u′i| ≥ 3 : u′i = u′′i U for some U ∈ U .Then set U ′ := U ∪ U ′, where U ′ is a new variable, I ′ := I ∪ ı where ı /∈ I

vj := uj for all j ∈ I − i, v′j := u′j for all j ∈ I − i,

vi := ui, v′i := U ′U, vı := U ′, v′ı := u′′i .

C′(U) = C(U) for all U ∈ U , C

′(U ′) = G.

40

Rational constraints Let C : U → B(Rat(G)). By Proposition 1, for every vari-able U ∈ U , one can construct some strict normal partitioned finite t-automatonAU over the labelling set B(Rat(H)) such that:

C(U) = L(AU ).

Let A be the direct product of the fta (AU )U∈U . This fta A is strict, normalpartitioned and, for every u ∈ U , there exists IU ⊆ QA, TU ⊆ QA such that

C(U) = g ∈ G | (IU × TU ) ∩ µA,G(g) 6= ∅.

Let us define, for every U ∈ U , the subset B(U) ⊆ B(Q) by:

B(U) := ρ ∈ B(Q) | (IU × TU ) ∩ ρ 6= ∅.

Note that B(U) is upwards-closed i.e., if ρ ⊆ ρ′ and ρ ∈ B(U), then ρ′ ∈ B(U).It is then clear that A recognizes the constraint C in the sense that, for everyU ∈ U ,

C(U) = µ−1A,G(B(U)).

Let us define

M(C) := µ : U → B(Q) | ∀U ∈ U , µ(U) ∈ B(U).

We can thus express the initial system of equations with constraint as follows:

∨

µU∈M(C)

((ui, u′i)i∈I , µA,G, µU ). (108)

A solution of the system ((ui, u′i)i∈I , µA,G, µU) is now any monoid-homomorphism

σG : U∗ → G fulfilling both conditions:

∀i ∈ I, σG(ui) = σG(u′i) (109)

∀U ∈ U , µA,G(σG(U)) = µU (U). (110)

An over-solution of the system ((ui, u′i)i∈I , µA,G, µU ) is any monoid-homomorphism

σG : U∗ → G fulfilling the above condition (109) together with the condition

∀U ∈ U , µA,G(σG(U)) ⊇ µU (U). (111)

Note that, for the disjunction of systems (108) built above, since each set B(U) isupwards-closed, every over-solution of the disjunction of systems is also a solutionof the disjunction of systems. We have thus proved the following proposition.

Proposition 3. Given a system S0 of equations with rational constraints overG over a set of variables U0, one can compute a finite family of systems (S1,j)j∈Jof equations with rational constraints over G with set of variables U1 ⊇ U0, ofthe form

S1,j = ((ui, u′i)i∈I , µA,G, µU1,j) (112)

41

such that1- every equation (ui, u

′i) is quadratic,

2- µA,G is the map associated with a strict normal finite t-automaton A andµU1,j is a map U1 → B(QA),3- every solution σ : U∗

0 → G of S0 extends into a solution σ1 : U∗1 → G of∨

j∈J S1,j

4- every solution σ1 : U∗1 → G of

∨j∈J S1,j induces a restriction σ|U∗

0which is a

solution of S0.5- every over-solution of

∨j∈J S1,j is also a solution of

∨j∈J S1,j.

Every system S = ((ui, u′i)i∈I , µA,G, µU) where the equations are quadratic and

the map µU is defined through a strict normal partitioned finite t-automaton Aover the labelling set B(Rat(H)) is said to be in quadratic normal form.Every finite disjunction of systems

∨j∈J ((ui, u

′i)i∈I , µA,G, µU ,j) where the equa-

tions are quadratic, the map µU is defined through a strict normal partitionedfinite t-automaton A over the labelling set B(Rat(H)) and such that every over-solution of the disjunction is also a solution of the disjunction is said to be inclosed quadratic normal form.

4.2 Equations and disequations with rational constraints

Let us recall that a finite system of equations and disequations S over G is afamily:

((ui, ci, u′i))i∈I (113)

where I is a finite set, U is a finite set of variables, ui ∈ U∗, u′i ∈ U∗ andci ∈ =, 6=; let us consider also a rational constraint C : U → B(Rat(G)) (see§2.4). Such a system is said to be quadratic iff it has the form:

((ui = u′i)1≤i≤n, (ui 6= u′i)n+1≤i≤2nµA,G, µU ) (114)

where, for every i ∈ [1, n], |ui| = 1, |u′i| = 2 and, for every i ∈ [n + 1, 2n], |ui| =|u′i| = 1.

Using the same kind of tricks as in the previous subsection, we can prove thefollowing normalization statement.

Proposition 4. Given a system S0 of equations and disequations with rationalconstraints over G over a set of variables U0, one can compute a finite familyof systems (S1,j)j∈J of equations with rational constraints over G with set ofvariables U1 ⊇ U0, of the form

S1,j = ((ui = u′i)1≤i≤n, (ui 6= u′i)n+1≤i≤2n, µA,G, µU1,j) (115)

such that1- every system S1,j is quadratic,2- µA,G is the map associated with a strict normal finite t-automaton A andµU1,j is a map U1 → B(QA),

42

3- every solution σ : U∗0 → G of S0 extends into a solution σ1 : U∗

1 → G of∨j∈J S1,j

4- every solution σ1 : U∗1 → G of

∨j∈J S1,j induces a restriction σ|U∗

0which is a

solution of S0.5- every over-solution of

∨j∈J S1,j is also a solution of

∨j∈J S1,j.

Every finite disjunction of systems∨j∈J((ui, u

′i)i∈I , µA,G, µU ,j) where the

equations/disequations are quadratic, the map µU is defined through a strictnormal partitioned finite t-automaton A over the labelling set B(Rat(H)) andsuch that every over-solution of the disjunction is also a solution of the disjunc-tion is said to be in closed quadratic normal form.

43

5 Equations over Ht

5.1 t-equations

A system of t-equations is a family of ordered pairs

S = (wi, w′i)i∈I (116)

where wi, w′i ∈ Wt, γ(wi) = γ(w′

i) 6= ∅. A solution of S is anyAB-homomorphismψt : Wt → Ht such that, for every i ∈ I

ψt(wi) = ψt(w′i). (117)

Notice that here, the rational constraints are replaced by the even more restric-tive conditions that define the notion of AB-homomorphism: beside preservationof µ the map ψt must also preserve I, γ and δ.

5.2 From G-equations to t-equations

Let us start with a system of equations with rational constraints , over G, whichis in quadratic normal form (see Proposition 3):

((ui, u′i)i∈I , µA,G, µU ) (118)

We suppose I = [1, n], we denote by Ui,1( resp. Ui,2, Ui,3) the unique letter of ui(resp. the first, second letter of u′i). therefore, the equations of S take the form

Ei : (Ui,1, Ui,2Ui,3) for all 1 ≤ i ≤ n (119)

We define here a reduction of the satisfiability problem for such systems to thesatisfiability problem for systems of t-equations. The leading idea is simply that,since πG : Red(H, t)/ ∼→ G is a bijection, every solution in G correspondsto a map into Ht. Nevertheless the product in G corresponds to a somewhatcomplicated operation over Red(H, t)/ ∼ that we must deal with. Let us considerthe alphabets V0 := I×[1, 3]×[1, 9] and the alphabet W constructed from this V0

in §3.6. We consider all the vectors (Wi,j,k) where 1 ≤ i ≤ n, 1 ≤ j ≤ 3, 1 ≤ k ≤ 9

44

of elements of W ∪ 1 and all triple (ei,1,2, ei,2,3, ei,3,1) ∈ (A ∪B)3 such that:

p1(Wi,j,k) = (i, j, k) ∈ V0 for Wi,j,k 6= 1 (120)

γ(

9∏

k=1

Wi,j,k) = (1, H, b, 1, 1) for some b ∈ 0, 1

(121)

µ1(

9∏

k=1

Wi,j,k) = µU (Ui,j) (122)

Wi,2,k,Wi,3,10−k ∈ W ∪ 1 for 6 ≤ k ≤ 9 (123)

γ(

4∏

k=1

Wi,1,k) = γ(∏4k=1Wi,2,k) (124)

γ(

9∏

k=6

Wi,2,k) = γ(∏9k=6W i,3,10−k) (125)

γ(

9∏

k=6

Wi,1,k) = γ(∏9k=6Wi,3,k) (126)

Wi,j,5 ∈ W ∧ γ(Wi,j,5) is a H-type (127)

ei,1,2 ∈ Gi(Wi,1,5) = Gi(Wi,2,5) (128)

ei,2,3 ∈ Ge(Wi,2,5) = Gi(Wi,3,5) (129)

ei,3,1 ∈ Ge(Wi,3,5) = Ge(Wi,1,5). (130)

A vector (W , e) fulfilling all the properties (120-130) is called an admissiblevector. For every admissible vector (W , e) we define the following equations:

(9∏

k=1

Wi,j,k,∏9k=1Wi′,j′,k) if Ui,j = Ui′,j′

(131)

(Wi,1,1Wi,1,2Wi,1,3Wi,1,4ei,1,2, Wi,2,1Wi,2,2Wi,2,3Wi,2,4) (132)

(Wi,2,6Wi,2,7Wi,2,8Wi,2,9, ei,2,3W i,3,4W i,3,3W i,3,2W i,3,1) (133)

(Wi,1,6Wi,1,7Wi,1,8Wi,1,9, ei,3,1Wi,3,6Wi,3,7Wi,3,8Wi,3,9) (134)

(Wi,1,5, ei,1,2Wi,2,5ei,2,3Wi,3,5ei,3,1) (135)

for all 1 ≤ i, i′ ≤ n, 1 ≤ j, j′ ≤ 3. Equations (132-135) correspond to a decom-position of the planar diagram associated with the relation Ui,1 ≈ Ui,2Ui,3 intofour pieces, as indicated on Figure 8. Equation (131) expresses the fact that somevariables from U are common to several equations of S.

We denote by St(S,W , e) the sequence of equations (131-134) and by SH(S,W , e)the sequence of equations (135). For every (i, j) ∈ [1, n]× [1, 3] we denote by ı, the smallest pair such that Ui,j = Uı,. By σW ,e : U∗ → W we denote the unique

45

Ui,1

Ui,2Ui,3

Wi,1,1

Wi,2,1

Wi,3,1

ei,1,2 ei,3,1

ei,2,3

Wi,2,9

Wi,3,9

Wi,1,9Wi,1,5

Wi,2,5Wi,3,5

· · · · · ·

· · · · · ·

···

···

1

1 1

Fig. 8. Equation cut into 4 pieces

monoid-homomorphism such that,

σW ,e(Ui,j) =

9∏

k=1

Wı,,k.

Notice that, by the conditions imposed through the notion of “admissible vec-tor”, the equations of St(S,W , e) are really t-equations, while some of therighthand-sides of the equations of SH(S,W , e) might have an empty imageby γ.

Lemma 19. Let S = ((Ei)1≤i≤n, µA,G, µU) be a system of equations over G,with rational constraint. Let us suppose that S is in normal form. A monoidhomomorphism

σ : U∗ → G

is a solution of S if and only if, there exists an admissible choice (W , e) ofvariables of Wt and elements of A ∪B and an AB-homomorphism

σt : Wt → Ht

solving both systems St(S,W , e) and SH(S,W , e), such that

σ = σW ,e σt πG.

46

(We denote by πG : Ht → G the canonical projection; see Figure 9). We prove

σ σt

G HtπG

U∗

WtσW ,e

Fig. 9. Lemma 19

this lemma in the following two subsections.

From G-solutions to t-solutions Let σ : U∗ → G be a monoid homomorphismsolving the system S. Let us fix some equation from S, i.e. some integer 1 ≤ i ≤ n.Let us construct the vectors (Wi,∗,∗), (ei,∗,∗) corresponding to this equation. Letus choose, for every j ∈ [1, 3], some si,j ∈ Red(H, t) such that:

σ(Ui,j) = πG(si,j).

Let us consider decompositions of the form (5) for si,2, si,3:

si,2 = h0tα1h1 · · · t

αλhλ · · · tαℓhℓ, (136)

si,3 = k0tβ1k1 · · · t

βρkρ · · · tβmkm. (137)

We know that si,1 ≈ si,2si,3. There exist some integers λ ∈ [1, ℓ], ρ ∈ [1,m] suchthat

αλ + βρ = 0, (138)

tαλhλ · · · tαℓhℓ · k0t

β1k1 · · · tβρ ≈ ei,2,3 ∈ A(βρ), (139)

h0 · · · tαλ−1(hλ−1ei,2,3kρ)t

βρ+1 · · · tβmkm ∈ Red(H, t), (140)

We include in the above notation the following “degenerated” cases:• [Left-degenerated case]: λ = 1; then h0 · · · t

αλ−1 must be understood as 1• [Right-degenerated case]: ρ = m; then tβρ+1 · · · tβmkm must be understood as1,• [Middle-degenerated case]: αℓ + β1 6= 0 or (αℓ + β1 = 0 and hℓ · k0 /∈ A(β1));we then consider that λ = ℓ + 1, ρ = 0, ei,2,3 = 1. Equality (138) disappears,(139) becomes the trivial equation 1 = 1 while (140) remains valid.• [LM-degenerated case]: ℓ = 0;

47

We then consider that λ = ρ = 0, ei,2,3 = 1 and equation (139) becomes thetrivial equation 1 = 1. The lefthand-side of assertion (140) consists just of si,3.• [MR-degenerated case]: m = 0;Same notation as for LM. The l.h.s. of (140) consists just of si,2.Notice that these cases are not disjoint; in particular, when ℓ = m = 0 thethree kinds of degeneracy occur simultaneously. Each kind of degeneracy canbe visuallized on Figure 10 as one of the three triangular pieces consisting of atrivial relation.Let us consider the following factors of the reduced sequences si,2, si,3:

Pi,2 = h0 · · · tαλ−1 , Mi,2 = hλ−1, Si,2 = tαλ · · · tαℓhℓ,

Pi,3 = k0 · · · tβρ , Mi,3 = kρ, Si,3 = tβρ+1 · · · tβmkm.

Since si,1 ≈ si,2si,3 ≈ Pi,2(Mi,2ei,2,3Mi,3)Si,3 and si,1, Pi,2(Mi,2ei,2,3Mi,3)Si,3are reduced sequences, by Lemma 1 we get:

si,1 ∼ Pi,2(Mi,2ei,2,3Mi,3)Si,3.

Case 1[standard case] λ ∈ [2, ℓ], ρ ∈ [1,m− 1].There must exist Pi,1,Si,1 ∈ Red(H, t),Mi,1 ∈ H and connecting elements ei,1,2 ∈B(αλ−1), ei,2,3 ∈ A(αλ) such that:

Pi,1ei,1,2 ∼ Pi,2, (141)

ei,1,2Mi,2ei,2,3Mi,3ei,3,1 =H Mi,1 (142)

ei,3,1Si,1 ∼ Si,3, (143)

while relation (139) can be rewritten as:

Si,2 ∼ ei,2,3It(Pi,3), (144)

see Figure 10. Let πT : H ∗ t, t∗ → t, t∗ be the natural projection. By (141),(resp.(143),(144))we know that

πT (Pi,1) = πT (Pi,2), πT (Si,1) = πT (Si,3), πT (Si,2) = πT (It(Pi,3)).

Hence the t-automaton R6 has computations of the following forms:

(1, H)Pi,1

→ qi,1Mi,1

→ ri,1Si,1

→ (1, 1) (145)

(1, H)Pi,2

→ qi,1Mi,2

→ ri,2Si,2

→ (1, 1) (146)

(1, H)Pi,3

→ IR(ri,2)Mi,3

→ ri,1Si,3

→ (1, 1). (147)

Since qi,1 6= (1, 1), an easy inspection of the automaton R6 shows that the

computation (1, H)Pi,1

→ qi,1 can be factored into four subcomputations

(1, H) = qi,1,0Pi,1,1

→ qi,1,1Pi,1,2

→ qi,1,2Pi,1,3

→ qi,1,3Pi,1,4

→ qi,1,4 = qi,1 (148)

48

ei,1,2 ei,3,1

ei,2,3

si,1

si,2si,3

Mi,1

Mi,2Mi,3

Pi,1

Pi,2

Si,1

Si,3

Si,2 Pi,3

1 1

1

Fig. 10. Cutting the solution into three factors

such that each (qi,1,k, ‖|Pi,1,k+1‖|, qi,1,k+1) is an edge of R. Any decompositionhaving this property w.r.t. its projection on R will be called R-compatible. Sim-

ilarly the computations ri,1Si,1

→ (1, 1) ( resp. ri,2Si,2

→ (1, 1)) have R-compatibledecompositions:

qi,1,5Pi,1,6

→ qi,1,6Pi,1,7

→ qi,1,7Pi,1,8

→ qi,1,8Pi,1,9

→ qi,1,9 = (1, 1), (149)

qi,2,5Pi,2,6

→ qi,2,6Pi,2,7

→ qi,2,7Pi,2,8

→ qi,2,8Pi,2,9

→ qi,2,9 = (1, 1). (150)

Combining decomposition (148) with equation (141), (149) with (143), (150)with (144), we get the three R-compatible decompositions:

(1, H) = qi,1,0Pi,2,1

→ qi,1,1Pi,2,2

→ qi,1,2Pi,2,3

→ qi,1,3Pi,2,4

→ qi,1,4 = qi,1 (151)

ri,1 = qi,1,5Pi,3,6

→ qi,1,6Pi,3,7

→ qi,1,7Pi,3,8

→ qi,1,8Pi,3,9

→ qi,1,9 = (1, 1), (152)

(1, H) = I(qi,2,9) = qi,3,0Pi,3,1

→ I(qi,2,8) = qi,3,1Pi,3,2

→

I(qi,2,7) = qi,3,2Pi,3,3

→ I(qi,2,6) = qi,3,3Pi,3,4

→ I(ri,2) = qi,3,4. (153)

49

Finally, we define Pi,j,5 := Mi,j . We summarize on Figure 11 the above decom-positions and relations. Let us extract from these the vector (W , e) and theAB-homomorphism σt:- we choose for Wi,j,k a letter from W with γ(Wi,j,k) = (qi,j,k−1, ‖|Pi,j,k‖|, qi,j,k)and which can be mapped by some AB-homomorphism on the t-sequence Pi,j,k.- we define σt(Wi,j,k) := [Pi,j,k]∼; the choice of the letters Wi,j,k together withLemma 11 imply that any extension of σt on the alphabet Wt (respecting condi-tions 1-5 of Lemma 11), will posess a unique extension as an AB-homomorphismfrom Wt to Ht.One can check that (W , e) is an admissible vector, that σt solves the systems ofequations St(S,W , e) and SH(S,W , e) and that

σ = σW ,e σt πG.

We indicate now how these arguments must be adapted to the degenerated

ei,1,2 ei,3,1

ei,2,3

si,1

si,2si,3

Pi,2,5Pi,3,5

qi,1

ri,2

ri,1

ri,1qi,1

IR(ri,2)

Pi,1,1

Pi,2,1

Pi,1,9

Pi,3,9

(1, 1)(1, H)

(1,H)

(1,H)

(1, 1)

(1, 1)

Pi,2,9 Pi,3,1

Pi,1,5

1 1

1

Fig. 11. Cutting the solution into nine factors

cases.Case 2[L]:λ = 1.We take: Pi,2 = Pi,1 = 1, ei,1,2 = 1 and, accordingly

Pi,2,k = Pi,1,k = 1, Wi,2,k = Wi,1,k = 1

50

for 1 ≤ k ≤ 4.Case 3[R]:ρ = m.We take: Si,3 = Si,1 = 1, ei,3,1 = 1 and, accordingly

Si,3,k = Si,1,k = 1, Wi,3,k = Wi,1,k = 1

for 6 ≤ k ≤ 9.Case 4[M]:αℓ + β1 6= 0 or (αℓ + β1 = 0 and hℓ · k0 /∈ A(β1)).We take: Si,2 = Pi,3 = 1, ei,2,3 = 1 and, accordingly

Si,2,k = Pi,3,10−k = 1, Wi,2,k = Wi,3,10−k = 1

for 6 ≤ k ≤ 9.Case 5[LM]:ℓ = 0.We choose all the special values chosen in Case 2 and Case 4 i.e. Pi,2 = Pi,1 =Si,2 = Pi,3 = 1, ei,1,2 = ei,2,3 = 1 and the resulting choices for Wi,∗,∗.Case 6[MR]:m = 0.We choose all the special values chosen in Case 3 and Case 4.

From t-solutions to G-solutions Let σt : Wt → Ht be an AB-homomorphismsolving both systems St(S,W , e) and SH(S,W , e).1- Let us show that, for every i ∈ [1, n],

σt(σW ,e(Ui,1)) ≈ σt(σW ,e(Ui,2Ui,3)). (154)

Let us fix such an integer i. The conjunction of equivalences (132-135), impliesthat

σt(

9∏

k=1

Wi,1,k) ≈ σt(

9∏

k=1

Wi,2,k

9∏

k=1

Wi,3,k).

(Figure 8 gives a decomposition of the Van-Kampen diagram corresponding tothe above equivalence into four diagrams corresponding to (132-135)). Usingequation (131) we get:

σt(

9∏

k=1

Wı,1,k) ≈ σt(

9∏

k=1

Wı,2,k

9∏

k=1

Wı,3,k).

Taking into account the definition of σW ,e and the fact that σt is a monoid-homomorphism, we get a proof of (154).2- Let us show that, for every i ∈ [1, n]

µA,G(πG(σt(σW ,e(Ui,j)))) = µU (Ui,j).

By definition (14) this means that the value of

µA,1((1, H, b, 1, 1), σt(σW ,e(Ui,j))), (155)

51

where b = ‖|σt(σW ,e(Ui,j))‖| is equal to µU (Ui,j).We first remark that

σt(σW ,e(Ui,j)) = σt(∏9k=1 Uı,,k) by definition of σW ,e

= σt(∏9k=1 Ui,j,k) by condition (131) (156)

As σt is an AB-homomorphism

µA,1((1, H, b, 1, 1), σt(

9∏

k=1

Ui,j,k)) = µA,1((1, H, b, 1, 1),

9∏

k=1

Ui,j,k). (157)

Using now the conditions defining the notion of admissible vector we get:

µA,1((1, H, b, 1, 1),

9∏

k=1

Ui,j,k) = µA,1(∏9k=1 Ui,j,k) by (121)

= µU (Ui,j) by (122) (158)

Putting together (156)(157)(158), we obtain that the value of term (155) isexactly µU (Ui,j), as required.

52

6 Equations over W

We suppose here that a system of equations with rational constraints over G isfixed. Thus the AB-algebras W and Ht are completely defined (from the vari-able alphabet of the system and the t-automaton expressing the constraints).Given an H-involution I′ (this notion is defined at the end of subsubsection 3.6)we abbreviate as (W, I′) the AB-algebra obtained from the AB-algebra W byreplacing the standard involution I by I′ (see Lemma 10, point (4)).

6.1 W-equations

A system of W-equations is a family of ordered pairs together with an involution:

S = ((wi, w′i)i∈I , I

′) (159)

where wi, w′i ∈ Wt, γ(wi) = γ(w′

i) 6= ∅, I′ ∈ I.A solution of S is any AB-homomorphism σW : (Wt, I) → (Wt, I

′) such that, forevery i ∈ I

σW(wi) = σW(w′i). (160)

6.2 From t-equations to W-equations

From t-solutions to W-solutions

Lemma 20 (factorisation of t-solutions). Let S = ((wi, w′i))1≤i≤n be a sys-

tem of t-equations of the form (116). Let us suppose that σt : Wt → Ht is anAB-homomorphism solving the system S. Then there exists an involution I′ ∈ Iand AB -homomorphisms

σW : (Wt, I) → (Wt, I′), ψt : (Wt, I

′) → (Ht, It)

such that, σt = σW ψt and

σW(wi) = σW(w′i) for all 1 ≤ i ≤ n.

In other words: every solution in Ht of a system of t-equations factorizes througha solution in Wt of the same system of equations, with an involution in I.This factorization lemma is obtained via the more technical Lemma 21 below.

For every w ∈ W let us note

A(w) := CardW ∈ W | |w|W 6= 0 +1

2CardW ∈ W | |w|W,W 6= 0.

Lemma 21. Let K0 be an integer such that K0 < Card(V0) and ((wi, w′i))1≤i≤n+m

be a sequence of pairs (wi, w′i) ∈ W×W such that γ(wi) = γ(w′

i) 6= ∅. Let us sup-pose that λi : (Wt, I) → (Wt, I) ( for 1 ≤ i ≤ n+m) and θt : (Wt, I) → (Ht, It)are AB-homomorphisms such that

θt(λi(wi)) = θt(λi(w′i)) for all 1 ≤ i ≤ n+m,

53

ψt

(Ht, It)G

σW

(Wt, I′)

(Wt, I)σW ,e

σ σt

πG

U∗

Fig. 12. Lemma 20

λi = λj for all 1 ≤ i ≤ n, 1 ≤ j ≤ n,

A(

n+m∏

i=1

λi(wiw′i)) ≤ K0.

Then there exists an involution I′ ∈ I and AB homomorphisms

λ′i : (Wt, I) → (Wt, I′), θ′t : (Wt, I

′) → (Ht, It)

such that,λi θt = λ′i θ

′t, (161)

λ′i(wi) = λ′i(w′i) for all 1 ≤ i ≤ n+m, (162)

λ′i = λ′j for all 1 ≤ i ≤ n, 1 ≤ j ≤ n. (163)

Proof: Let S = ((wi, w′i))1≤i≤n+m be a sequence of pairs (wi, w

′i) ∈ Wt ×

Wt such that γ(wi) = γ(w′i) and let λ = (λi)1≤i≤n+m be a sequence of AB-

homomorphisms from Wt to Wt.

Distinguishing pair For every i ∈ [1, n + m] we define ≡i as the least monoidcongruence over W containing (λj(wj), λj(w′

j)) | i+ 1 ≤ j ≤ n+m. For everyi ∈ [1, n+m] let us consider some decompositions

λi(wi) = Pi · Si, λi(w′i) = P ′

i · S′i (164)

such that Pi ≡i P ′i , and this choice of the decomposition (164) minimizes the

integer∆(Pi, Si, P

′i , S

′i, θt).

(this integer was defined by equation (69)). Such a (Pi, P′i ) is called a distin-

guishing pair for (λi(wi), λi(w′i)) and we denote by

∆i(S,λ, θt)

the corresponding value of ∆(Pi, Si, P′i , S

′i, θt).

54

Size We call size of the triple (S,λ, θt) the multiset of natural integers:

‖(S, λ, θt)‖ = ∆1(S, λ, θt)), . . . , ∆i(S, λ, θt), . . . , ∆n+m(S, λ, θt). (165)

For every w ∈ W we use the notation

Alph(w) := W ∈ W | |w|W 6= 0 ∪ W ∈ W | |w|W,W 6= 0,

w(S,λ) :=

n+m∏

i=1

λi(wiw′i),

Alph(S,λ) := Alph(w(S,λ)),

A(S,λ) := A(w(S,λ)).

Induction Let us prove Lemma 21 by induction over ‖(S,λ, θt)‖, with respectto the partial ordering over multisets of integers induced by the natural orderingover N (it is known that this ordering is well-founded). Let (S,λ, θt) fulfill thehypothesis of the lemma.Case 1: In this case we suppose that, for every i ∈ [1, n + m], one of the twofollowing situations occurs:

λi(wi) ≡i λi(w′i) (166)

λi(wi) = eiWfi, λi(w′i) = c−1

i Wd−1i (167)

for some W ∈ W, (ei, fi), (di, ci) ∈ (Gi(W ),Ge(W )).Let us consider the partition

W = W0 ∪W1 ∪ W1,W1 = W1, · · · ,Wp

where W1 ∪ W1 is exactly the set of variables occuring in the equations of type(167). We can modify the system S in such a way that in every equation oftype (167), ei = fi = 1, with preservation of the hypothesis of the lemma (withthe same morphisms) and also of the size of (S,λ, θt). We can also supposethat W1 is exactly the set of lefthand-sides of the equations (167). For everyk ∈ [1, p], let (Wk, c

−1i(k)Wkd

−1i(k)) be the equation of type (167) with smallest

index, i(k) ∈ [1, n+m], such that λi(k)(wi(k)) = Wk. We thus have λi(k)(w′i(k)) =

c−1i(k)Wkd

−1i(k). Let us notice that, since θt is an AB-homomorphism and θt(Wk) =

θt(λi(k)(wi(k))) = θt(λi(k)(w′i(k))) = θt(c

−1i(k)Wkd

−1i(k)) we know that

θt(Wk) = θt(ci(k)Wkdi(k)). (168)

As θt preserves δ, for every e ∈ Gi(Wk), e′ ∈ Ge(Wk),

eWk = Wke′ ⇔ eci(k)Wkdi(k) = ci(k)Wkdi(k)e

′. (169)

It follows that there exists a unique monoid homomorphism λ′ : W → W fulfiling:

λ′(e) = e for all e ∈ ιA(A) ∪ ιB(B) (170)

λ′(W ) = W for all W ∈ W −W1 (171)

λ′(Wk) = Wk for all 1 ≤ k ≤ p (172)

λ′(Wk) = ci(k)Wkdi(k) for all 1 ≤ k ≤ p. (173)

55

The involutory anti-isomorphism It defined on Ht, maps θt(Wk) to θt(ci(k)Wkdi(k))

and θt(Wk) to θt(c−1i(k)Wkd

−1i(k). The arguments used for proving points (1)(3)(4)

of Lemma 10 show that the tuple (ci(1), di(1), . . . , ci(k), di(k), . . . , ci(p), di(p)) ful-fils the three conditions (76-77-78). Hence, by Lemma 10, there exists a unique

involutive monoid anti-isomorphism I′ : W → W such that

I′(e) = e−1 for all e ∈ ιA(A) ∪ ιB(B) (174)

I′(W ) = I(W ) for all W ∈ W0 (175)

I′(Wk) = ci(k)Wkdi(k) for all 1 ≤ k ≤ p (176)

I′(Wk) = c−1

i(k)Wkd−1i(k) for all 1 ≤ k ≤ p. (177)

and which makes 〈W, ιA, ιB, I′, γ, µ, δ〉 into an AB-algebra. Using (168), we can

see that this involution I′ also meets condition (82) with the choice hk :=θt(Wk)). Hence I′ belongs to the set I (this set of “H-involutions” is definedat the end of §3.6). It is clear that λ′ preserves ιA, ιB . The fact that

I λ′ = λ′ I′ (178)

can be checked over the generators of W. The only non-trivial verification is forW = Wk:

λ′(I(Wk)) = λ′(Wk) = Wk

I′(λ′(Wk)) = I

′(ci(k)Wkdi(k)) = d−1i(k)ci(k)Wkdi(k)c

−1i(k),

and by condition (77), the righthand-side of this last equality is Wk. Thus (178)is established. The fact that λ′ preserves µ, γ, δ is ensured by:-the hypothesis that θt does so,-the fact that all the monoid generatorsW ∈ W are mapped by γ into a singleton-the fact that, for e ∈ ιA(A)∪ιB(B), γt(e) = γW(e), and for all θ ∈ γ(e), µt(θ, e) =µW(θ, e), δt(θ, e) = δW(θ, e).We have thus established that

λ′ : (W, I) → (W, I′) is an AB-homomorphism.

Let us defineλ′i = λi λ

′, θ′t = θt.

As every λ′i is a composite of twoAB-homomorphisms, it is anAB-homomorphismfrom (Wt, I) to (Wt, I

′). By hypothesis, θt : Wt → Ht is a monoid-homomorphismwhich preserves ι, µ, γ, δ. It is clear that θt I is equal to I′ θt over ιA(A) ∪ιB(B) ∪ W0. Moreover

I(θt(Wk)) = θt(I(Wk)) = θt(Wk), while

θt(I′(Wk)) = θt(c

−1i(k)Wkd

−1i(k)).

But the hypothesis that θt(λi(k)(wi(k))) = θt(λi(k)(w′i(k))) ensures that θt(Wk) =

θt(c−1i(k)Wkd

−1i(k)). We have established that

θt : (Wt, I′) → (Ht, It) is an AB-homomorphism.

56

Let us check now that (λ′i, θ′t) fulfill conclusions (161),(162),(163) of the lemma.

Let us show thatθt = λ′ θt. (179)

The only non-trivial verification is for generators of the form Wk: from (168) weget that θt(Wk) = θt(λ

′(Wk)), which establishes (179). This equality (179) andthe fact that λ′i = λi λ

′ prove (161).Let 1 ≤ i ≤ n +m. Suppose that (wi, w

′i) is one of the pairs of the form (167)

(recall we reduced to the case where ei = fi = 1): there exists some k ∈ [1, p]such that

λi(wi) = Wk, λi(w′i) = c−1

i Wkd−1i

Thereforeλ′(λi(wi)) = Wk, λ′(λi(w

′i)) = c−1

i ci(k)Wkdi(k)d−1i (180)

Applying λi θt on (wi, w′i), on one hand, on (wi(k), w

′i(k)) on the other hand,

we obtainθt(Wk) = c−1

i θt(Wk)d−1i = c−1

i(k)θt(Wk)d−1i(k)

henceθt(Wk) = c−1

i ci(k)θt(Wk)di(k)d−1i .

This shows that (c−1i ci(k), did

−1i(k)) ∈ δ(θt(Wk)) which, as θt is anAB-homomorphism,

implies that (c−1i ci(k), did

−1i(k)) ∈ δ(Wk), which finally proves that both righthand-

sides of the two equalities in (180) are equal. We have thus established (162).By hypothesis λi = λj for all 1 ≤ i, j ≤ n, which immediately implies (163).Case 2: We suppose here that, there exists some i ∈ [1, n+m], such that noneof conditions (166)(167) does hold.Let i ∈ [1, n+m] be the minimal integer fulfilling ¬(166) ∧ ¬(167).Let (Pi, P

′i ) be a distinguishing pair for (λi(wi), λi(w

′i)). The hypothesis that

Pi ≡i P ′i implies that γ(Pi) = γ(P ′

i ), θt(Pi) = θt(P′i ). By Lemma 2, G is left-

cancellative, so that θt(Pi) = θt(P′i ) implies that πG(θt(Si)) = πG(θt(S

′i)). But,

since θt(Si), θt(S′i) have a non-empty set of types they are classes of reduced

sequences, henceθt(Si) = θt(S

′i). (181)

Lemma 13 applied on Pi, Si, P′i , S

′i ∈ W asserts that γ(Si) = γ(S′

i). As θt re-stricted to ιA(A) (resp. to ιB(B)) is injective, the value ‖Si‖ = 0 would lead toSi = S′

i, which contradicts the choice of i. Finally we must have:

‖Si‖ ≥ 1, ‖S′i‖ ≥ 1, γ(Si) = γ(S′

i).

Decomposing Si, S′i over the set of generators GW we must have

Si = cW · Li; S′i = c′W ′ · L′

i (182)

where

W,W ′ ∈ W , γ(W ) = γ(W ′) ∈ TA0, γ(Li) = γ(L′i) ∈ TR, c, c′ ∈ Gi(W ).

57

Subcase 2.1:W ′ = W ,γ(W ) is a H-type.Equation (181), decomposition (182) and Lemma 8 imply that there exists d, d′ ∈Ge(W ) such that

θt(cWd) = θt(c′Wd′); θt(d

−1Li) = θt(d′−1L′

i).

As θt is δ-preserving, this implies that cWd = c′Wd′. Taking Qi = PicWd, Ti =d−1Li, Q

′i = P ′

i c′Wd′, T ′

i = d′−1L′i, we obtain a decomposition of λi(wi), λi(w

′i)

such that 2‖ψt(Ti)‖+2‖ψt(T ′i )‖ ≤ 2‖ψt(Si)‖+2‖ψt(S′

i)‖ and χH(Ti)+χH(T ′i ) <

χH(Si) + χH(S′i) (because Ti, T

′i cannot begin with a letter having a H-type).

This is enough to entail

∆(Qi, Ti, Q′i, T

′i , θt) < ∆(Pi, Si, P

′i , S

′i, θt)

and Qi ≡i Q′i, violating the hypothesis of minimality in the choice of decompo-

sition (164). This subcase is thus impossible.Subcase 2.2: W ′ = W , γ(W ) is a H-type.Reasoning as in subcase 2.1, we obtain d, d′ ∈ Ge(W ) such that

θt(cWd) = θt(c′Wd′); θt(d

−1Li) = θt(d′−1L′

i).

Let us define a new equation

wn+m+1 = cWd; w′n+m+1 = c′Wd′,

and the AB-morphisms:

λ′i = λi for all 1 ≤ i ≤ n+m,λ′n+m+1 = IdWt, θ′t = θt.

The new system S′ = ((wi, w′i))1≤i≤n+m+1 together with λ′ = (λi)1≤i≤n+m+1

and θ′t fulfills the hypothesis of Lemma 21.

PicWd ≡i P′i c

′Wd′

‖θt(λi(Li))‖ = ‖θt(λi(Si))‖ (183)

2χH(Li) = 0 ≤ 2χH(Si) − 2 (184)

1 − χAB(PicWd) = 1 ≤ (1 − χAB(Pi)) + 1. (185)

Adding up comparisons (183)(184)(185), we obtain:

∆(PicWd, P ′i c

′W ′d′, d−1Li, d′−1L′

i, θt) < ∆(Pi, P′i , Si, S

′i, θt)

hence, by definition of ∆i,

∆i(S′,λ′, θ′t) < ∆i(S,λ, θt). (186)

The integer i was supposed to do not fullfil (167), hence

1 ≤ 1 − χAB(Pi) + 4‖θt(Li)‖,

58

which implies∆n+m+1(S

′, λ′, θt) < ∆i(S, λ, θt). (187)

The above inequalities (186)(187) prove that

∆i(S′,λ′, θ′t), ∆n+m+1(S

′,λ′, θ′t) < ∆i(S,λ, θt).

Hence‖(S′,λ′, θ′t)‖ < ‖(S,λ, θt)‖.

By induction hypothesis, the conclusion of the lemma holds for (S ′,λ′, θ′t). Thisproves that it holds for (S,λ, θt) too.Subcase 2.3: W ′ /∈ W, W,γ(W ) is a H-type.As in subcase 2.1 we obtain that there exists d, d′ ∈ Ge(W ) such that

θt(cWd) = θt(c′W ′d′); θt(d

−1Li) = θt(d′−1L′

i). (188)

Let us consider the monoid homomorphism λ′ : W → W fulfilling:

λ′(e) = e for all e ∈ ιA(A) ∪ ιB(B) (189)

λ′(W ) = c−1c′W ′d′d−1 (190)

λ′(W ) = dd′−1W ′c′−1c (191)

(this definition is written for the case where W ∈ W , in the case where W /∈ W ,last line of this definition must be cancelled). Such an homomorphism existsbecause (188) ensures that (W, c−1c′W ′d′d−1) (resp. (W , dd′−1W ′c′1c) in caseW ∈ W) have the same image by δ. Unicity of λ′ is straightforward. Suchan homomorphism λ′ also preserves µ, γ, δ because θt does so. It preserves thepartial involution because dd′−1W ′c′1c = I(c−1c′W ′d′d−1). Hence λ′ is an AB-homomorphism. Let us define:

λ′i = λi λ′ for all 1 ≤ i ≤ n+m, θ′t = θt.

Since Alph(S,λ′) = Alph(S,λ) − W, W, the inequality A(S,λ′) ≤ K0 stillholds.In this sytem we now have:

PicWd ≡i P′i c

′W ′d′

‖θt(λi(d−1Li))‖ = ‖θt(λi(Si))‖ (192)

2χH(d−1Li) = 0 ≤ 2χH(Si) − 2 (193)

The conjunction of comparisons (192) (193) imply

∆i(S′, λ′, θ′t) < ∆i(S, λ, θt). (194)

and finally‖(S′,λ′, θ′t)‖ < ‖(S,λ, θt)‖.

59

The system S′ = S together with λ′ = (λ′i)1≤i≤n+m and θ′t fulfills the hypothesisof Lemma 21 and has smaller size. By induction hypothesis the conclusion of thelemma holds for (S′,λ′, θ′t). This proves that it holds for (S,λ, θt) too.Subcase 2.4:W ′ = W ,γ(W ) is a T-type.Let us observe that ‖θt(cWd)‖ = ‖θt(c′Wd′)‖. In view of equation (181), de-composition (182) and the above equality, point (1) of Lemma 9 applies: thereexists d, d′ ∈ γ3(W ) such that

θt(cWd) = θt(c′Wd′); θt(d

−1Li) = θt(d′−1L′

i).

We can then conclude as in subcase 2.1.Subcase 2.5:W ′ = W ,γ(W ) is a T-type.Then γ(W ) = γ(W ) = IT (γ(W )). But the two atomic T-types are exchangedby the involution IT , which makes this subcase impossible.Subcase 2.6:W ′ /∈ W, W,γ(W ) is a T-type θ′.By (181) θt(cWLi) = θt(c

′W ′L′i). Using that γ(W ) = γ(W ′) is a T-type, we

can apply Lemma 9 on P = θt(cW ), P ′ = θt(cW′), S = θt(Li), S

′ = θt(L′i). We

distinguish 3 subsubcases according to which point of Lemma 9 occurs.subsubcase 2.6.1: ‖θt(cW )‖ = ‖θt(cW ′)‖.This corresponds to point (1) of Lemma 9: there exists some d ∈ Ge(W ) suchthat,

θt(cW ) = θt(cW′)d, dθt(Li) = θt(L

′i). (195)

We can end this subsubcase as for case 2.3.subsubcase 2.6.2: ‖θt(cW )‖ < ‖θt(cW ′)‖.This corresponds to point (2) of Lemma 9: ∃d ∈ Ge(W ), P ′

1, P′2, P

′3 ∈ Ht,P

′1 has

a T-type θ′1,P

′3 has a T-type θ′

3,P′2 has a H-type θ′

2 such that

θt(cW ) = P ′1d, θt(c

′W ′) = P ′1 · P

′2 · P

′3, dθt(Li) = P ′

2P′3θt(L

′i), θ′ = θ′

1θ′2θ

′3

(196)Let us apply Lemma 14 over the AB-homomorphism θt, the equality θt(dLi) =P ′

2·(P′3θt(L

′i)), the H-type θ = θ′

2 and the T-type ρ such that γt(P′3θt(L

′i)) = ρ.

We obtain some P2, S ∈ W such that:

dLi = P2 · S, θt(P2) = P ′2, θt(S) = P ′

3θt(L′i). (197)

First stepLet us assume, in this step, that W ′ ∈ W.By axiom (25), c′ ∈ W, by axiom (26) c′W ′ ∈ W. Axiom (36) on AB-homomorphismsimplies that P ′

1 ·P′2 ·P

′3 ∈ dom(It). Axiom (26) implies that all the P ′

i (1 ≤ i ≤ 3)belong to dom(It) and axiom (36) implies that

W,P2 ∈ W. (198)

We saw above that γ(P ′3) is a T-type and, by hypothesis, A(S,λ) ≤ K0 <

Card(V0).Hence, we can choose a letter W3 ∈ W such that

W3 /∈ Alph(S,λ), γ(W3) = γ(P ′3), µ(W3) = µ(P ′

3), δ(W3) = δ(P ′3).

60

1 1c

P ′

1 P ′

3P ′

2

d 1

Li

θt(cW )

θt(L′

i)

θt

θt(Li)

θt(c′W ′)

11

SP2

Fig. 13. subsubcase 2.6.2

We define a monoid-homomorphism λ′ : W → W by

λ′(e) = e for all e ∈ ιA(A) ∪ ιB(B)

λ′(W ′′) = W ′′ for all W ′′ ∈ W − W ′, W ′

λ′(W ′) = c′−1cWd−1P2W3 (199)

λ′(W ′) = I(c′−1cWd−1P2W3) . (200)

and we define a new monoid-homomorphism θ′ : Wt → Ht by

θ′(ιA(a)) = a for all a ∈ A

θ′(ιB(b)) = b for all b ∈ B

θ′(W ′′) = W ′′ for all W ′′ ∈ W − W3, W3

θ′(W3) = P ′3 (201)

θ′(W3) = It(P′3). (202)

As in the above cases we can check that λ′, θ′t are AB-homomorphisms. Let uscheck that, for every W ′′ ∈ Alph(S,λ)

θ′t(λ′(W ′′)) = θt(W

′′). (203)

SinceW3 /∈ Alph(S,λ),W 3 /∈ Alph(S,λ), for everyW ′′ ∈ Alph(S,λ)−W ′, W ′,θ′t(λ

′t(W

′′)) = θ′t(W′′) = θt(W

′′).

61

Moreover

θ′t(λ′(W ′)) = θ′t(c

′−1cWd−1P2W3) by (201)

= c′−1θt(cW )d−1θ′t(P2)P′3 by (201)

= c′−1θt(cW )d−1θt(P2)P′3 ( no occurrence of W3, W3 in P2)

= c′−1θt(cW )d−1P ′2P

′3 by (197), second equation

= c′−1P ′1dd

−1P ′2P

′3 by (196), first equation

= θt(W′) by (196), second equation.

Finally, since λ′, θ′t, θt preserve the involutions IW, It, we also get

θ′t(λ′(W ′)) = θt(W

′).

Let us defineλ′i = λi λ

′ for all 1 ≤ i ≤ n+m.

Let us notice that γ(W ′) is a T-type while γ(P2) is a H-type. Hence ‖θt(P2)‖ =0 < ‖θt(W ′)‖ which proves that

W ′, W ′ /∈ Alph(P2)

Hence Alph(S′,λ′) = Alph(S,λ) ∪ W3, W3 − W ′, W ′. We thus obtain

A(S′,λ′) ≤ K0. (204)

Equality (203) and inequality (204) ensure that the new triple (S,λ′, θ′) fulfillsthe hypothesis of Lemma 21.Let us evaluate now the size of this new triple.

(λ′i(wi), λ′i(w

′i)) = (λ′(PicWLi), λ

′(P ′i c

′W ′L′i))

= (λ′(Pi)cWλ′(d−1P2S), λ′(P ′i )c

′(c′−1cWd−1P2W3)λ′(L′

i))

= (λ′(Pi)cWd−1P2 · λ′(S)), λ′(P ′i )cWd−1P2 ·W3λ

′(L′i)) by (6.2)

Let us set

Qi = λ′(Pi)cWd−1P2, Ti = λ′(S), Q′i = λ′(P ′

i )cWd−1P2, T ′i = W3λ

′(L′i)

∆(Qi, Ti, Q′i, T

′i , θ

′t) ≤ 3 + 4‖θ′t(W3)θ

′t(λ

′(L′i))‖

= 3 + 4‖P ′3‖ + 4‖θt(L′

i)‖ ( by (201), (203))

< 4(‖P ′3‖ + ‖P ′

3‖ + ‖P ′3‖) + 4‖θt(L′

i)‖ ( because ‖P ′1‖ ≥ 1)

≤ ∆(Pi, Si, P′i , S

′i, θt). (205)

The hypothesis that Pi, P′i are related by the monoid-congruence generated

by the set of pairs (λj(wj), λj(w′j)) | i + 1 ≤ j ≤ n + m implies that

62

λ′(Pi), λ′(P ′

i ) are related by the monoid-congruence generated by the set of pairs(λ′(λj(wj)), λ′(λj(w′

j))) | i+ 1 ≤ j ≤ n+m. It follows that

∆i(S,λ′, θ′t) ≤ ∆(Qi, Ti, Q

′i, T

′i , θ

′t) < ∆(Pi, Si, P

′i , S

′i, θt) = ∆i(S,λ, θt)

and thus:‖(S′,λ′, θ′)‖ < ‖(S,λ, θ)‖.

By induction hypothesis, conclusions (161-163) are true for (S,λ′), which alsoimplies that they hold for (S,λ).second stepLet us handle the situation where W ′ does not belong to W.We just cancel the last line of (200) and can conclude as in first step.subsubcase 2.6.3: ‖θt(cW )‖ > ‖θt(c′W ′)‖.Symetric to subsubcase 2.6.2. 2

Let us prove Lemma 20.Proof: Let us suppose that σt : Wt → Ht is an AB-homomorphism solvingthe system S. Let us define m := 0, λi := Id for 1 ≤ i ≤ n, θt := σt,K0 :=A(

∏ni=1 wiw

′i). Since

A(

n∏

i=1

wiw′i) ≤ Card(W) < Card(V0),

the integers n,m, the maps (λi)1≤i≤n+m, θt and the integer K0 are fulfilingthe hypothesis of Lemma 21. Let us consider the maps I′, λ′i, θ

′t given by the

conclusion of Lemma 21 and choose:

σW := λ′1, ψt := θ′t.

These objects are fulfiling the properties asserted in Lemma 20, as required. 2

From W-solutions to t-solutions Conversely, every W-solution of a givensystem S of the form (116) provides a t-solution of the same system. Let usstate this formally.

Lemma 22 (W-solutions provide t-solutions). Let S = ((wi, w′i))1≤i≤n be

a system of t-equations of the form (116). Let us suppose that there exists anH-involution I′ ∈ I and an AB-homomorphism σW : (Wt, I) → (Wt, I

′) suchthat

σW(wi) = σW(w′i) for all 1 ≤ i ≤ n.

Then, there exists an AB-homomorphism ψt : (Wt, I′) → (Ht, It) such that σW

ψt solves the system S.

Proof: Let I′ ∈ I and σW : (Wt, I) → (Wt, I′) fulfil the hypothesis of the lemma.

We define a monoid-homomorphism ϕ : W∗t ∗A ∗B → Ht in the following way:

-for every a ∈ A,ϕ(ι(a)) = a

63

-for every b ∈ B,ϕ(ι(b)) = b-for every W ∈ Wt \ W, we choose some t-sequence s ∈ H ∗ t, t∗ which realizesW in the sense that

s /∈ dom(It), γ(W ) ⊆ γ(s),

∀θ ∈ γ(W ), µ(θ,W ) = µ(θ, s), δ(θ,W ) = δ(θ, s),

and we set ϕ(W ) = [s]∼.-for every W ∈ Wt ∩ W0, we choose some t-sequence s ∈ H ∗ t, t∗ such that

s ∈ dom(It), γ(W ) ⊆ γ(s),

∀θ ∈ γ(W ), µ(θ,W ) = µ(θ, s), δ(θ,W ) = δ(θ, s),

and we set ϕ(W ) = [s]∼.-for every k ∈ [1, p], we choose some element hk ∈ H such that

hk ∈ I(H), akhkbk = h−1k , γ(W ) ⊆ γ(hk),

∀θ ∈ γ(W ), µ(θ,W ) = µ(θ, hk), δ(θ,W ) = δ(θ, hk),

and we set ϕ(Wk) = [h]∼, ϕ(Wk) = [h−1k ]∼. Since, for every W ∈ W and every

(c, d) ∈ δ(W ), cϕ(W ) ∼ ϕ(W )d, the map ϕ induces a monoid-homomorphismψt : Wt → Ht. This monoid homomorphism clearly fulfils conditions (35,36,38,39,40)by our choices. It is also compatible with the involutions (condition (37)) be-cause,

ψt(I′(Wk)) = ψt(akWkbk) = [akhkbk]∼ = [h−1

k ]∼ = It(ψt(Wk))

and, analogously

ψt(I′(Wk)) = ψt(a

−1k Wkb

−1k ) = [a−1

k h−1k b−1

k ]∼ = [hk]∼ = It(ψt(Wk)).

Thus ψt is an AB-homomorphism. Since σW solves S it is clear that σW ψtsolves S too.2

64

7 Equations over U

7.1 The group U

Let us adjoin to W an alphabet of inverses W . The extended alphabet W ′ :=

W ∪ W ∪ W is now endowed with a total involution W 7→ W , which extendsthe partial involution IW. The maps γ, µ, δ are extended to W ′ in such a waythat the axiom (33) of AB-algebras is fulfilled by W ′∗ ∗A ∗B endowed with thisinvolution. We define the group

U := 〈A ∗B,W ′; WeW = δ(W )(e) (e ∈ Gi(W ),W ∈ W)〉 (206)

i.e. it is an HNN-extension of the free product A ∗ B with, as stable letters, allthe letters W from W ′ and as partial isomorphisms, the maps δ(W ). We identifyιA (resp. ιB) with the natural embedding of A (resp. B) into A ∗B. We denoteby ≡U the monoid-congruence over W ′∗ ∗A ∗B generated by the set of relations(206). We denote by

πU : W ′∗ ∗A ∗B → U

the homomorphism z 7→ [z]≡U. All the pairs of (70) also belong to ≡U, hence

≡⊆≡U. Thus, there exists a unique map πU : W → U such that

πU |W∗∗A∗B = π≡ πU .

An element z ∈ W ′∗ ∗ A ∗ B is said to be a reduced sequence iff it does notcontain any factor of the form WeW with W ∈ W ′, e ∈ Gi(W ). We denote byRed(A ∗B,W ′) the subset of W ′∗ ∗A ∗B consisting of all reduced sequences.

Lemma 23. Let z, z′ be some reduced sequences in W∗ ∗ A ∗ B. Then z ≡U z′

if and only if z ≡ z′.

This lemma is obtained via the analogue of lemma 1, but in the case of an HNN-extension with a set W ′ of stable letters (instead of just t, t). Such an analoguecan be obtained from lemma 1, by induction over the number of stable letters.

Lemma 24. Let z, z′ ∈ W∗ ∗A ∗B such that γ(z) = γ(z′) 6= ∅. Then z ≡U z′ if

and only if z ≡ z′.

Every z such that γ(z) 6= ∅ is a reduced sequence. This lemma is thus a directcorollary of lemma 23.

7.2 From W-equations to U-equations

We use here the notion of system of equations with rational constraint over U,as defined in §2.4 for any monoid. Let us consider a system of W-equations ofthe form described in (159), together with a morphism:

SW = ((wi, w′i)i∈I , I

′); σH ∈ HomAB((WH , I), (WH , I′)).

65

The involution I′ is given by formulas (75). Let us notice that every letterWk ∈ W1, i.e. on which I′ has a “non-standard”value, fulfills (78), so that γ(Wk)must be a H-type and Gi(Wk) = Ge(Wk).Let zi, z

′i ∈ W∗

t ∗A ∗B be some representatives, modulo ≡, of wi, w′i. By lemma

18, we can also choose an AB-homomorphism σH : W∗t ∗A∗B → W∗

t ∗A∗B suchthat π≡ σH = σH π≡. For every W ∈ Wt we consider the following rationalsubsets of W∗

t ∗A ∗B:

RI,W := z ∈ W∗t ∗A ∗B | γ(z) = γ(W ) ∧ z ∈ W∗

t ∗A ∗B ⇔W ∈ Wt,

Rµ,W := z ∈ W∗t ∗A ∗B | γ(z) = γ(W ) ∧ µ(z) = µ(W ),

Rδ,W := z ∈ W∗t ∗A ∗B | γ(z) = γ(W ) ∧ δ(z) = δ(W ).

RH,W := σH(W ) if W ∈ WH ; RH,W := W∗t ∗A ∗B if W /∈ WH

We define the rational constraint C : W∗t ∗A ∗B → Rat(U) by:

∀W ∈ Wt, C(W ) := πU(RI,W ) ∩ πU(Rµ,W ) ∩ πU(Rδ,W ) ∩ πU(RH,W ),

∀e ∈ ιA(A) ∪ ιB(B), C(e) := πU(e).

Let us define a system of equations over U with rational constraint:

SU(σH) := ((zi, z′i)1≤i≤n,C),

(note that SU(σH) does really depend on SW and σH , but not on the choice ofσH).

Lemma 25. The map Φ : HomAB((Wt, I), (Wt, I′)) → Hom(W∗

t ∗ A ∗ B,U),σW 7→ πW σW πU induces a bijection from the set of solutions of SW whichextend σH , into the set of solutions of SU(σH).

We prove this lemma in the subsequent three subsections; see figure 14 for thegeneral context and figure 15 for the details of the proof.

From W-solutions to U-solutions Let σW be a solution of SW, extendingσH .Let σU := Φ(σW) i.e.

σU := π≡ σW πU.

From the definitions of zi, z′i we get

σU(zi) = πU(σW(wi) = πU(σW(w′i) = σU(z′i), (207)

thus σU is a solution of the system of equations (zi, z′i)1≤i≤n.

The map π≡ σW is an AB-homomorphism, hence it maps I on I′ and preservesγ, µ, δ. It follows that, for every W ∈ Wt:

σW(π≡(W )) ∈ π≡(RI,W ) ∩ π≡(Rµ,W ) ∩ π≡(Rδ,W ).

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ψt

(Ht, It)G

σtσ

(Wt, I′)

(Wt, I)σW ,e

πG

π≡

W∗

t ∗ A ∗ B

U

σU

σW

WH

WH

σH

U∗

πU

Fig. 14. lemma 25: the context

As σW extends σH we get, for every W ∈ W :

σW(π≡(W )) ∈ π≡(RH,W ).

Applying πU on both sides of the above membership relations we obtain that,for every W ∈ Wt:

σU(W ) ∈ C(W ). (208)

Moreover the three maps π≡, σW, πU are fixing every element of A ∪B, so that,for every e ∈ A ∪B:

σU(e) ∈ C(e). (209)

By (207)(208)(209) σU is a solution of SU(σH).

From U-solutions to W-solutions Let σU be a solution of SU(σH).Since, for every W ∈ Wt, σU(W ) ∈ C(W ) ⊆ πU(Rµ,W ), there is a choice mapσU : Wt → W∗

t ∗A ∗B fulfilling:

∀W ∈ Wt, σU(W ) ∈ Rµ,W , πU(σU(W )) = σU(W ). (210)

Let us denote by σU the unique monoid homomorphism fixing every element ofA ∗ B and extending the above choice map. Since σU(W ) ∈ C(W ), there existzI,W ∈ RI,W , zδ,W ∈ Rδ,W , zH,W ∈ RH,W fulfilling

πU(σU(W )) = πU(zI,W ) = πU(zδ,W ) = πU(zH,W ).

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W∗

t ∗A ∗B

U

Wt

WH

Wt

WH

π≡ π≡

σW

σH

σU

πU

πU

W∗

t ∗A ∗ B

σU

σH

W∗

H ∗ A ∗ B W∗

H ∗A ∗B

π≡ π≡

Fig. 15. lemma 25: the proof

All these zI,W , zδ,W , zH,W have a non-empty image by γ and are equivalent withσU(W ) modulo ≡U. By lemma 24:

σU(W ) ≡ zI,W ≡ zδ,W ≡ zH,W .

The equivalence ≡ preserves the AB-structure of W∗∗A∗B (see (71-72)), hence,for every W ∈ Wt,

σU(W ) ∈ Wt ∗A ∗B ⇔W ∈ Wt

δ(σU(W )) = δ(W )

π≡(σU(W )) = π≡(σH(W )) if W ∈ WH . (211)

The map σU : Wt → W∗t ∗ A ∗B defined by (210) can thus be extended into an

AB-homomorphism σU : W∗t ∗ A ∗ B → W∗

t ∗ A ∗ B . By lemma 16 it defines aunique AB-homomorphism σW : Wt → Wt fulfilling:

π≡ σW = σW π≡. (212)

By (211)(212) σW extends σH .By hypothesis σU πU is a solution of SU(σH) hence:

πU(σU(zi)) = πU(σU(z′i))

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i.e. σU(zi) ≡U σU(z′i). We know that γ(zi) = γ(z′i) 6= ∅ and that σU preserves γ.Using lemma 24 we conclude that σU(zi) ≡ σU(z′i), hence

σU(wi) = σU(w′i).

Using (212), π≡ σW πU = σW π≡ πU = σW πU = σU. Finally, σW is asolution of SW which extends σH and

σU = Φ(σW).

Bijection Φ Subsubsection 7.2 established that Φ is surjective. Let us check itis injective.Suppose that σW, σ

′W

∈ HomAB(Wt,Wt) fulfill:

π≡ σW πU = π≡ σ′W πU

As π≡ is surjective we get

σW πU = σ′W πU.

By lemma 24, πU is injective over z ∈ Wt | γ(z) 6= ∅, hence, for every g ∈Wt ∪ ιA(A) ∪ ιB(B),

σW(g) = σ′W(g),

which impliesσW = σ′

W.

By the above three subsubsections, Lemma 25 is proved.

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8 Transfer of solvability

8.1 The structure of U

Since U is built from two finite groups A,B by a finite number of operationswhich are either a free product or an HNN-extension, U is a virtually free group.Let us notice that, for the particular virtually free groups K of the form:

K = A ∝ F (V) (213)

i.e. a semi-direct product of a finite group A by a free group with finite rank, thedecidability of the satisfiability problem for equations with rational constraintsin K is Turing-reducible to the same problem in the free group F (V). Hence, bythe main theorem of [DHG05], this problem is decidable. By successive sequencesof Tietze transformations, we show that U can be constructed from a group K ofthe form (213) by a finite number of HNN-extensions, with associated subgroupsof strictly smaller cardinality than A.

First transformation Let t ∈ W such that δ(t) is a full isomorphism A →B, for example we can choose t such that δ(t) = ϕ. Let us apply the Tietzetransformation:

b−− > t−1ϕ−1(b)t for all b ∈ B \ 1

we obtain a group-presentation with A ∪ W ∈ W ′ | Gi(W ) = A as set ofgenerators and relations of the form:

W−1aW = δ(W )(a); for Gi(W ) = Ge(W ) = A

W−1aW = t−1ϕ−1(δ(W )(a))t; for Gi(W ) = A,Ge(W ) = B,W 6= t

aa′ = a′′; for a, a′, a′′ ∈ A such that aa′ = a′′

Second transformation Taking as new set of generators:

A∪t∪W ∈ W ′ | Gi(W ) = Ge(W ) = A∪tW−1 | Gi(W ) = A,Ge(W ) = B,W 6= t

we obtain a set of generators of the form A∪V and a set of relations of the form

V aV = δ′(V )(a) for V ∈ V ,

with δ′ : V → PIs(A,A) together with the basic relations

aa′ = a′′; for a, a′, a′′ ∈ A such that aa′ = a′′.

Let us denote by PHNN(A) the set of all such presentations of a multiple HNN-extension of a group A.

70

Decomposition Let us consider a presentation P ∈ PHNN(A). Let VA := V ∈V , dom(δ′(V )) = im(δ′(V ) = A and

K := 〈A; V aV = δ′(V )(a) for V ∈ VA〉.

We claim that K is of the form (213) and the group U presented by P is ob-tained from K by a finite number of HNN-extension operations, with associatedsubgroups of cardinality < |A|.

8.2 Transfer results

We prove now a general transfer theorem for systems of equations with rationalconstraints. We first treat the case of groups since it is technically simpler.

Transfer for groups

Proposition 5. Let H be a group and G an HNN-extension of H with finiteassociated subgroups A,B. The satisfiability problem for systems of equations withrational constraints in G is Turing-reducible to the pair of problems (Q1, Q2),where1- Q1 is the SAT-problem for systems of equations with rational constraints ina group defined by a presentation in PHNN(A)2- Q2 is the SAT-problem for systems of equations with rational constraints inH

Proof: Let us consider a system S0 of equations with rational constraints in G.By Proposition 3 it can be reduced to a finite disjunction of systems in quadraticnormal form. By Lemma19, solving one system SG in quadratic normal form re-duces to problem P1 and the auxiliary problem AP1:P1 Compute the alphabet Wt.AP1 Solve the disjunction of all the pairs of systems St(S,W , e),SH(S,W , e)by a common σt ∈ HomAB(Wt,Ht).For every value of W , e, by Lemma 20 AP1 reduces to:AP2 Find an involution I′ ∈ I and a solution σW ∈ HomAB((Wt, I), (Wt, I

′))to the system St(S,W , e), and find a ψt ∈ HomAB((Wt, I

′), (Ht, It)) such thatσW ψt solves SH(S,W , e).By Lemma 25, and a (finite) enumeration of all the I′ ∈ I, and σH ∈ HomAB(WH ,WH),AP2 reduces toP2P2.1 Solve the system SU(σH),P2.2 Find a ψH,t ∈ HomAB((WH , I

′), (Ht, It)), solving σH(SH(S,W , e)). Let usexamine now the remaining problems P1,P2.1,P2.2.Problem P1 consists, for every letter W ∈ W , in deciding whether there existsan AB-homomorphism from the induced sub-AB-algebra 〈W, W 〉 into Ht. Let us

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suppose I′ is given by the formulas (75) of §3.6. For every W ∈ WH we definethe sets

CI(W ) := H; if W ∈ W0,

CI(Wk) := h ∈ H | akhbk = h−1; for k ∈ [1, p],

CI(Wk) := h ∈ H | a−1k hb−1

k = h−1; for k ∈ [1, p],

Cγ(W ) := h ∈ H | γ(W ) ⊆ γt(h) ,

Cµ(W ) := h ∈ H | µ(W ) = µt(h),

Cδ(W ) := h ∈ H | ∀θ ∈ γ(W ), δ(θ,W ) = δt(θ, h),

C(W ) := CI(W ) ∩ Cγ(W ) ∩ Cµ(W ) ∩ Cδ(W ).

Recall the notions of equational and positively definable subsets of a monoidwere defined in §2.4 . Next lemma uses these notions as well as the notationintroduced for them.

Lemma 26. For every group H, B(EQ(H,B(Rat(H)))) ⊆ Def∃+(H,B(Rat(H))).

In words: a boolean combination of equational subsets of H (with rational con-straints) is positively definable (with rational contraints).Proof: Every disequation u 6= v in H can be translated as the formula

∃v′ · ∃w · vv′ = 1 ∧ uv′ = w ∧ w ∈ (H \ 1) (214)

Hence, if P is defined by a positive boolean combination of equations and ra-tional contraints, its complement is definable by a positive boolean combinationof statements of the form (214) above, of equations and of rational constraints.This positive boolean combination can be put in existential prenex form, show-ing that H \ P belongs to Def∃+(H,B(Rat(H))). 2

We observe that CI take values in EQ(H,B(Rat(H))), Cγ take values in B(1, A,B,H)and Cµ take values in B(Rat(H)) . The map Cδ take values which are intersec-tions of subsets of the form

CDH(c, d) := h ∈ H | ch =H hd (215)

for c, d ∈ A ∪ B and of subsets of the form H − CDH(c, d). It is clear thatCDH(c, d) ∈ EQ(H,B(Rat(H))) and, by Lemma 26, H − CDH(c, d) belongs toDef∃+(H,B(Rat(H))). Therefore, each of the four subsets CI(W ),Cγ(W ),Cµ(W ),Cδ(W )belongs to Def∃+(H,B(Rat(H))), so that C(W ) also belongs to Def∃+(H,B(Rat(H))).Deciding whether C(W ) = ∅ is thus an instance of Q2.Let W ∈ W −WH . We set now

CI(W ) := H ∗ t, t∗,

Cγ(W ) := s ∈ H ∗ t, t∗ | γ(W ) ⊆ γt(s),

Cµ(W ) := s ∈ H ∗ t, t∗ | µ1(W ) = µA(s),

Cδ(W ) := s ∈ H ∗ t, t∗ | ∀θ ∈ γ(W ), δ(θ,W ) = δt(θ, s),

C(W ) := CI(W ) ∩ Cγ(W ) ∩ Cµ(W ) ∩ Cδ(W ).

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Claim 1 (γ-constraint). For every W ∈ W −WH , Cγ(W ) is recognized by somet-automaton with labels in EQ(H,B(Rat(H))).

For every type θ ∈ T6, the subset CG(θ) := s ∈ H ∗ t, t∗ | θ ∈ γ(s) isrecognized by a variant of the t-automaton R6 where we choose as initial vertexthe initial vertex-type of θ, as terminal vertex the ending vertex-type of θ, andon which we apply a direct product with a finite automaton that memorizeswhether at least one letter t or t has been encountered. The subset Cγ(W ) canbe described as an intersection of the subsets CG(θ):

Cγ(W ) = (⋂

θ∈γ(W )

CG(θ)).

Each setCG(θ) is recognized by some fta with labelling set EQ(H,B(Rat(H))).By Lemma 3 of [LS08] (property of closure under intersection) the set Cγ(W ) isthus recognized by a fta with labelling set EQ(H,B(Rat(H))).

Claim 2 (µ-constraint). For every W ∈ W −WH , Cµ(W ) is recognized by somet-automaton with labels in Def∃+(H,B(Rat(H))).

For every states q, r ∈ Q the set

CM(q, r) := s ∈ H ∗ t, t∗ | (q, r) ∈ µA(s)

is recognized by the variant of the strict automaton A obtained just by takingq as initial state and r as terminal state. Simailarly, the set

CM(q, r) := s ∈ H ∗ t, t∗ | (q, r) ∈ (µA)−1(It(s))

is recognized by the variant of the fta A obtained by replacing the labels by theirinverses, the vertex-type of every state by its image by IR, taking q as initialstate and r as terminal state. The subset Cµ(W ) can be described as:

Cµ(W ) = (⋂

(q,r)∈µ1(W )

CM(q, r)) ∩ (⋂

(q,r)/∈µ1(W )

(H ∗ t, t∗ \ CM(q, r)))

(⋂

(q,r)∈(µ1(W ))−1

CM(q, r)) ∩ (⋂

(q,r)/∈(µ1(W ))−1

(H ∗ t, t∗ \ CM(q, r)))

By Proposition 1 and Proposition 2 of [LS08], each CM(q, r) is recognizedby a partitioned, saturated, deterministic and complete fta with labelling setB(EQ(H,B(Rat(H)))). Hence every subset H ∗ t, t∗ \CM(q, r) or H ∗ t, t∗ \CM(q, r) is also recognized by a partitioned, saturated, deterministic and com-plete fta with labelling set B(EQ(H,B(Rat(H)))). By Lemma 26 this labeling setis included in Def∃+(H,B(Rat(H))) and by Lemma 3 of [LS08], we conclude thatCµ(W ) is recognized by some t-automaton with labels in Def∃+(H,B(Rat(H))).

Claim 3 (δ-constraint). For every W ∈ W −WH , Cδ(W ) is recognized by somet-automaton with labels in Def∃+(H,B(Rat(H))).

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The subset Cδ(W ) is a boolean combination of subsets of the form:

CD(c, d) := s ∈ H ∗ t, t∗ | cs ∼ sd (216)

for some c, d ∈ A ∪ B. Each set CD(c, d) is recognized by some t-automatonwith A ∪B as set of states, with c as initial state, d as terminal state, and withdeterministic transitions such that, the state reached after reading the sequences ∈ H ∗ t, t∗, either is undefined (if s−1cs /∈ A ∪ B) or is equal to s−1cs (ifs−1cs ∈ A ∪ B). Such a fta has labels in EQ(H,B(Rat(H))). The claim followsby the same arguments as for the previous claim.Finally, due to the claims 1,2,3 and to Lemma 3 of [LS08], the subset C(W )is recognized by some t-automaton D with labels in Def∃+(H,B(Rat(H))). Theemptiness problem for C(W ) = L(D) reduces to the emptiness problem for el-ements of Def∃+(H,B(Rat(H))) (given by a system of equations with rationalconstraints), which are themselves instances of Q2.The problem P2.1, first line, is an instance of Q1, while P2.2 is an instance ofQ2.2

Proposition 6. Let H be a group with decidable SAT-problem for systems ofequations with rational constraints and let G an HNN-extension of H with finiteassociated subgroups. Then, the satisfiability problem for systems of equationswith rational constraints in G is decidable.

Proof: We prove this proposition by induction over the size of the finite associ-ated subgroups A,B used in the HNN-extension leading from H to G.Basis: |A| = 1.In this case G is the free product of H by Z. It is known that the additivegroup Z has a decidable SAT-problem for systems of equations with rationalconstraints (see, for example, [ES69]). It is proved in [DL03] that a free prod-uct of two groups having decidable SAT-problem for systems of equations withrational constraints has also a decidable SAT-problem for systems of equationswith rational constraints. Hence G has decidable SAT-problem for systems ofequations with rational constraints. 1

Induction step:|A| > 1.By Proposition 5, the SAT-problem for systems of equations with rational con-straints reduces to two other decision problems Q1, Q2. By hypothesis, here,problem Q2 is decidable.Q1 is the SAT-problem for systems of equations with rational constraints in thegroup U , which has a presentation in PHNN(A). By §8.1, U is obtained fromthe group K by a finite number of HNN-extension operations, with associated

1 the original formulation of [DL03, Theorem 2] deals with the existential first ordertheory, with rational constraints, of a group G = H1 ∗ H2. By the trick of formula(214), decidability of this fragment is equivalent to the decidability of the SAT-problem for systems of equations with rational constraints in G.

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subgroups of cardinality < |A|. We know that K has a decidable satisfiabil-ity problem for equations with rational constraints (see subsection 8.1) and, byinduction hypothesis, each of the HNN-extension operations preserves this de-cidability property. Hence U has a decidable satisfiability problem for equationswith rational constraints. Thus problem Q1 is decidable too, so that G has adecidable satisfiability problem for equations with rational constraints. 2

Theorem 1. Let H be a group and G an HNN-extension of H with finite associ-ated subgroups. The satisfiability problem for systems of equations with rationalconstraints in G is Turing-reducible to the SAT-problem for systems of equationswith rational constraints in H.

Proof: By Proposition 5 the satisfiability problem for systems of equations withrational constraints in G is Turing-reducible to a pair of problems Q1, Q2 (pre-cisely defined in the proposition). But Q1 is the SAT-problem for systems ofequations with rational constraints in U, where U is a group obtained from afinite group A by a finite number n of successive HNN-extension operationswith finite associated subgroups. It is clear that any finite group has a decidableSAT-problems for equations with rational constraints. Starting from H := Aand applying Proposition 6 n times, we obtain that Q1 is decidable. Thus thesatisfiability problem for systems of equations with rational constraints in G isTuring-reducible to the single problem Q2 , i.e. to the SAT-problem for systemsof equations with rational constraints in H. 2

Transfer for cancellative monoids We will prove the extension of Proposi-tion 5 to cancellative monoids:

Proposition 7. Let H be a cancellative monoid and G an HNN-extension of H

with finite associated subgroups. The satisfiability problem for systems of equa-tions with rational constraints in G is Turing-reducible to the pair of problems(Q1, Q2), where1- Q1 is the SAT-problem for systems of equations with rational constraints ina group defined by a presentation in PHNN(A)2- Q2 is the SAT-problem for systems of equations with rational constraints inH

What becomes difficult here is the computation of Wt: it requires to determine,for a given symbol W ∈ W whether there exists a t-sequence s such that: s ∈ H

( if γ(W ) is a H-type), s is not invertible ( if γ(W ) specifies that W does notbelong to the domain of IW), and cs 6= sd for some c, d ∈ A ∪B ( if the value ofδ(W ) imposes this). The non-invertibility condition is expressed via an univer-sally quantified formula (∀h′, h ·h′ 6= 1), and the non-commutation condition is adisequation. Since no hypothesis ensures that the satisfiability of such formulasover H is decidable, we give up the hope to compute Wt. Instead, we enumerate

75

all the subalphabets W ′ ⊆ W ( closed under IW ) and apply the above method inthe induced sub-AB-algebra W′ generated by W ′ instead of the sub-AB-algebraWt. But the above difficulty arises in the computation of ψt : W′ → Ht. Wehave to compute, for every W ′ ∈ W ′, an image ψt(W

′) having the same be-haviour as W ′ w.r.t. I, γ, µ, δ, while Lemma 26 is no more ensured when H isnot assumed to be a group. We avoid here this difficulty by computing a weakAB-homomorphism ψt : W′ → Ht.We call weak AB-homomorphism from M1 to M2 any map ψ : M1 → M2 ful-filling the seven properties (217-223) below:

ψ : (M1, ·, 1M1) → (M2, ·, 1M2

) is a monoid homomorphism (217)

∀a ∈ A, ∀b ∈ B,ψ(ιA,1(a)) = ιA,2(a), ψ(ιB,1(b)) = ιB,2(b) (218)

∀m ∈ M1 − γ−11 (∅), m ∈ dom(I1) ⇒ ψ(m) ∈ dom(I2) (219)

∀m ∈ M1, I2(ψ(m)) = ψ(I1(m)) (220)

∀m ∈ M1, γ2(ψ(m)) ⊇ γ1(m) (221)

∀m ∈ M1, ∀θ ∈ γ1(m), µ2(θ, ψ(m)) ⊇ µ1(θ,m), (222)

∀m ∈ M1, ∀θ ∈ γ1(m), δ2(θ, ψ(m)) ⊇ δ1(θ,m). (223)

We have thus replaced the axioms (36),(39),(40) by the weaker axioms (219),(222),(223).It remains true that the above list of axioms can be checked on the generatorsonly, i.e. the following analogue of Lemma11 is true.

Lemma 27. Let M2 = 〈M2, ·, 1M2, ιA,2, ιB,2, I2, γ2, µ2, δ2〉 be some AB-algebra.

Let W′ be the sub-AB-algebra generated by some subalphabet W ′ ⊆ W , which isclosed under IW. Let ψ : W′ → M2 be some monoid-homomorphism. This mapψ is a weak AB-homomorphism if and only if,1- ιA ψ = ιA,2, ιB ψ = ιB,2and for every g ∈ W ′ ∪A ∪B,θ ∈ γ(g):2- g ∈ dom(I) ⇒ ψ(g) ∈ dom(I2)2’- I2(ψ(g)) = ψ(I(g))3- γ2(ψ(g)) ⊇ γ(g)4- µ2(θ, ψ(g)) ⊇ µ(θ, g)5- δ2(θ, ψ(g)) ⊇ δ(θ, g).

Given two AB-algebras M1,M2, we denote by WHomAB(M1,M2) the set ofall weak AB-homomorphisms from M1 to M2.Accordingly, we weaken the notion of H-involution. An involution I′ of the form(74,75), fulfiling conditions (76-78) is called weakly H-realizable if, for everyk ∈ [1, p]

∃hk ∈ I(H), γ(Wk) ⊆ γ(hk), µ(Wk) ⊆ µt(hk), δ(Wk) ⊆ δ(hk), akhkbk = h−1k .

(224)We denote by WI the set of all partial involution I′ of the form (74,75) satisfyingthe four conditions (76)(77)(78) and (224). We call the elements of WI the weakH-involutions of W. We can then express the solutions of the original equationover G via some AB− (resp. weak AB−) homomorphisms.

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Lemma 28. Let DS =∨j∈J DSj be a finite disjunction of systems of equa-

tions over G, with rational constraint, where Sj := ((Ei)1≤i≤n, µA, µU ,j). Let ussuppose that DS is in closed quadratic normal form. A monoid homomorphism

σ : U∗ → G

is a solution of S if and only if, there exists and index j ∈ J , an admissiblechoice (W , e) of variables of W (resp. elements of A∪B), an alphabet W ′ ⊆ Wposessing all these variables and closed under IW, an involution I′ ∈ WI, anAB-homomorphism σW : (〈W ′〉, I),→ (〈W ′〉, I′) and a weak AB-homomorphismψt : (〈W ′〉, I′) → Ht such that:(S1) σW is a solution of St(Sj ,W , e),(S2) σW ψt is a solution of SH(Sj ,W , e),(S3) σ = σW ,e σW ψt πG.

(These maps were summarized on Figure 12 and form the left-lower part ofFigure 16).Proof: 1- Suppose that a monoid-homomorphism

σ : U∗ → G

is a solution of S. By Lemma19, there exists an admissible choice (W , e) ofvariables of Wt and an AB-homomorphism σt : Wt → Ht such that:

σt solves St(S,W , e) ∧ SH(S,W , e) (225)

σ = σW ,e σt πG (226)

By Lemma20 σt can be decomposed as

σt = σW ψt (227)

whereσW ∈ HomAB((Wt, I), (Wt, I

′)) (228)

I′ ∈ I. (229)

such thatσW solves St(S,W , e) (230)

Let us choose W ′ := Wt. By (230), (S1) is true; by (225) and (227) (S2) is true;by (226) and (227) (S3) is true.2- Conversely, suppose that (W , e),W ′, I′ ∈ WI, σW, ψt are fulfiling (S1)(S2)(S3)and let us define σt := σW ψt, σ := σW ,e σt πG.By (S1)(S2), the map σt is a weak-AB-homomorphism solving St(S,W , e) ∧SH(S,W , e). The arguments given in subsubsection 5.2, adapted to aweak-H-involution I′ and a weak-AB-homomorphism ψt from a sub-AB-algebra (〈W ′〉, I′)into Ht show that σ is an over-solution of DS, which is assumed in closedquadratic normal form, thus σ is a solution of DS. 2

By Proposition 3, every system S0 of equations with rational constraints can be

77

reduced to a disjunction DS =∨j∈J Sj in closed quadratic normal form where

Sj = (S, µA, µU ,j) (i.e. the different systems are differing by their map µU ,j

only). This disjunction DS can thus be solved by the followingAlgorithm Scheme:1- Enumerate the subalphabets W ′ which are closed under the involutin IW.2- Enumerate the involutions I′ ∈ WI, the indices j ∈ J and the admissiblevectors W , e.3- For every value of W ′, I′, j,W , e :4- W′ := 〈W ′〉.5- Find a solution σW ∈ HomAB((W′, I), (W′, I′)) to the system St(Sj ,W , e)by the following procedure5.1- Enumerate all the σH ∈ HomAB((W′

H , I), (W′H , I

′)),5.2- Find a solution σU for the system SU(σH),5.3- σW := Φ−1(σU)6- Find a ψt ∈ WHomAB((W′, I′), (Ht, It)) such that σWψt solves SH(S,W , e).7- Endfor

8- If some pair σW, ψt is found then DS is satisfiable9- else DS is unsatisfiable.(We summarize on Figure 16 the different maps to be found; doted arrows cor-

ψt

(Ht, It)G

σtσ

πG

π≡

U

σU

σW

σH

U∗

πU

W ′∗ ∗A ∗B

(W′, I′)

(W′, I)σW ,e

W′

H

W′

H

Fig. 16. the algorithm: monoid case

respond to weak AB-homomorphisms). Let us precise how one can achieve everyline of this scheme.Line 5.2 is an instance of Q1. Let us show that line 6 can be achieved by aTuring-reduction to Q2.

78

By Lemma 27, line 6 amounts to find some tuple (ψt(W ))W∈W′ such thatconditions 1-5 of Lemma 27 are fulfilled and (ψt(W ))W∈W′

His a solution of

σW(SH(S,W , e)). Conditions 1-5 can be expressed by the following constraints.For every W ∈ W ′

H :

CI(W ) := H if W ∈ W ′ − W , (231)

CI(W ) := I(H) if W ∈ W0, (232)

CI(Wk) := h ∈ I(H) | akhbk = h−1, (233)

CI(Wk) := h ∈ I(H) | a−1k hb−1

k = h−1, if Wk ∈ W0, (234)

Cγ(W ) := h ∈ H | γ(W ) ⊆ γt(h), (235)

Cµ(W ) := h ∈ H | µ(W ) ⊆ µt(h), (236)

Cδ(W ) := h ∈ H | ∀θ ∈ γ(W ), δ(θ,W ) ⊆ δt(θ, h). (237)

C(W ) := CI(W ) ∩ Cγ(W ) ∩ Cµ(W ) ∩ Cδ(W ). (238)

For every W ∈ W ′ −W ′H :

CI(W ) := s ∈ H ∗ t, t∗ |W ∈ dom(I′) ⇒ s ∈ dom(It), (239)

Cγ(W ) := s ∈ H ∗ t, t∗ | γ(W ) ⊆ γt(s), (240)

Cµ(W ) := s ∈ H ∗ t, t∗ | µ(W ) ⊆ µt(s), (241)

Cδ(W ) := s ∈ H ∗ t, t∗ | ∀θ ∈ γ(W ), δ(θ,W ) ⊆ δt(θ, s), (242)

C(W ) := CI(W ) ∩ Cγ(W ) ∩ Cµ(W ) ∩ Cδ(W ). (243)

The values of the C∗(W ) have been modified in such a way that, now, everysubset C(W ) with W ∈ W ′

H belongs to EQ(H,B(Rat(H))) while every subsetC(W ) with W ∈ W ′ − W ′

H is recognized by some t-automaton with labels inEQ(H,B(Rat(H))). Line 6 amounts thus to:- find a solution in H to the system σW(SH(S,W , e)) with the additional con-straints C(W ) for W ∈ W ′

H - this is an instance of Q2

- find an element in the set C(W ) for W ∈ W ′ − W ′H - this reduces to finitely

many instances of Q2.We have thus proved Proposition 7.

Theorem 2. Let H be a cancellative monoid and G an HNN-extension of H

with finite associated subgroups A,B. The satisfiability problem for systems ofequations with rational constraints in G is Turing-reducible to the SAT-problemfor systems of equations with rational constraints in H

Proof: Let H be a cancellative monoid and G an HNN-extension of H with finiteassociated subgroups. By Proposition 7 the satisfiability problem for systems ofequations with rational constraints in G is Turing-reducible to a certain pair ofproblems (Q1, Q2), where Q1 is the SAT-problem for systems of equations withrational constraints in a group U having a presentation in PHNN(A). But, dueto the structure of U (see §8.1) and to Theorem 1, this problem Q1 is decidable.Thus the pair (Q1, Q2) is Turing-reducible to the single problem Q2, i.e. the

79

SAT-problem for systems of equations with rational constraints in H. 2

The two following sections are stating some variants of the main Theorem 2.The variations consist in considering:

– other kinds of constraints: positive rational constraints, positive subgroupconstraints, constant constraints,

– not only equations but also disequations– the operation of free product with amalgamation instead of the operation of

HNN-extension.

It turns out that all these variations lead to an analogue of Theorem 2.

80

9 Equations with positive rational constraints over G

9.1 Positive rational constraints

We consider here constraints over the variables consisting of a rational subsetof the monoid (while in Theorem 2 the constraints were consisting in booleancombinations of rational subsets). More formally, we call system of equationswith positive rational constraints in the monoid M, any system of equations Swith constraints in C = Rat(M).

Theorem 3. Let H be a cancellative monoid and G an HNN-extension of H withfinite associated subgroups. The satisfiability problem for systems of equationswith positive rational constraints in G is Turing-reducible to the SAT-problemfor systems of equations with positive rational constraints in H.

Sketch of proof: It suffices to adapt the reduction given in the proof of Theorem2:- the strict normal partitioned finite t-automaton A with labeling set B(Rat(H))is replaced by, merely, a normal partitioned finite t-automaton A with labelingset Rat(H); the existence of such a t-automaton recognizing the given positiveconstraints is ensured by Proposition 1, point (1).- the AB-algebra Ht is replaced by the AB-algebra Ht+.- the notion of weakly H-realizable involution is replaced by the following notionof weakly positively H-realizable involution:an involution I′ of the form (74,75), fulfiling conditions (76-78) is called weaklypositively H-realizable if, for every k ∈ [1, p],

∃hk ∈ I(H), γ(Wk) ⊆ γ+(hk), µ(Wk) ⊆ µt(hk), δ(Wk) ⊆ δt(hk), akhkbk = h−1k .

(244)- for every W ∈ W ′

H : Cγ(W ) := h ∈ H | γ(W ) ⊆ γ+(h),- for W ∈ W ′ \W ′

H : Cγ(W ) := s ∈ H ∗ t, t∗ | γ(W ) ⊆ γ+(s),- for W ∈ W ′ \W ′

H , each of the sets CI(W ),Cγ(W ),Cµ(W ),Cδ(W ) is recognizedby a t-automaton with labels in EQ(H,Rat(H)). 2

9.2 Positive subgroup constraints

We consider here the case where G is the HNN-extension of a group H by anisomorphism ϕ : A→ A from a finite subgroup A of H into itself. Let C1, . . . , Cnbe finitely generated subgroups of H containing A; let us consider the sets:

CH := C1, . . . , Cn; CG := C1, . . . , Cn, 〈C1, t〉, . . . , 〈Cn, t〉.

The set of constraints CG turns out to be useful for the decidability of the positivefirst-order theory of G (see [LS05]).

81

Theorem 4. Let H be a group and G an HNN-extension of H with finite asso-ciated subgroups A = B. The satisfiability problem for systems of equations overG with constraints in CG is Turing-reducible to the SAT-problem for systems ofequations over H with constraints in CH.

Sketch of proof: It suffices to adapt the reduction given in the proof of Theo-rem 2:- the strict normal partitioned finite t-automaton A with labeling set B(Rat(H))is replaced by a normal partitioned finite t-automaton A with labeling seta | a ∈ A ∪ CH.- the AB-algebra Ht is replaced by the AB-algebra Ht+.- the notion of weakly H-realizable involution is replaced by the notion of weaklypositively H-realizable involution.Let us describe more precisely the necessary adaptations. Given a finitely gen-erated subgroupCi, we consider the fta Ai that posesses exactly two states (1, H)(its initial state) and (1, 1) (its terminal state) and one transition ((1, H), Ci, (1, 1)).This fta Ai is trivially partitioned, ≈-compatible ∼-saturated, unitary and it rec-ognizes the subgroup Ci. Let us consider now the fta Bi obtained from the ftaG6 by replacing each occurence of the label H by the label Ci. This fta Bi is par-titioned, ≈-compatible (because A ⊆ Ci), ∼-saturated (because each of the fourstates (A, T ), (B,H), (B, T ), (A,H) has a loop labeled by A) unitary (because,for every vertex-type θ, there exists only one state mapped to the type θ by τ)and recognizes the subgroup 〈Ci, t〉.

Let A be the direct product of all these partitioned fta Ai and Bi.The fta A is a partitioned fta which is also ≈-compatible and ∼-saturated (byLemma 5 of [LS08]) and unitary (because unitarity is also preserved by directproduct). Given a finite set of constraints µ : U∗ → CG, there exists IU ⊆QA, TU ⊆ QA such that

C(U) = g ∈ G | (IU × TU ) ∩ µA,G(g) 6= ∅.

Any system of equations S0 over G with constraints in CG can thus be reducedto a disjunction

∨j∈J S1,j of systems of equations with rational constraints over

G with set of variables U1 ⊇ U0, of the form (115), fulfiling points (1-3-4-5) ofProposition 3 and point (2), with the modification that the finite t-automatonA is a normal partitioned finite t-automaton (which might be non-strict) overthe labelling set a | a ∈ A ∪ CH. Line 6 of the Algorithm Scheme amountsto:- find a solution in H to the system σW(SH(S,W , e)) with the additional con-straints C(W ) for W ∈ W ′

H ; this is an instance of the SAT-problem for systemsof equations with constraints in CH

- find an element in the set C(W ) for W ∈ W ′−W ′H ; the set C(W ) is recognized

by a finite t-automaton where each label is the set of solutions of a system ofequations in H with constraints in CH ; this problem thus reduces to finitelymany instances of the SAT-problem for systems of equations with constraints inCH.

82

2

9.3 Constants

The sets of constraints corresponding to so-called equations with constants are

CH := h | h ∈ H ∪ H; CG := g | g ∈ G ∪ G.

Theorem 5. Let H be a cancellative monoid and G an HNN-extension of H

with finite associated subgroups. The satisfiability problem for systems of equa-tions with constants in G is Turing-reducible to the SAT-problem for systems ofequations with constants in H.

Sketch of proof: It suffices to adapt the reduction given in the proof of Theo-rem 2:- the strict normal partitioned finite t-automaton A with labeling set B(Rat(H))is replaced by, merely, a normal partitioned finite t-automaton A with labelingset CH.- the AB-algebra Ht is replaced by the AB-algebra Ht+.- the notion of weakly H-realizable involution is replaced by the notion of weaklypositively H-realizable involution.2

83

10 Equations and disequations with rational constraints

over G

We recall that the notion of systems of equations and disequations with rationalconstraints over a monoid has been defined in §2.4.

10.1 Rational constraints

Let us show how to reduce a system of equations and disequations (with rationalconstraints) over G to a systems of equations (with rational constraints) overHt together with a system of equations/disequations (with rational constraints)over H.Let us start with a system of equations/disequations with rational constraints,over G, which is in normal form (see Proposition 4):

((Ei)1≤i≤n, (Ei)n+1≤i≤2n, µA,G, µU ) (245)

The equations Ei have the form

Ei : (Ui,1, Ui,2Ui,3) for all 1 ≤ i ≤ n (246)

while the disequations Ei have the form

Ei : (Ui,1, Ui,2) for all n+ 1 ≤ i ≤ 2n (247)

where, for every i ∈ [1, n], the symbols Ui,1, Ui,2, Ui,3, Un+i,1, Un+i,2 belong to thealphabet of unknowns U . Let us consider the alphabet V0 := [1, 2n]× [1, 3]× [1, 9]and the alphabet W constructed from this V0 in §3.6. We consider all the vectors(Wi,j,k) where 1 ≤ i ≤ 2n, 1 ≤ j ≤ 3, 1 ≤ k ≤ 9 of elements of W ∪ 1 and alltriple (ei,1,2, ei,2,3, ei,3,1) ∈ (A ∪B)3 such that: the vectors

(Wi,j,k)1≤i≤n,1≤j≤3,1≤k≤9, (ei,1,2, ei,2,3, ei,3,1)1≤i≤n

fulfill conditions (120-130) and their counterpart for disequations

(Wi,j,k)n+1≤i≤2n,1≤j≤2,1≤k≤9, (ei,1,2)n+1≤i≤2n

fulfill the analogous conditions:

p1(Wi,j,k) = (i, j, k) ∈ V0 for Wi,j,k 6= 1 (248)

γ(9∏

k=1

Wi,j,k) = (1, H, b, 1, 1) for some b ∈ 0, 1(249)

µ(9∏

k=1

Wi,j,k) = µU (Ui,j) (250)

γ(

4∏

k=1

Wi,1,k) = γ(∏4k=1Wi,2,k) (251)

Wi,j,5 ∈ W ∧ γ(Wi,j,5) is a H-type (252)

ei,1,2 ∈ Gi(Wi,1,5) = Gi(Wi,2,5) (253)

84

A vector (W , e) fulfilling (120-130) for the indices i ∈ [1, n] and (248-253) forthe indices i ∈ [n + 1, 2n], is called an admissible vector. For every admissiblevector (W , e) we define the following equations and disequations:

9∏

k=1

Wi,j,k =∏9k=1Wi′,j′,k if Ui,j = Ui′,j′ (254)

Wi,1,1Wi,1,2Wi,1,3Wi,1,4ei,1,2 = Wi,2,1Wi,2,2Wi,2,3Wi,2,4 (255)

for all 1 ≤ i ≤ 2n

Wi,2,6Wi,2,7Wi,2,8Wi,2,9 = ei,2,3W i,3,4W i,3,3W i,3,2W i,3,1 (256)

Wi,1,5Wi,1,6Wi,1,7Wi,1,8 = ei,1,3Wi,3,6Wi,3,7Wi,3,8Wi,3,9 (257)

Wi,1,5 = ei,1,2Wi,2,5ei,2,3Wi,3,5ei,3,1 (258)

for all 1 ≤ i ≤ n

∧

d∈Ge(Wi,1,5)

Wi,1,5 · d 6= ei,1,2 ·Wi,2,5 (259)

for all n+ 1 ≤ i ≤ 2n such that τe(Wi,1,5) = τe(Wi,2,5).We denote by St(S,W , e) the sequence of equations (254-257), by SH(S,W , e)the sequence of equations and disequations (258-259). For every (i, j) ∈ [1, n] ×[1, 3]∪ [n+ 1, 2n]× [1, 2] we denote by ı, the smallest pair such that Ui,j = Uı,.By σW ,e : U∗ → W we denote the unique monoid-homomorphism such that,

σW ,e(Ui,j) =

9∏

k=1

Wı,,k.

Lemma 29. Let S = ((Ei)1≤i≤n, (Ei)n+1≤i≤2n, µA,G, µU) be a system of equa-tions and disequations over G, with rational constraint. Let us suppose that S isin quadratic normal form. A monoid homomorphism

σ : U∗ → G

is a solution of S if and only if, there exists an admissible choice (W , e) ofvariables of Wt and elements of A ∪B and an AB-homomorphism

σt : Wt → Ht

solving simultaneously the system St(S,W , e) of equations over Ht and the sys-tem SH(S,W , e) of equations and disequations over H, and such that

σ = σW ,e σt πG.

85

Ui,1

Ui,2

Wi,1,1

Wi,2,1

ei,1,2

Wi,2,9

Wi,1,9Wi,1,5

Wi,2,5

· · ·

· · · · · ·

···

d

Fig. 17. Disequation cut into 3 parts

From G-solutions to t-solutions Let σ : U∗ → G be a monoid homomorphismsolving the system S. For every 1 ≤ i ≤ n we construct the vector (Wi,∗,∗, ei,∗,∗)as in §5.2. Let us fix now some disequation from S, i.e. some integer n+1 ≤ i ≤2n. Let us choose, for every j ∈ [1, 2], some si,j ∈ Red(H, t) such that:

σ(Ui,j) = πG(si,j).

Let us consider some decomposition of the form (5) for si,2, si,1:

si,2 = h0tα1h1 · · · t

αλhλ · · · tαℓhℓ, (260)

si,1 = h′0tα′

1h′1 · · · tα′

λh′λ · · · tα′

ℓ′h′ℓ′ , (261)

We know that si,1 6≈ si,2. Let us distinguish the possible forms for si,1, asrepresented on figures 18-20.Case 1: there exists some integer λ ∈ [2, ℓ], some ei,1,2 ∈ B(αλ−1) such that

h0 · · · tαλ−1hλ−1 = h′0 · · · t

αλ−1h′λ−1ei,1,2, αλ = α′λ

ei,1,2hλ−1 6= h′λ−1d for all d ∈ A(αλ).

We consider the following factors of si,2, si,1:

Pi,2 = h0 · · · tαλ−1 , Mi,2 = hλ−1, Si,2 = tαλ · · · tαℓhℓ.

86

ei,1,2···

d

· · ·

· · ·

si,2

si,1

· · ·tαλ

hλ−1

h′

λ−1

tαλ

Fig. 18. Disequations, case 1

Pi,1 = h′0 · · · tαλ−1 , Mi,1 = h′λ−1, Si,1 = tαλ · · · tα

′

ℓ′h′ℓ′ .

Following the lines of §5.2, these reduced sequences can be cut into nine factors(Pi,j,k) , 1 ≤ j ≤ 2, 1 ≤ k ≤ 9, and subsequently lifted to nine letters (Wi,j,k) ,1 ≤ j ≤ 2, 1 ≤ k ≤ 9, such that the vector (Wi,∗,∗, ei,1,2) fulfills conditions (248-253) and the classes ([Pi,∗,∗]∼) fulfill equations and disequations (255),(259). Wecan define σt(Wi,j,k) = [Pi,j,k]∼; σt can be extended into an AB-homomorphismsolving both systems St(S,W , e), SH(S,W , e) and such that

σ = σW ,e σt πG.

Case 2: there exists λ ∈ [2, ℓ], some ei,1,2 ∈ B(αλ−1) such that

h0 · · · tαλ−1hλ−1 = h′0 · · · t

αλ−1h′λ−1ei,1,2, αλ = −α′λ

We consider the factors Pi,2,Mi,2, Si,2, Pi,1,Mi,1 defined by the same formulasas in case 1, and define

Si,1 = t−αλ · · · tα′

ℓ′h′ℓ′ .

This time we obtain a vector (Wi,∗,∗) such that τe(Wi,1,5) 6= τe(Wi,2,5). It followsthere is no disequation (259) associated to this index i. The vector (Wi,∗,∗, ei,1,2)fulfills conditions (248-253) and the classes [Pi,∗,∗]∼) fulfill equation (255).

87

ei,1,2···

· · ·

· · ·

si,2

si,1

· · ·

hλ−1

h′

λ−1

tαλ

t−αλ

Fig. 19. Disequations, case 2

Case 3: there exists λ ∈ [2, ℓ], such that

si,1 = h0 · · · tαλ−1hλ−1.

This case can be treated similarly as case 2. We just define Si,1 = 1 and, corre-spondingly Wi,1,k = 1, for 6 ≤ k ≤ 9.Case 4: there exists some integer λ ∈ [2, ℓ′], such that

si,2 = h0 · · · tαλ−1hλ−1.

This case is obtained from case 3 by exchanging si,1 with si,2.It remains to treat some degenerated cases.Case 5: λ = 1 ≤ ℓ, fulfills one of the conditions defining cases 1-3 (except beeingsmaller than 2) or λ = 1 ≤ ℓ′ fulfills the conditions defining case 4 (except beeingsmaller than 2).We take: Pi,1 = Pi,2 = 1, ei,1,2 = 1, Si,1 = 1 (under the condition of case 3),Si,2 = 1 (under the condition of case 4).The construction is ended as in the degenerated cases of §5.2.Case 6: ℓ = ℓ′ = 0.We take: Pi,1 = Pi,2 = 1, ei,1,2 = 1, Si,1 = Si,2 = 1.The construction is ended as in the degenerated cases of §5.2.

88

ei,1,2

···

· · ·

· · ·

si,2

hλ−1

h′

λ−1

tαλ

si,1

Fig. 20. Disequations, case 3

From t-solutions to G-solutions Let σt : Wt → Ht be an AB-homomorphismsolving both systems St(S,W , e) and SH(S,W , e).Owing to the proofs of §5.2, we just have to prove that for every i ∈ [n+ 1, 2n],

σt(σW ,e(Ui,1)) 6≈ σt(σW ,e(Ui,2)).

Using equation (254) and the definition of σW ,e, the above inequalities are equiv-alent with

σt(

9∏

k=1

Wi,1,k) 6≈ σt(

9∏

k=1

Wi,2,k). (262)

Equation (255) states that

σt(

4∏

k=1

Wi,1,k)ei,1,2 ≈ σt(

4∏

k=1

Wi,2,k). (263)

Let us distinguish several cases according to the values of τe(Wi,j,5). Since, by(252), γ(Wi,j,5) are H-types, one of the following cases must occur.Case 1: τe(Wi,2,5) = τe(Wi,1,5) = (1, 1)

By (249), γ(∏9k=6Wi,j,k) = (1, 1, 0, 1, 1) for j ∈ [1, 2], which implies that, for

every j ∈ [1, 2]:

σt(

9∏

k=1

Wi,j,k) = σt(

5∏

k=1

Wi,j,k), (264)

89

But disequation (259) reads:

σt(Wi,1,5) 6=Htei,1,2 · σt(Wi,2,5). (265)

The conjunction of (263)(264)(265) proves inequality (262).Case 2: τe(Wi,2,5) = τe(Wi,1,5) = (A, T )Since σt is a solution of equation (255) and disequation (259), two representatives

of σt(∏5k=1Wi,2,k), ( resp. σt(

∏5k=1Wi,1,k)) cannot be prefixes of two reduced

sequences which are equivalent modulo ∼. Thus (262) is established.Case 3: τe(Wi,2,5) = τe(Wi,1,5) = (B, T )Same argument as for case 2.Case 4: τe(Wi,2,5) 6= τe(Wi,1,5)

This shows that the projections of σt(∏9k=1Wi,2,k), σt(

∏9k=1Wi,1,k) on t, t∗

are non-equal, so that, a fortiori, inequality (262) holds.

Theorem 6. Let H be a cancellative monoid and G an HNN-extension of H

with finite associated subgroups A,B. The satisfiability problem for systems ofequations and disequations with rational constraints in G is Turing-reducible tothe SAT-problem for systems of equations and disequations with rational con-straints in H

In order to prove this theorem we must, as for proving Theorem 2, cope withthe fact that we do not know an algorithm testing the satisfiability over H of anequation with the constraint H \ I(H) (the set of non-units of the monoid H).Therefore we use again the notion of weak-AB-homomorphism 2. We first adaptLemma 28 into the following

Lemma 30. Let DS =∨j∈J Sj be a finite disjunction of systems of equations

and disequations over G, with rational constraint defined by a strict normal fta Awith labelling set B(Rat(H)) and where Sj := ((Ei)1≤i≤n, (Ei)n+1≤i≤2n, µA, µU ,j).Let us suppose that DS is in closed quadratic normal form. A monoid homomor-phism

σ : U∗ → G

is a solution of DS if and only if, there exists and index j ∈ J , an admissiblechoice (W , e) of variables of W (resp. elements of A∪B), an alphabet W ′ ⊆ Wposessing all these variables and closed under IW, an involution I′ ∈ WI, anAB-homomorphism σW : (〈W ′〉, I) → (〈W ′〉, I′) and a weak AB-homomorphismψt : (〈W ′〉, I′) → Ht such that:(S1) σW is a solution of the system of equations St(Sj ,W , e),(S2) σWψt is a solution of the system of equations and disequations SH(Sj ,W , e),(S3) σ = σW ,e σW ψt πG.

2 though we might also use a notion weaker than AB-hom and stronger than weak-AB-hom,since we are able to translate the conditions over γ, µ, δ by equations anddisequations with rational constraints.

90

This lemma can be proved in the same way as Lemma 28.Sketch of proof of Theorem 6: It suffices to adapt the reductions given in theproof of Theorem 2 as follows:- Lemma 28 is replaced by Lemma 30- Instead of a system of equations with rational constraints in H we use thesystem SH(S,W , e) of equations and disequations with rational contraints in H

supplied by Lemma 29.2

10.2 Positive rational constraints

Here we consider CG := Rat(G) and CH := Rat(H).

Theorem 7. Let H be a cancellative monoid and G an HNN-extension of H

with finite associated subgroups A,B. The satisfiability problem for systems ofequations and disequations with positive rational constraints in G is Turing-reducible to the SAT-problem for systems of equations and disequations withpositive rational constraints in H

In order to prove this theorem we must, as for proving Theorem 2, cope with thefact that we do not know an algorithm testing a set of the form s ∈ H ∗ t, t∗ |µt(s) = µ(W ) for emptiness: such a set is not known to be recognized bya t-automaton with labels which are defined by equations, disequations andpositive rational constraints over H. Therefore we use the notion of weak-AB-homomorphism.

Lemma 31. Let DS =∨j∈J Sj be a finite disjunction of systems of equations

and disequations over G, with positive rational constraint defined by a normal ftaA with labelling set Rat(H) and where Sj := ((Ei)1≤i≤n, (Ei)n+1≤i≤2n, µA, µU ,j).Let us suppose that DS is in closed quadratic normal form. A monoid homomor-phism

σ : U∗ → G

is a solution of DS if and only if, there exists and index j ∈ J , an admissiblechoice (W , e) of variables of W (resp. elements of A∪B), an alphabet W ′ ⊆ Wposessing all these variables and closed under IW, an involution I′ ∈ WI, anAB-homomorphism σW : (〈W ′〉, I) → (〈W ′〉, I′) and a weak AB-homomorphismψt : (〈W ′〉, I′) → Ht such that:(S1) σW is a solution of the system of equations St(Sj ,W , e),(S2) σWψt is a solution of the system of equations and disequations SH(Sj ,W , e),(S3) σ = σW ,e σW ψt πG.

Sketch of proof of Theorem 7: It suffices to adapt the reductions given in theproof of Theorem 2 as follows:- Lemma 28 is replaced by Lemma 31

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- Instead of a system of equations with rational constraints in H we use the sys-tem SH(S,W , e) of equations and disequations with positive rational contraintsin H supplied by Lemma 29.- for every W ∈ W ′

H , C(W ) belongs to DEQ(H,Rat(H)).- for every W ∈ W ′ \ W ′

H , C(W ) is recognized by a finite t-automaton withlabels in DEQ(H,Rat(H)): this is clear for CI(W ),Cµ(W ),Cδ(W ); for Cγ(W ),the trick consists just in seeing the subset H \A (resp. H \B) as defined by thesystem of disequations

∧a∈A v 6= a. 2

10.3 Constants

The set of constraints here are CG := g | g ∈ G ∪ G and CH := h | h ∈H ∪ H

Theorem 8. Let H be a cancellative monoid and G an HNN-extension of H withfinite associated subgroups. The satisfiability problem for systems of equationsand disequations with constants in G is Turing-reducible to the SAT-problem forsystems of equations and disequations with constants in H.

Sketch of proof: We adapt the reductions given in the proof of Theorem 2 asfollows:- Lemma 28 is replaced by Lemma 30- for every W ∈ W ′, C(W ) is recognized by a finite t-automaton with labelingset DEQ(H, CH), i.e. the set of subsets of H which are definable by systems ofequations and disequations with constants. 2

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11 Equations and disequations over an amalgamated

product

In this section we adapt the transfer result about the operation of HNN-extension(i.e. Theorem 6) as a transfer result about free product with amalgamation (seeTheorem 10 below).

11.1 Free product

In [LS08, section 6.1] we defined a notion of (H1,H2)-automata which is thedirect adaptation to the operation of free product ( possibly with amalgama-tion) of the notion of t-automaton. Let us formulate with the help of theseautomata a result about free-products which is a particular case of [DL03, The-orem 2] (where the more general operation of graph product is treated). Giventwo sets CH1

∈ P(H1), CH2∈ P(H2), we denote by L(CH1

, CH2) the set of subsets

of Red(H1,H2) which are recognized by partitioned, ≈-compatible and finite(H1,H2)-automata with labelling set (CH1

, CH2). Informally, such an automaton

A is a finite automaton over the alphabet CH1∪CH2

which has transitions labeledalternatively by elements of CH1

and by elements of CH2; ≈-compatibility means

that, if it recognizes some (H1, H2)-sequence s it also recognizes the reducedsequence s′ such that s ≈ s′; the language recognized by A is obtained by sub-stituting, in the ordinary language recognized by A, each letter by its value inP(H1) ∪ P(H2).

Theorem 9. Let us consider two monoids H1,H2. The satisfiability problem forsystems of equations and disequations with constraints in B(L(CH1

, CH2)) over the

free product H1 ∗ H2 is Turing-reducible to the pair of problems (S1, S2) where1- S1 is the SAT-problem for systems of equations and disequations with con-straints in B(CH1

)2- S2 is the SAT-problem for systems of equations and disequations with con-straints in B(CH2

)

Note that when CHi= Rat(Hi) (for i ∈ 1, 2), then L(CH1

, CH2) = Rat(H1 ∗H2).

11.2 Free product with amalgamation

Theorem 10. Let us consider two cancellative monoids H1,H2, two finite sub-groups A1 ≤ H1, A2 ≤ H2, and an isomorphism ϕ : A1 → A2. The satisfiabilityproblem for systems of equations and disequations with rational constraints inthe amalgamated product G := 〈H1,H2; a = ϕ(a)(a ∈ A1)〉 is Turing-reducible tothe pair of problems (S1, S2) where1- S1 is the SAT-problem for systems of equations and disequations with rationalconstraints in H1

2- S2 is the SAT-problem for systems of equations and disequations with rationalconstraints in H2

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Proof: Let us use the embedding η : G → G defined in subsubsection 2.2by formula (13), where G := 〈H1 ∗ H2, t; t

−1at = ϕ(a)(a ∈ A1)〉. For everyR ∈ Rat(G), the constraint R is translated by the the rational constraint η(R)

over G and the constraint G \ R is translated by the constraint η(G) \ η(R),

which is a constraint in B(Rat(G)). The SAT problem for systems of equationsand disequations with rational constraints in G is thus reduced to the sameproblem in G. By Theorem 2 this problem reduces to the same problem in theproduct H1∗H2 and by Theorem 9 this last problem reduces to the same problemin every factor i.e. (S1, S2). 2

Note that when CHi= h | h ∈ Hi ∪ Hi then L(CH1

, CH2) ⊇ g | g ∈

G ∪ G. Hence, by similar arguments as above, one can prove the variant ofTheorem 10 where equations and disequations with constants are considered.

11.3 Equations over an amalgamated product

We strongly believe that one can adapt the main transfer result about equationsand the operation of HNN-extension (i.e. Theorem 2) as a transfer result aboutfree product with amalgamation . Since we lack such a theorem even in the case ofa free product, the most natural method would consist in adapting all the methoddeveloped in sections 3-6 to the case of a free product with amalgamation overfinite subgroups. As well, analogues of Theorem 3, dealing with equations withpositive rational constraints, and of Theorem 5, dealing with equations withconstants, should hold for free products with amalgamation.

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12 Equations and disequations over a graph of groups

Let us recall a graph of groups G is a directed graph G = (V,E) endowed witha family of groups (Gv)v∈V and a family (ϕe)e∈E of maps, such that, for everye ∈ E, ϕe is a partial isomorphism from Gι(e) into Gτ(e) (here ι(e) denotes theinitial vertex of edge e and τ(e) denotes the terminal vertex of edge e). For everyvertex v, we denote by π1(G, v) the fundamental group of G with base-point v(see [Ser77] or [DD90] for background on graphs of groups).

Theorem 11. Let G := (V,E, (Gv)v∈V , (ϕe)e∈E) be a finite graph of groupswhere all the edge partial isomorphisms ϕe have finite domain and let v0 ∈ V .The satisfiability problem for systems of equations and disequations with rationalconstraints in the fundamental group π1(G, v0) is Turing-reducible to the join ofthe problems (Sv)v∈V where Sv is the SAT-problem for systems of equations anddisequations with rational constraints in Gv

Proof: We can assume that (V,E) is connected (otherwise adding some edgeswith trivial partial isomorphisms would preserve the fundamental group andmake the graph connected). Let T ⊆ E be a covering tree. The group π1(G, v0)can be obtained as a k-fold HNN-extension of a ℓ-fold free product with amal-gamation of the vertex groups (Gv), where the associated (or amalgamated)subgroups are pairs of the form (dom(ϕe), im(ϕe)) for e ∈ E, k = |T | − |E| andℓ = |T |. Hence, using k times Theorem 6 and ℓ times Theorem 10, we obtain thedesired Turing reduction. 2

Corollary 1. If G is a virtually-free group of finite type, then the satisfiabilityproblem for systems of equations and disequations with rational constraints in G

is decidable.

Proof: Recall that the virtually-free groups of finite type are exactly the fun-damental groups of finite graphs of groups with finite vertex-groups ([DD90]).Corollary 1 follows from Theorem 11 since, in a finite group, systems of equa-tions and disequations with rational constraints are algorithmically solvable. 2

Note that another proof of this result has been exposed in [DG07].

Related worksIn [DG07] F. Dahmani and V. Guirardel are proving Corollary 1 by geometricalmethods. They claim that, using this result, they get an algorithm that solvesequations in any word hyperbolic group (even with torsion). As mentioned in theintroduction, A. Myasnikov and O. Karlampovich have shown that the full first-order theory of a free group of finite rank is decidable. Their solution includes([KM05a,KM05b]) methods for solving equations in so-called fully residuallyfinite groups (these groups generalize free groups).

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PerspectivesWe think that part of the techniques exposed here will be extendable to theHNN-extensions where the subgroups A,B are infinite but assumed to be nicelyembedded in the base group (or monoid) H.We extend in [LS05] our transfer theorems to the positive first-order theory ofHNN-extensions (or free products with amalgamation) of groups. Whether anextension to the full first-order theory is true (or not) is a fascinating openquestion.

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References

[DD90] W. Dicks and M.J. Dunwoody. Groups acting on graphs. Cambridge Univer-sity Press, 1990.

[DG07] F. Dahmani and V. Guirardel. Geometric Makanin algorithm for solvingequations in virtually free groups. In Talk given on November 08, 2007.MSRI,Lecture nr 12581, 2007.

[DHG05] V. Diekert, C. Hagenah, and C. Gutierrez. The existential theory of equationswith rational constraints in free groups is PSPACE-complete. TOCS, pages1–45, 2005.

[DL03] Volker Diekert and Markus Lohrey. Word equations over graph products. InProceedings FSTTCS, pages 156–167. LNCS 2914, 2003.

[DL04] Volker Diekert and Markus Lohrey. Word equations over graph products.Theory of Computing Systems, 37(1):133–156, 2004.

[DM06] Volker Diekert and Anca Muscholl. Solvability of equations in graph groupsis decidable. Internat. J. Algebra Comput., 16(6):1047–1069, 2006.

[ES69] Samuel Eilenberg and M. P. Schutzenberger. Rational sets in commutativemonoids. J. Algebra, 13:173–191, 1969.

[Gre90] Elisabeth R. Green. Graph Products of Groups. PhD thesis, The Universityof Leeds, 1990.

[HNN49] Graham Higman, B. H. Neumann, and Hanna Neumann. Embedding theo-rems for groups. J. London Math. Soc., 24:247–254, 1949.

[HU79] J.E. Hopcroft and J.D. Ullman. Introduction to Automata Theory, Languagesand Computation. Addison-Wesley, Reading, Mass., 1979.

[KM98] Olga G. Kharlampovich and Alexei Myasnikov. Tarski’s problem about theelementary theory of free groups has a positive solution. Electronic ResearchAnnouncements of the American Mathematical Society, 4:101–108, 1998.

[KM05a] Olga Kharlampovich and Alexei Myasnikov. Implicit function theorem overfree groups. J. Algebra, 290(1):1–203, 2005.

[KM05b] Olga Kharlampovich and Alexei G. Myasnikov. Effective JSJ decompositions.In Groups, languages, algorithms, volume 378 of Contemp. Math., pages 87–212. Amer. Math. Soc., Providence, RI, 2005.

[KMS05] B. Khan, A. G. Myasnikov, and D. E. Serbin. On positive theories of groupswith regular free length function. Manuscript, 2005.

[KP98] Antoni Koscielski and Leszek Pacholski. Makanin’s algorithm is not primitiverecursive. Theoretical Computer Science, 191(1-2):145–156, 1998.

[LS77] Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Springer-Verlag, Berlin, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete,Band 89.

[LS05] M. Lohrey and G. Senizergues. Positive theories of HNN-extensions andamalgamated free products. Manuscript, 2005.

[LS06] M. Lohrey and G. Senizergues. Theories of HNN-extensions and amalgamatedproducts. In Proceedings ICALP’06, volume 4052 of LNCS, pages 504–515.Springer-Verlag, 2006.

[LS08] Markus Lohrey and Geraud Senizergues. Rational subsets in HNN-extensionsand amalgamated products. Internat. J. Algebra Comput., 18(1):111–163,2008.

[Mak83] Gennadiı Semyonovich Makanin. Equations in a free group. Izv. Akad. NaukSSR, Ser. Math. 46:1199–1273, 1983. In Russian; English translation in Math.USSR Izvestija 21, 1983.

97

[Mak84] Gennadiı Semyonovich Makanin. Decidability of the universal and positivetheories of a free group. Izv. Akad. Nauk SSSR, Ser. Mat. 48:735–749, 1984.In Russian; English translation in Math. USSR Izvestija, 25, 75–88, 1985.

[Mer66] Y. I. Merzlyakov. Positive formulas on free groups. Algebra i Logika Sem.,5(4):25–42, 1966. In Russian.

[MS83] D.E. Muller and P.E. Schupp. Groups, the theory of ends and context-freelanguages. JCSS vol 26, no 3, pages 295–310, 1983.

[Pla99] Wojciech Plandowski. Satisfiability of word equations with constants is inPSPACE. In 40th Annual Symposium on Foundations of Computer Science(FOCS 99), pages 495–500. IEEE Computer Society Press, 1999.

[Rog87] Hartley Rogers, Jr. Theory of recursive functions and effective computability.MIT Press, Cambridge, MA, second edition, 1987.

[RS95] E. Rips and Z. Sela. Canonical representatives and equations in hyperbolicgroups. Inventiones Mathematicae, 120:489–512, 1995.

[Ser77] Jean-Pierre Serre. Arbres, amalgames, SL2. Societe Mathematique de France,Paris, 1977. Avec un sommaire anglais, Redige avec la collaboration de Hy-man Bass, Asterisque, No. 46.

[Sta71] J. R. Stallings. Group Theory and Three-Dimensional Manifolds. Number 4in Yale Mathematical Monographs. Yale University Press, 1971.

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