weakly representable but not representable relation algebras

Algebra Universalis, 32 (1994) 31 43 0002-5240/94/010031 13501.50 +0.20/0 �9 1994 Birkh/iuser Verlag, Basel

Weakly representable but not representable relation algebras

H . ANDR]~KA l

Abstract. We prove the existence of non-representable relation algebras the union and complementation free reducts of which can be represented, i.e. which are weakly representable. This answers Problem 3 in Jdnsson [4], and has consequences concerning the complexity of the equational theory of representable relation algebras.

A relation algebra (A, + , . , - , 0, 1, ;,'~, 1') is weakly representable (is a wRRA) if it is representable as an algebra o f binary relations where all the

operat ions except perhaps + and - have their natural set theoretic meanings (i.e.

. , 0, 1, ;, ~, I ' denote set theoretic intersection, empty set, biggest set, relation

composit ion, inverse or converse, and identity relation respectively but -I- and - do

not necessarily denote union and complementat ion). This not ion was introduced in

J6nsson [4], where an infinite set o f quasi-equations was given to axiomatize the

class o f all weakly representable relation algebras. Problem 3 in J6nsson [4] asks if

every wRRA is representable such that every operat ion including + and - is

s tandard (is an RRA) or not. This amounts to asking whether there is a cause o f

nonrepresentabil i ty that can be attr ibuted to "un ion" solely. In this sense, the

subject belongs to the investigation o f reducts o f relation algebras, a survey paper

on which is Schein [10]. On the other hand, an answer to Problem 3 gives

informat ion on the "complexi ty" o f the equational theory o f RRA, see Corol lary 2

at the end of the present paper. More mot ivat ion to Problem 3 can be found in

J6nsson [4] and in N6meti [9]. The answer to Problem 3 in J6nsson [4] is no,

moreover the following stronger theorem holds.

T H E O R E M 1. RRA is not finitely axiomatizable over wRRA. Le. for no finite set Ax of first order formulas is RRA = M o d ( A x ) n wRRA true.

Presented by B. Jonsson. Received September 7, 1990; accepted in final form April 18, 1991.

Research supported by Hungarian National Foundation for Scientific Research grants No. 1911 and No. T7255.

31

32 H. ANDRI~KA ALGEBRA UNIV.

P r o o f We will give a sequence of weakly representable but not representable relation algebras the u l t raproduct o f which is representable.

T h r o u g h o u t this paper , let n > 5 be a natural number . Let the finite, integral, symmetr ic relation algebra 92 = g[ n be defined as follows. The a toms of 92 are

I ' , P l , . . �9 , P n

and

p i ; p ~ = l - p ~ , p~;pj = 0' i f l < - i < j < n .

Variants o f this a lgebra have been used to prove nonfinite axiomatizabi l i ty results, see M o n k [8], M a d d u x [6, 7]. A representat ion of 92 gives rise to a coloring

of a full g raph with colors Pl . . . . . Pn and wi thout m o n o c h r o m a t i c triangles. Thus 9.I can have a representat ion over a finite set only. Our first observat ion is that 92 can have a weak representat ion over an infinite set.

We say that h : 92 ~ 91e(U) is a weak representat ion if h is a ho rnomorph i sm

with respect to all operat ions of 92, except possibly for + and - . I.e., h is one- to-one and h is a h o m o m o r p h i s m with respect to the opera t ions �9 0, 1, ;, u, 1'. Here 9te(U) denotes the full set relat ion algebra on U.

L E M M A 1. 92 has a weak representation h : 92.1 ~ 9te(U) over an infinite set U

such that {w ~ U: (u, w) ~ h(pi)} is infinite f o r all u ~ U and 1 < i < n.

P r o o f We call (G, 7) a colored graph if G is a set, ? : G x G --* A, where A is the universe of 92, and 7 satisfies the following for all u, v, w e G:

7(u, u) = 1', ?(u, v) :~ 1' if u # v,

and

~(u, v) <- ~(u, w); 7(w, v), 7(u, v) = 7(v, u).

Let (G, 7) be a colored graph, u, v e G, and let y, z E A be such that y, z ;~ 1' and x < y; z, y < x; z, z < y; x where x -- y(u, v). We show that there is a colored graph

(G' , ? ' ) such that

G'=Gu{w} for s o m e w C G

7' I G = 7, 7'(u, w) = y, 7'(w, v) = z.

Vot. 32, 1994 Weakly relation algebras 33

I.e (G', ~') is an extension o f (G, ~) with a new point w such that 7'(u, w) = y, 7"(w, v) = z.

Let At = { l ' , p i . . . . . p , } denote the set o f all a toms of 9.I. We now construct (G', ~'). Let w r G, G' = G w {w}, 7' r G = 7, and 7'(w, w) = 1'. For a ~ G let

t(a, w) = (~(a, u); y ) . (~,(a, v); z),

and we define

f t(a, W)

'(a, w) = I [ . t(a, w) . O'

if ( 1' -< z and y = 7(a, u) e At)

or (1' < y and z = 7(a, v) e At) or a e {u, v }

otherwise.

' (w, a) = ~'(a, w).

Then ~'(u, w) --= y and ~;'(w, v) = z. We want to show that (G', ;~') is a colored graph. Fo r any a ~ G, 7"(a, w ) ~ t" because 7'(u, w ) = y :~ 1", }"(v, w ) = z ~ 1', and i f

a r {u, v} then 0 ' . (ql; q2) �9 (q3; q4) ~ ~,'(a, w) for some diversity a toms qt . . . . . q4 and it can be checked that

(q~; q J " (q3; q4) ~ 1' for all diversity a toms q~ . . . . , q4-

(A diversity a tom is an a tom which is not l t ) It remains to check "commuta t iv i ty o f triangles".

Let a, b ~ G be arbitrary. We have to show that

7"(a, w) < 7(a, b); ~'(b, w) and 7(a, b) ~ },'(a, w); ~'(w, b).

34 ft. ANDRI~KA ALGEBRA UNIV.

6 ) -

lta, b)

4L t X

I f {a, b} = {u, v}, then we are done by our hypotheses on x ,y , z. Assume now a r {u, v} and b = v. N o w 7'(a, w) -< 7(a, v); 7'(v, w) = 7(a, v); z holds by definition. To check 7(a, v) < 7'(a, w); z we first notice that

O' -< [0 ' - (ql; q J " ( q 3 ; q4)]; q5 for all diversity a toms ql . . . . , qs of 9l. ( , )

Thus 0 ' < 7'(a, w); z. Therefore it is enough to show that if 1 ' < ~,(a, v), then

1' < 7'(a, w); z, i.e. that 7'(a, w) �9 z # 0. Assume 1' < y(a, v). Since the equat ion

x �9 ( y ; z) -< x �9 [(y - x ; z); (z . y ; x)] (**)

holds in every symmetr ic relation algebra, and since (G, 7) is a colored graph, we have 7(a, v) < 7(a, u); ~(u, v) < (y(a, u); y); z <- [(7(a, u) ; y )" (7(a, v); z)]; z = t(a, w); z. Thus if t(a, w) = 7'(a, w) then we are done. Assume therefore t(a, w) ~ 7'(a, w). By 1'<-t(a, w);z we have t(a, w ) . z 4=0. I f 1 ' ~ z then we are done because then

0 4= O" �9 t(a, w) �9 z = 7'(a, w) �9 z. Assume 1' < z, and let q < z for some diversity a t o m q of 9.I. Then by t(a, w) # y'(a, w) it is not true that y = ~,(a, u) = q, and therefore

q -< 7(a, u);y. Also, by 1' < 7(a, v), q < z we have q -< 7(a, v); z, hence q < 7'(a, w) showing 7'(a, w ) . z # 0 . The a rgument showing z < 7'(a,w); 7(a,v) is similar.:

O'<_7"(a,w);7(a,v ) by ( , ) , so assume l ' < z . Again by (**) we have z < t(a, w); 7(a, v), so assume t(a, w ) # 7'(a, w). Let q < 7(a, v) be a diversity a tom. Then by t(a, w) # 7'(a, w) and 1' <- z we have that q = y = 7(a, u) does not hold, therefore q < t(a, w) �9 O' = 7'(a, w) and we are done.

Assume now {a, b} c~ {u, v} = ~ . N o w 7(a, b) < 7'(a, w); 7'(w, b) because by n >- 5 it is not difficult to check that

[0 ' ' (q,; q2) " (q3; q4)]; [0 ' ' (qs; q6) " (q7; q8)] = 1

for all diversity a toms q l , . . . , q8 ***)

Vol. 32, 1994 Weakly relation algebras 35

holds in 92. I t remains to check 7'(a, w ) < 7(a, b); 7'(b, w). By ( , ) assume that

1' -< 7'(a, w), and we have to show 7(a, b) �9 7'(b, w) # 0. By 1' < 7'(a, w) we m a y assume, wi thout loss o f generality, that 1' < z and y = y(a, u) = q is a diversity a t o m o f 9.I. Let q ' < 7(a, b) be any diversity a tom. Then q ' < 7(a, b) < 7(a, u); 7(u, b) = q; ~(u, b) = y; 7(b, u) = 7(b, u); y and also q ' < 7(b, v); z because

O' -< (1 ' + q t); q2 for any diversity a toms ql, q2 (,4)

holds in 9.1. Thus 0 # q ' -< 7'(b, w)'7(a, b) and we are done with showing that

(G' , 7') is a colored graph.

CI. t ~ + . . .

Now, by using the above a rgument and (**) we can build a colored graph (G, 7)

with the following properties:

(i) Fo r any x e A, x # 0 there are u, v e G with 7(u, v) = x.

(ii) Fo r any u, v e G and x, y e A, if 7(u, v) < x; y then there is w e G such that

~(u, w) -< x, 7(w, v) - y. (iii) {w e G: 7(u, w) = P i } is infinite for all u e G and 1 < i < n.

We use (**) to achieve (ii) as follows: Assume 7(u, v) < x; y. Let x ' = x �9 (7(u, v); y),

y ' = y �9 (x; 7(u, v)). Then by (**), 7(u, v) -< x ' ; y ' , x ' < 7(u, v); y ' , and y ' < x ' ; 7(u, v). Thus x ' # 0 # y ' and if x ' # 1' # y ' then we can add a new point w to the g raph with 7(u, w) = x ' , 7(w, v) = y ' , otherwise if x ' = 1' then y ' = 7(u, v) and w = u will do, and similarly for y ' = 1'. To achieve (i) and (iii) we use that 9.1 is integral: Let u ~ G and x e A, x ~ 1' be arbi t rary. Take v = u, then 7(u, v) = 1' < x; x, x -< 1'; x, and therefore we can add w with 7(u, w ) = x. Let (G, 7) be a colored graph satisfying ( i ) - ( i i i ) . Define for all x e A:

h(x) -- {(u, v) ~ G x G: ~(u, v) -< x}.

36 1f. ANDREKA ALGEBRA UNIV.

Now it is not dil~cult to check that h : 91 ~-~ ~te(G) is a weak representation with the desired properties. QED

Now we want to use Lemma 1 to construct nonrepresentable but weakly representable relation algebras. Our idea is that we consider two copies of 91 and then we split "their direct product" into infinitely many parts. This splitting then forces the representations of the two copies of 91 to be on infinite sets, and this will rule out "normal" representations but will allow weak representations. More about the general method of "splitting" in relation algebras can be read in Andr6ka- Maddux-N6meti [ 1]. The definition below will not rely on that paper.

In order to have fewer atoms, we make our algebra ~3 integral and symmetric at the same time. Let n, m -< co, where co denotes the least infinite ordinal.

We now define the algebra ~3 = ~3(n, m). The Boolean part of ~3 is the complete Boolean algebra with atoms

1 ' , P o , � 9 P i , � 9 � 9 to . . . . . 0 , " " " i < n , j < m .

Let B denote the universe of ~3. Composition is defined as follows: For all i , j < n and r, s < m

p~;p,= ~ p k + l ' , p ~ ; p ~ = G ; L = ~ p~. i f i r k < n k < n k r

tr;t~= ~ pk + l', p i ; t r = t r ; p l = ~ tk. k < n k < m

and r # s

Further, 1'; x = x; 1' = x for all x ~ B, and ; is additive, i.e.

(E X); ( E Y) = E {x;y: x e X , y ~ Y}

for any sets X, Y of the atoms of ~3. Converse is defined by .~ = x for all x ~B. It is not difficult to check that ~3(n, m) is a relation algebra.

LEMMA 2. (i) ~3(n, m) is weakly representable i f m <- m, 5 <__ n < co.

(ii) ~3(n, m) is not representable if m >_ 3. n!, 5 <- n <co, (iii) Any nontrivial ultraproduct of the ~(n, co)'s with n finite is representable.

Proof Proof o f (i): Let (G, 7) be a colored graph satisfying (i)-(iii) in the proof of Lemma 1. Let (G', y') be a disjoint copy of (G, ?). We may assume that G is countable. We will "color" the edges between G and G' with colors 0, J < m. Now


by proper ty (iii) o f (G, ~) in the p r o o f o f Lemma 1, we can define, step-by-step, a

funct ion q : G x G'--*{ty:j < m} with the following properties:

(i) For all u, v e G, u # v and i , j < m there is w e G ' such that q(u, w) = t~, q(v, w) = tj.

(ii) For all u, v e G', u # v and i , j < m there is w e G such that q(w, u) = t~, ~(w, v) = tj.

(iii) For all u e G, v e G ' and i < n, j < m there is w ~ G such that 7(u, w) = Pi

and q(w, v) = tj.

(iv) For all u e G, v e G ' and i < n , j < m there is w e G ' such that 7"(v, w) =pi and r/(u, w) = tj.

Let U = G u G ' a n d p : U x U ~ B b e d e f i n e d b y

p(u, ~) =

"y(u,v) ifu, v e G

7"(u,v) i fu , v e G "

rl(u, v) {f u e G, v e G"

r/(v,u) i f u e G ' , v e G .

Define for all x e B

h(x) = {(u, v) s U • U: p(u, v) < x}.

N o w it is easy to check that h : ~ ~ ~ e ( U ) is a weak representation o f ~ = ~(n , m).

Proof of (iO: Let m > 3 - n!. Assume that h : ~(n , m) ~ 9~e(U) is a representation. We will derive a contradiction.

For u e U define G(u) = {w e U: (u, w) e h(pi) for some i < n } . Then (w, v) h(pi) for some i < n if w, v e G(u), w # v. Define

~,(w, v) = P i if (w, v) ~ h(pi).

Then 7 is a coloring o f G(u) with colors Po, - �9 �9 P , - 1 and without monochromat i c

triangles. Thus IG(u)[ < 3 �9 n! by G r e e n w o o d - G l e a s o n [3]. Let a e U be such that

(u, a) e h(to). Such an a exists, because 1' < to; to in ~B(n, m). Then for all j < m

there is uj e U with (u, uj) e h(pl ) and (uj, a) e h(tj), by to -< pl ; tj. Clearly, u; # uj if i # j , and uj e G(u). This shows [G(u)l > m > 3 �9 n!, a contradict ion.

Proof o f (iii): Let I be a set, F a nonprincipal ultrafilter o f I and let <ni: i e I>

be a nonbounded sequence o f natural numbers (i.e. {i e I : n i = k } C F for any k < co). Fo r any relation algebra if, let At ff denote the set o f all a toms o f ft.

Let ~3=I-]i~l~.B(ni, co)/F. Then ~ is a tomic because each ~(ni, co) is a tomic

38 H. ANDREKA ALGEBRA UNIV.

a n d A t 13 = H i ~ 1 A t O 3 ( n ~ , co)) /F. L e t P ' = [ I i s l {Po . . . . . Pni - 1 }/F~ T ' =

~I,~z{t~:j<oo}/F. Then le ' l= l r ' l and A t 1 3 = { l ' } u P ' u T ' . Let It =l r ' l and P ' = {p~: i < It}, T ' = {t~: i < It}. Also,

p~;p~= ~ p ~ + l ' , p~;p~=t ; ; t j= ~. p'~ ifi, j<I t , i r k<p, k < # k;a i

t ; ; t ~ = ~ p ~ + l ' and p ~ ; t ~ = t j ; p ~ = ~ t'~. k < # k<,u

N o w to give a representat ion for 13, we will build a colored graph (U, rl) with the following properties: U is a set, q : U x U ~ At 13 and the following hold for all u,v, w ~ U, u r and i , j ,k <It

(i) U = G w H , G n H = ( ~ ~/(u, u) = 1', ~/(u, v) E P ' if u, v e G or u, v e H,

q ( u , v ) ~ T ' i f u s G , v ~ H o r u e H , v e G , t/(u, v) --- ~/(v, u) (ii) q(u, v) -< q(u, w); t/(w, v)

(iii) if q(u,v)=p; and [{i , j ,k}[r then there is w such that ~ / ( u , w ) = p j ,

~(w, v) =p~ (iv) if ~/(u, v) = p ~ then there is w such that ~/(u, w) = t~, rt(w, v) = t ; (v) if q(u, v) = tj then there is w such that r/(u, w) = p ; and q(w, v) = t;,.

We can define (U, r/) e.g. as follows: Let p be an "enumera t ion of the requirements

( i i i ) - ( v ) " , e.g. let p : It ~ It x It x It x # x It x {0, 1, 2} be an onto funct ion such that p takes each value at arbitrari ly high places, i.e. (Va e Rng p ) ( W < It)(32 ->

~c)p~. = a. Here It = {j:j < #}. Let U o = Go = H0=~/o = ~ . Then (Uo, qo) satisfies ( i ) - ( i i ) . Let K < i t and

assume that (U~., ~b.), G~, H~ have already been defined for all 2 < ~, such that they

satisfy ( i ) - ( i i ) , and for all v < ). < It we have ]Uz[ < )., U~ _ It, Gv _ G;., H~ _ H~,

~/v = r h I U~. Let U'~ = U{U~: ~ <~c}, G~ = U{G~: ,t. < ~}; H~, = U{/-/~: 2 < ~ } , and q'~ = U { ~ : 2 < ~}. Let p(~) = (u, v, i,j, k, r). Consider the following condi t ion (*):

u, v e U ' ~ , u r

if r = 0 then ~/'(u, v) =p~ and [{i,j, k}[ r 1

if r = 1 then ~/2(u, v) = p ;

t if r = 2 then ~/~ (u, v) = tj.

I f ( * ) does not hold then we let U~ = U'~, G~ = G~,, H~ = H2, ~/~ = t/'~. Assume that (*) holds. Let now w ~it , w r U ; be arbi t rary and let v < # be such that p ; r Rng ~/~. Such a v exists because IU21 < ~ < #.


We let G~ = G;,:w{w} and H~ =H~, if r = 0 and u,v e G~ or if r = 1 and u,v e H~ or if r = 2 and u s G~, otherwise G~ = G ~ and H~ =H'~w{w} . Let U~ = G~ uH~, ~/~ I U2 = q~,, and define for a e U~, a r {u, v}

rl~(a,w)=p~, i fa , w 6 G ~ o r a , w 6 H ~

~/~(a, w) = t6 otherwise,

r/~(w, a) = q~(a, w), q~(w, w) = 1',

~/~(u, w) =p ) , r/~(v, w) =p~ if r = 0,

f l~(u,w)=t~, ~/~(v, w,) = t~ i f r = 1,

q~(u, w) =p~, ~l~(V, w) = t~ if r = 2,

and

~/~(w, b) = r/~(b, w) if b ~ {u, v}.

Let now U = U{U~: < # } , r /= U{r/K: ~c </~}. Then it is not difficult to check that (U, ~) satisfies (i) (v), and hence the function

h(x) =((u , v) e U x U: q(u, v) ~ x}

gives a representation for ~ . QED

Remark. We added the condition m < o~ in Lemma 2(i) and we used the algebras ~3(n, ~9) in Lernma 2(iii) only for convenience. Lemma 2(i) is true for any m, and Lemma 2(iii) is true for any nontrivial ultraproduct of the algebras ~3(n;, mi) if the sequences (n;: i ~ I ) and (mi: i ~ I ) of natural numbers are not bounded with respect to the ultrafilter by finite numbers.

Theorem 1 implies that the symbols + or - have to occur infinitely many times in any axiomatization of RRA (using the symbols + , �9 - , 0, 1, ;, ~, 1'). But there is a stronger corollary:

C OR OLLARY 2. Let Z be any set of universal formulas such that no formula in S contains both + and . . Assume that RRA = Mod(Z)c~wRRA. Then there are infinitely many elements in Z in which of all of ;, - and one of +, �9 occur.

Proof Let S be as in the statement of Corollary 2. Let 2; ~;) denote the set of all universal formulas valid in RRA in which ; does not occur, and similarly, let 2; (- '+), S ~-'') and S ~+" ) denote the sets of all universal formulas valid in RRA in which

40 H, ANDREKA ALGEBRA UNIV.

neither - nor + , neither - n o r . , and neither + nor �9 occur, respectively. Let

X o = r~\(X c) u 2 (-" + ) ~ Z ( - ' ) u . S (+, >). Then Zo consists of those formulas of

in which all of ;, - and one of + , �9 occur. Therefore we have to show that S o is

infinite.

Consider the algebras ~3(n, co) in the proof of Theorem 1 with 0 < n < co, n even.

An ultraproduct ~ of these algebras is representable by Lemma 2(iii), hence

~ So. Let 0 < n <co, n even. Then ~(n, c o ) e w R R A by Lemma 2(i), and

~(n, co) r RRA by Lemma 2(ii). Thus ~3(n, co) ~ Z by RRA -- Mod(Z) n wRRA.

Below, we will show that

~ ( n , co) ~ 22 c) ~ Z ( - ' + ) u Z ( - , ' ) u Z (+'").

Then ~3(n, co) ~ i;0, for all 0 < n < co, n even. This shows that 2; 0 is not finite, since

in the ultraproduct ~3 of these algebras Z0 is valid. Let ~[ = (A, + , �9 - , 0, 1, ;, ~, 1') be an algebra and let F be a subset of the

operations of 9.I. By the F-free reduct of ~I we understand the algebra we obtain

from ~1 by omitting the operations belonging to F. We say that the F-free reduct

9.I F of ~ is representable if there is a homomorphic embedding of 9.U into the F-free

reduct of an RRA. Let 0 < n < co, n even and let 9.1 = ~3(n, co). Now 9.I e wRRA means that the - ,

+-f ree reduct 9.1 - ' + of 9.1 is a subalgebra of the - , +-f ree reduct ~ - ' + of a

~3 e RRA. Then by RRA ~ Z (- '+) we have ~ ~ Z ~-'+), then ~3 - ,+ ~ X (- '+) since

neither - nor + occur in S (-,+). Then 91 - ,+ ~ Z (- '+) because 9.I - ,+ is a

subalgebra of ~3 - ' + and S ( - '+) consists of universal formulas, and then 9.[ ~ Z (-,+).

We have seen that N ~ S (- '+), By J6nsson-Tarsk i [52] Thin. 4.22 we have that the - , �9 reduct of any

relation algebra is representable, therefore 9.1 ~ S ( - , ' ) by an argument similar to the

above. It is easy to see that the ;-free reduct of ~I is representable. (For example, let U

be an infinite set and let Pi, i < n, T], j < co be a partition of U x U \ I d into

symmetric relations. Let h(1') = Idv, h(pi) = Pi, h(tj) = Tj for i < n, j < co and let h(x) = (j{h(a): a ~ At g[, a -<-- x}. Then h is a homomorphism with respect to the

operations + , - , 0, 1; u 1'.) Thus 9.I ~ Z (,). Finally we show that the + , �9 -free reduct of 9.1 is representable. It is easy to see

that ~3(co, w) is completely representable. Let g : ~3(co, co) ~ ~fle(U) be a complete

representation of ~(co, co). We now define a function h : ~ ~ 0te(U) as follows.

Recall that the atoms of ~ are l ' , p o , . . . , P , - - ~ , t o , h , - . - , and the atoms of ~B(co, co) are l ' , p o , . . . , P ~ l , P ~ , . . . , t o , q . . . . . Let P = { p ~ : i < n } w { l ' } , At = P w { t ~ : i <co}, Q = U{g(pi): i -> n} and n = 2 k . Then Q = U • U \ g ( ~ At),


and A = {2 X: X _c At}. We define for any X _ At

h(~X)=~g(ZX) if Ixnp l <- k, ~g(~ X) u Q otherwise.

We have to show that h is one-to-one and that h preserves - , 0, 1, ;, ~, l'. Let a, b ~ A, a :/: b. Then g(a) r g(b), therefore h(a) ~ h(b), because it is easy to see that g(x) = h(x) n g ( ~ At) for all x e A. Also, h(0) = g(0) = ~ and h(l ' ) = g( l ' ) = Idu by l < k , a n d h ( 1 ) = h ( Z A t ) = g ( ~ ' A t ) u Q = g ( 1 ) = U • U b y [ a t n P I = ] P l ~ k . We have Q I = Q since /~i=pi and thus g(pi)-l=g(pi) for all i. Therefore h(?t) = h(a)= h(a)-1. These show that h is a one-to-one homomorphism with respect to 0, 1, 1', ~

We are going to check that h is a homomorphism with respect to - . Let

Xc_At , Y = A t \ X , a = Z X . Then - a = ~ Y . By n = Z k we have I P I = 2 k + l , therefore ] X n P ] - < k iff IYnPI 4~ k. Thus Q c_h(a)uh(-a), therefore h(a) u h ( - a ) ~ _ Q u g ( ~ X ) w g ( ~ Y ) = Q w g ( ~ A t ) = U x U. By Q c~g(~ At) = ~ we have Q n h ( ~ X ) = ~ if [Xc~PI<_k. Thus Qnh(a)~h(-a)=~j . Also, g(~At) nh(a)=g(a) for all a, hence g ( ~ A t ) n h ( a ) n h ( - a ) = g ( ~ A t ) n g ( ~ X ) n g ( ~ r ) = ~ by x n r = ~ . We have seen h ( a ) u h ( - a ) = U x U , h(a) n h ( - a ) = ~ . Thus h ( - a ) = U • U\h(a), i.e. h preserves - .

Next we show that h preserves ;. Let T -- {t~: i < co}. It is not difficult to check that

h(y~ x) = h(~ (x n ~)) ~h(~ ( x~ v)),

and

h ( Z X u Y ) = h ( Z X ) u h ( Z Y ) i f X u Y _ T w { l ' } .

Let X, Y ~ A t , X ' = X ~ P , X " = X n T , Y'= YnP, Y " = Y ~ T , a = ~ X , a'= Z X', a" =E X', b =y' Y, b' = E Y', b"=E Y". We want to show h(a; b) = h(a); h(b). Now a'; b' + a"; b" < ~ P and a'; b" + a"; b' < ~ T + 1', therefore

h(a; b) =h(a ' ; b'+a"; b")wh(a'; b')uh(a"; b'),

and

h(a); h(b) =h(a ' ) ; h(b')uh(a"); h(b")wh(a'); h(b")wh(a"); h(b').

42 ft. ANDREKA ALGEBRA UNIV.

First we show h(a'; b") = h(a ' ) ; h(b"). I f X ' , Y" _~ {1'} then we are done, therefore

we may assume X' , Y ' ~ { I ' } . Then a ' ; b " = ~ T , and thus h(a';b") = h ( ~ T) = g ( ~ T) = g ( a ' ) ; g(b") = (g(a') ~ Q ) ; g(b") = h(a ' ) ; (b"). The p r o o f of h(a"; b ' ) = h(a"); h(b') is completely analogous.

A s s u m e X " # ~ Y " a n d l e t z = l ' i f a - b r z = 0 i f a . b = 0 . Then

h(a';b + a , =h + z g Pk u g ( z ) k k

= h(a'); h(b') u h(a"); h(b").

Assume therefore X " = ~ or Y"= ~ . Then a"; b"= O, h(a"); h(b") = ~ , thus we

have to prove h(a'; b ' ) = h(a ' ) ; h(b'). Assume X ' = Y' = {pi } for some i < n. Then

k < n k < n k < r

= g(Pi) ; g(P~) = h(a ' ) ; h(b').

Assume now tha t X ' = Y' = {Pi} holds for no i < n and that X ' , Y' ~ {1'}. Assume X ' n Y ' = ~ . Then h(a') ~ {g(a ') ,g(a ' )~Q}, h(b') e {g(b ' ) ,g(b ' )uQ} but if

h(a') = g(a') u Q then h(b') = g(b'). N o w

= g(a ' ) ; (g(b') u Q) = h(a ' ) ; h(b').

Assume X ' n Y' vL ~ . Then

h(a'; b ' ) = h ( ~ P ) = g ( k <~o~ p k + 1 ' ) = g ( a ' ) ; g ( b ' )

= (g(a') u a ) ; (g(b') u Q) = h(a ' ) ; h(b').

We have seen that h preserves ;. Thus the + , �9 -free reduct o f 9.I is representable, and hence 9.1 ~ Z (+, ~ Q E D

We note that �9 is term-expressible with - , + and + is term-expressible with - , . in Boolean algebras ( e . g . x . y = - ( - x +--y) ) , thus Corol la ry 2 becomes false if we replace "one o f + , �9 " in it with " + " or with " �9 ". Also, - is expressible


with + and �9 namely x �9 y = 0/x x + y = 1 ~ y = - x holds in Boolean algebras. Hence the condit ion "no formula in v contains both + and - " cannot be omitted

in Corol lary 2. We note that this condit ion can be omitted if we replace "universal formulas" with "equat ions" in Corol lary 2. We also note that u, 1' do not necessarily have to occur infinitely many times in an axiomatizat ion o f R R A . : There

is an equational axiomatizat ion o f R R A in which ~ and l ' occur in finitely many formulas only. This follows f rom a theorem in A n d r 6 k a - N 6 m e t i [2].

Acknowledgements

I want to express my thanks to the organizers o f the J6nsson Symposium (Iceland, July 1 -6 , 1990), and to its chairman G. McNul ty for the warm, cheerful and stimulating a tmosphere in which the result reported in this paper was obtained. I also want to express my thanks to W. Craig, S. Givant and L. Henkin for organizing an algebraic logic seminar some weeks earlier in Oakland, California where I learned a lot, especially f rom R. Maddux to whom I also want to express my gratitude. I am also grateful to B. J6nsson for raising this problem and to R. Maddux for restating this problem in his plenary talk on the first day of the J6nsson Symposium.

! am indebted to the referee for useful suggestions and careful reading o f the

paper.

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Hungarian Academy of Sciences Budapest, Hungary

weakly representable but not representable relation algebras

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