the algebraic structure of...

The algebraic structure of semi-brace

Paola Stefanelli

[email protected]

Università del Salento

Spa, 22 June 2017

De�nitions and examples Additive and multiplicative structures Ideals and quotient structures

Notations

We introduce the semi-brace, an algebraic structure that allows us to obtainleft non-degenerate solutions of the Yang-Baxter equation.

If X is a set, a (set-theoretical) solution of the Yang-Baxter equationr : X × X → X × X is a map such that the well-known braid equation

r1r2r1 = r2r1r2

is satis�ed, where r1 = r × idX and r2 = idX ×r .

If a, b ∈ X , we denote r (a, b) = (λa (b) , ρb (a)) where λa, ρb are maps from Xinto itself.

In particular, we say that r is left (right, resp.) non-degenerate if λa (ρa, resp.)is bijective, for every a ∈ X . Moreover, r is non-degenerate if r is both left andright non-degenerate.

P. Stefanelli (UniSalento) The algebraic structure of semi-brace 1 / 15

Skew Braces

In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.

De�nition

Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.

We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).

If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.

Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Skew Braces

De�nition

a ◦ (b + c) = a ◦ b − a+ a ◦ c,

Semi-bracesWe introduced a generalization of skew braces.

De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)

Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if

a ◦ (b + c) = a ◦ b + a ◦(a− + c

holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.

Note that, if B is a skew brace, then it is a semi-brace. In fact

I (B,+) is a group and, in particular, a left cancellative semigroup;

I if a, b, c ∈ B then

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

a ◦ (b + c) = a ◦ b + a ◦(a− + c

a ◦ b + a ◦(a− + c

Examples of semi-braces

1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,

I (E ,+) is a left cancellative semigroup;

I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c

)= a ◦ c = a ◦ (b + c).

We call this semi-brace the trivial semi-brace.

2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set

a+ b := b ◦ f (a) ,

for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;

I (B,+) is left cancellative;

I if a, b, c ∈ B, then

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I if a, b, c ∈ B, then a+ (b + c) = a+ (c ◦ f (b)) = c ◦ f (b) ◦ f (a) =

c ◦ f (b) ◦ f 2 (a) = c ◦ f (b ◦ f (a)) = (a+ b) + cI (B,+) is left cancellative;

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

I if a, b, c ∈ B, such that a+ b = a+ c, then b ◦ f (a) = c ◦ f (a), i.e., b = c

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

)= a ◦ c = a ◦ (b + c).

a+ b := b ◦ f (a) ,

a ◦ b + a ◦(a− + c

)= a ◦ b + a ◦ c ◦ f

)= a ◦ c ◦ f

)◦ f (a ◦ b)

= a ◦ c ◦ f(a− ◦ a ◦ b

)= a ◦ c ◦ f (b)

= a ◦ (b + c) .

The additive structure - I

Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

and, by left cancellativity, we have that a = 0+ a.

Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .

If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

Hence, the additive structure (B,+) is a right group.

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a

), then

x + t = x + x ◦(x− + x− ◦ y

)= x ◦

(0+ x− ◦ y

)= x ◦ x− ◦ y = y .

The additive structure - II

Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E

In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and

B = G + E .

In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.

B = G + E .

Examples

I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.

I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

Examples

x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0

⇐⇒ x ∈ ker f .

x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .

The multiplicative group

Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.

I Clearly, we have that G ∩ E = {0}.

I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

b = g︸︷︷︸∈G

◦ g− ◦ (g + e)︸︷︷︸∈E

In fact,

g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,

i.e., g− ◦ (g + e) ∈ E .

Ideals of a skew brace

Guarnieri and Vendramin give the following de�nition of ideal for a skew brace.

De�nition

Let B be a skew brace. A subset I of B is said an ideal if

I I is a normal subgroup of (B, ◦);I I is a normal subgroup of (B,+);

I λa (I ) ⊆ I , for every a ∈ B, where λa (b) := −a+ a ◦b, for all a, b ∈ B.

In particular, if B is a brace, the second condition follows by the �rst and thirdones.

De�nition

Ideals of a semi-brace - I

De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)

Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if

I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);

I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;

I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .

Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

)− ◦ afor every b ∈ B.

As we expect, B and {0} are ideals of B that we call the trivial ideals of B.

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

λa (b) := a ◦(a− + b

)and ρa (b) :=

(b− + a

Comparison between the two de�nitions of ideal - I

If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,

I I is a normal subgroup of (B, ◦); 3

I I ∩ G is a normal subgroup of (G ,+); 3

I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3

I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; λg (0) = −g + g ◦ 0 = 0 ∈ I

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

ρb (n) =(n− + b

)− ◦ b =(b− ◦

(n− + b

))−=(b− ◦

(b − b + n− + b

))−=(b− ◦ b − b− + b− ◦

(−b + n− + b

))−= (λb−(−b + n− + b︸︷︷︸

))− ∈ I .

Comparison between the two de�nitions of ideal - II

Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,

I I is a normal subgroup of (B,+); 3

I λa (I ) ⊆ I , for every a ∈ B. 3

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

λa (x) = a ◦(a− + x

a− + x)− ◦ a−)− =

((a− + x − a− + a−

)◦ a−

)−= (ρa−(a− + x − a−︸︷︷︸

))− ∈ I .

Remark

In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.

Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,

I I ∩ G = {0} is a normal subgroup of (G ,+); 3

I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3

I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3

But if a ∈ B \ I , then

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

i.e., I is not λa - invariant, nevertheless I is an ideal of B.

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Remark

λa (0) = a ◦(a− + 0

)= a ◦ 0 = a /∈ I ,

Ideals of a semi-brace - II

Proposition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)

If B is a semi-brace and I is an ideal of B, then the relation ∼I on B given

∀ x , y ∈ B, x ∼I y ⇐⇒ y− ◦ x ∈ I

is a congruence of B.

Further, if B is a semi-brace and I is an ideal, then the quotient structure B/Iof B with respect to the relation ∼I is a right group with respect to the sum.

Therefore the quotient structure B/I is a semi-brace.

∀ x , y ∈ B, x ∼I y ⇐⇒ y− ◦ x ∈ I

The socle

Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.

De�nition

Let B be a skew brace. Then the ideal de�ned by

S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}

is said the socle of B.

We may generalize this de�nition for semi-braces in the following way:

De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)

If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by

Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .

the socle of the semi-brace B.

If B is a skew brace, then S (B) = Soc (B).

The socle

De�nition

The socle

De�nition

The socle

De�nition

The socle

De�nition

The socle

De�nition

The socle

De�nition

The socle

De�nition

The socle

De�nition

The socle

De�nition

The socle

De�nition

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