Zappa-Sz&amp;#233;p Products of Semigroups

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple (S, +, •) with (S, +) a semigroup and (S, •) an inverse semigroup satisfying the relation a (b + c) = ab + a a −1 + c , for all a, b, c ∈ S, where a −1 is the inverse of a in (S, •). In particular, we give several constructions of inverse semi-braces which allow for obtaining solutions that are different from those until known. Keywords Quantum Yang-Baxter equation • set-theoretical solution • inverse semigroups • brace • semi-brace • skew brace • asymmetric product MSC 2020 16T25 81R50 16Y99 16N20 20M18 r : S × S −→ S × S satisfying the relation (r × id S) (id S ×r) (r × id S) = (id S ×r) (r × id S) (id S ×r) A PREPRINT is said to be a set-theoretical solution of the Yang-Baxter equation, or briefly a solution. The map r is usually written as r(x, y) = (λ x (y), ρ y (x)), with λ x and ρ y maps from S into itself, for all x, y ∈ S. One says that a solution r is left non-degenerate if λ x is bijective, for every x ∈ S, right non-degenerate if ρ y is bijective, for every y ∈ S, and non-degenerate if r is both left and right non-degenerate. If r is neither left nor right non-degenerate, then it is called degenerate. Determining all the solutions is still an open problem and it has drawn the attention of several mathematicians. A large number of works related to this topic has been produced in recent years, actually. The milestones are the papers by Etingof, Schedler and Soloviev [28], Gateva-Ivanova and Van den Bergh [30], Lu, Yan, and Zhu [38], and Soloviev [53], where a greater attention has been posed on non-degenerate bijective solutions. Subsequently, involutive solutions have been profusely investigated by many authors, as mostly illustrated into details in the introduction of the paper by Cedó, Jespers, and Okniński [21]. The most used approach is based on left braces, algebraic structures introduced by Rump [47] that include the Jacobson radical rings. In particular, such structures are involved for obtaining non-degenerate solutions which are also involutive, i.e., r 2 = id. In this way, Rump traced a novel research direction and later fruitful results on these kind of solutions appeared, as one can see in the survey by Cedó [19], along the references therein. To classify involutive solutions, Rump [46] also involved another algebraic structure, that is the left cycle set. Interesting contributions in this framework have been obtained, for example, see [48,

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“…(cf [56,. Theorem 4]) Let S and T be inverse semigroups, σ : T → S S and δ : S → T T maps satisfying (Z1) and (Z2).…”

mentioning

confidence: 99%

Semi-braces and the Yang–Baxter equation

2017

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“…In this case, to obtain an inverse semigroup (S × T , +), we need that the codomain of the map δ is Aut(S), cf. [35,Theorem 4]. Moreover, the additional condition (15) ensures that it also is a Clifford semigroup, a necessary condition by Theorem 8.…”

Section: Definition 17mentioning

confidence: 99%

Set-theoretic solutions of the Yang–Baxter equation associated to weak braces

et al. 2022

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We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang–Baxter equation. Specifically, a weak (left) brace is a non-empty set S endowed with two binary operations $$+$$ + and $$\circ $$ ∘ such that both $$(S,+)$$ ( S , + ) and $$(S, \circ )$$ ( S , ∘ ) are inverse semigroups and $$\begin{aligned} a \circ \left( b+c\right) = \left( a\circ b\right) - a + \left( a\circ c\right) \qquad \text {and} \qquad a\circ a^- = - a + a \end{aligned}$$ a ∘ b + c = a ∘ b - a + a ∘ c and a ∘ a - = - a + a hold, for all $$a,b,c \in S$$ a , b , c ∈ S , where $$-a$$ - a and $$a^-$$ a - are the inverses of a with respect to $$+$$ + and $$\circ $$ ∘ , respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution r associated to an arbitrary weak brace S has a behavior close to bijectivity, namely r is a completely regular element in the full transformation semigroup on $$S\times S$$ S × S . In addition, we provide some methods to construct weak braces.

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“…In this case, to obtain an inverse semigroup (S × T, +), we need that the codomain of the map δ is Aut(S), cf. [35,Theorem 4]. Moreover, the additional condition (15) ensures that it also is a Clifford semigroup, a necessary condition by Theorem 10.…”

Section: Moreover We Obtainmentioning

confidence: 99%

Set-theoretic solutions of the Yang-Baxter equation associated to weak braces

Catino,

Mazzotta,

Miccoli

et al. 2021

Preprint

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We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang-Baxter equation. Specifically, a skew (left) inverse semi-brace is a non-empty set S endowed with two binary operations + and • such that both (S, +) and (S, •) are inverse semigroups and they holdfor all a, b, c ∈ S, where −a and a − are the inverses of a with respect to + and •, respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution r associated to an arbitrary skew inverse semi-brace S has a behavior close to bijectivity, namely r is a completely regular element in the full transformation semigroup on S × S. In addition, we provide some methods to construct skew inverse semi-braces.

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Zappa-Szép Products of Semigroups

Cited by 8 publications

References 13 publications

Semi-braces and the Yang–Baxter equation

Semi-braces and the Yang–Baxter equation

Set-theoretic solutions of the Yang–Baxter equation associated to weak braces

Set-theoretic solutions of the Yang-Baxter equation associated to weak braces

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