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,
We describe the class of all skew left braces with non-trivial annihilator through ideal extension of a skew left brace. The ideal extension of skew left braces is a generalization to the non-abelian case of the extension of left braces provided by Bachiller in [D. Bachiller, Extensions, matched products, and simple braces, J. Pure Appl. Algebra 222 (2018) 1670–1691].
Catino and Rizzo ['Regular subgroups of the affine group and radical circle algebras', Bull. Aust. Math. Soc. 79 (2009), 103-107] established a link between regular subgroups of the affine group and the radical brace over a field on the underlying vector space. We propose new constructions of radical braces that allow us to obtain systematic constructions of regular subgroups of the affine group. In particular, this approach allows to put in a more general context the regular subgroups constructed in Tamburini Bellani ['Some remarks on regular subgroups of the affine group' Int.
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