Abstract
Abstract This paper aims to introduce a construction technique of set-theoretic solutions of the Yang–Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known that braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions.
Highlights
The quantum Yang–Baxter equation appeared in the work of Yang [49] and Baxter [2]
This paper aims to introduce a construction technique of set-theoretic solutions of the Yang–Baxter equation, called strong semilattice of solutions
Brzeziński [4] focused on left semitrusses with a left cancellative semigroup (B, +) and a group (B, ∘), and showed that such a left semi-truss is equivalent with a left cancellative semi-brace, and providing set-theoretic solutions of the Yang–Baxter equation, albeit known ones
Summary
The quantum Yang–Baxter equation appeared in the work of Yang [49] and Baxter [2]. It is one of the basic equations in mathematical physics, and it laid the foundations of the theory of quantum groups [33]. Colazzo, and Stefanelli [7] and Jespers and Van Antwerpen [32] introduced the algebraic structure called left semi-brace to deal with solutions that are not necessarily non-degenerate or that are idempotent. It has been shown that left semi-braces, under some mild assumption, provide set-theoretic solutions of the Yang–Baxter equation. Brzeziński [4] focused on left semitrusses with a left cancellative semigroup (B, +) and a group (B, ∘), and showed that such a left semi-truss is equivalent with a left cancellative semi-brace, and providing set-theoretic solutions of the Yang–Baxter equation, albeit known ones. In [40], Miccoli introduced almost left semi-braces, a particular instance of left semi-trusses, and constructed set-theoretic solutions associated with this algebraic structure. As a corollary of this result, we prove that solutions associated with strong semilattices of left semi-braces are not bijective, so they are clearly different from solutions obtained by left semi-braces
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