Left semi-braces and solutions of the Yang–Baxter equation

Antwerpen, Arne Van; Jespers, Eric

doi:10.1515/forum-2018-0059

Cited by 25 publications

(36 citation statements)

References 25 publications

(53 reference statements)

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“…(The case when S is costrongly distributive is handled in a dual fashion.) We claim that S satisfies (14). We obtain (x ∧ y) ∨ ((x ∨ y) ∧ z) = (x ∧ y) ∨ (x ∧ z) ∨ (y ∧ z), and (x∨(y∧z))∧(y∨z) = (x∧(y∨z))∨(y∧z∧(y∨z)).…”

Section: The Left Hand Side Of This Equation Is Equal Tomentioning

confidence: 88%

“…Then S satisfies the identities (13) and (15). If in addition to being left handed, S is either strongly distributive or co-strongly distributive, then it also satisfies (14) and is thus a strong distributive solution.…”

Section: The Left Hand Side Of This Equation Is Equal Tomentioning

confidence: 97%

“…In general, we cannot omit either strong distributivity or co-strong distributivity from the assumptions of It is easy to check that 3 R,0 is a right handed skew lattice with two comparable D-classes {1, 2} > {0}, and it is strongly distributive (but not co-strongly distributive) by a result of Leech [21]. We claim that 3 R,0 is not a strong distributive solution, more specifically, it does not satisfy the identity (14). Take x = 0, y = 1 and z = 2.…”

Section: The Left Hand Side Of This Equation Is Equal Tomentioning

confidence: 97%

“…Later braces were generalized by Guarnieri and Vendramin [12] to study non-degenerate solutions that are not necessarily involutive. Catino, Colazzo, and Stefanelli [2], and Jespers and Van Antwerpen [14] introduced (left cancellative) left semi-braces to deal with solutions that are not necessarily non-degenerate or solutions that are idempotent or cubic. Some of these solutions are degenerate.…”

Section: Introductionmentioning

confidence: 99%

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Skew lattices and set-theoretic solutions of the Yang-Baxter equation

Cvetko-Vah

Verwimp

2020

Journal of Algebra

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In this paper we discuss and characterize several set-theoretic solutions of the Yang-Baxter equation obtained using skew lattices, an algebraic structure that has not yet been related to the Yang-Baxter equation. Such solutions are degenerate in general, and thus different from solutions obtained from braces and other algebraic structures.Our main result concerns a description of a set-theoretic solution of the Yang-Baxter equation, obtained from an arbitrary skew lattice. We also provide a construction of a cancellative and distributive skew lattice on a given family of pairwise disjoint sets.

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Section: The Left Hand Side Of This Equation Is Equal Tomentioning

confidence: 88%

Section: The Left Hand Side Of This Equation Is Equal Tomentioning

confidence: 97%

Section: The Left Hand Side Of This Equation Is Equal Tomentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Skew lattices and set-theoretic solutions of the Yang-Baxter equation

Cvetko-Vah

Verwimp

2020

Journal of Algebra

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“…Also non-bijective set-theoretic solutions are of importance and receive attention. For example Lebed in [31] shows that idempotent solutions provide a unified treatment of factorizable monoids, free and free commutative monoids, distributive lattices and Young tableaux and Catino, Colazzo, and Stefanelli [10], and Jespers and Van Antwerpen [27] introduced the algebraic structure called "(left) semi-brace" to deal with solutions that are not necessarily non-degenerate or that are idempotent.…”

Section: Introductionmentioning

confidence: 99%

The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation

Jespers

Kubat

Antwerpen

2019

Trans. Amer. Math. Soc.

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For a finite involutive non-degenerate solution (X, r) of the Yang-Baxter equation it is known that the structure monoid M (X, r) is a monoid of I-type, and the structure algebra K[M (X, r)] over a field K shares many properties with commutative polynomial algebras, in particular, it is a Noetherian PI-domain that has finite Gelfand-Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid M (X, r) and the algebra K[M (X, r)] is much more complicated than in the involutive case, we provide some deep insights.In this general context, using a realization of Lebed and Vendramin of M (X, r) as a regular submonoid in the semidirect product A(X, r) Sym(X), where A(X, r) is the structure monoid of the rack solution associated to (X, r), we prove that the structure algebra K[M (X, r)] is a module-finite normal extension of a commutative affine subalgebra. In particular, K[M (X, r)] is a Noetherian PI-algebra of finite Gelfand-Kirillov dimension bounded by |X|. We also characterize, in ring-theoretical terms of K[M (X, r)], when (X, r) is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of M (X, r).These results allow us to control the prime spectrum of the algebra K[M (X, r)] and to describe the Jacobson radical and prime radical of K[M (X, r)]. Finally, we give a matrix-type representation of the algebra K[M (X, r)]/P for each prime ideal P of K[M (X, r)]. As a consequence, we show that if K[M (X, r)] is semiprime then there exist finitely many finitely generated abelian-by-finite groups, G 1 , . . . , Gm, each being the group of quotients of a cancellative subsemigroup of M (X, r) such that the algebra, a direct product of matrix algebras. Vendramin [33] studied the structure group G(X, r) for arbitrary finite bijective non-degenerate solutions (i.e., not necessarily involutive). In [33,35,47] they associate, via a bijective 1-cocycle, to the structure group G(X, r) the structure group G(X, r ) of the structure rack (X, r ) of (X, r). As a consequence, it follows that again the groups G(X, r) are abelian-by-finite. Recall that a set X with a self-distributive operation is called a rack if the map y → y x is bijective, for any x ∈ X (cf. [29]). In contrast to the involutive case, the set X is not necessarily canonically embedded into G(X, r), the reason being that M (X, r) need not be cancellative in general (i.e., it is not necessarily embedded in a group). Hence, for an arbitrary solution (X, r) the structure monoid M (X, r) contains more information on the original solution. However, it is in general not true that two set-theoretic solutions (X, r) and (Y, s) are isomorphic if and only if the monoids M (X, r) and M (Y, s) are isomorphic. This does hold if one of both solutions (and thus both) is assumed to be an involutive non-degenerate set-theoretic solution.In this paper we give a structural approach of the study of the structure monoid ...

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The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order

2020

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In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. In particular, we prove that the matched product of two solutions rS and rT is of finite order if and only if rS and rT are. Furthermore, we show that with sufficient information on rS and rT we can precisely establish the order of the matched product. Finally, we prove that if B is a finite semi-brace, then the associated solution r satisfies r n = r, for an integer n closely linked with B.Mathematics Subject Classification (2010). Primary 16T25; Secondary 81R50, 16Y99, 16N20.

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Left semi-braces and solutions of the Yang–Baxter equation

Cited by 25 publications

References 25 publications

Skew lattices and set-theoretic solutions of the Yang-Baxter equation

Skew lattices and set-theoretic solutions of the Yang-Baxter equation

The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation

The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order

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