Colazzo, I., Koch, A., Martin-Lyons, I., & Truman, P. J. (2024). Skew bracoids and the Yang-Baxter equation. arXiv.
@unpublished{240415929,
title = {Skew bracoids and the {Y}ang-{B}axter equation},
author = {Colazzo, Ilaria and Koch, Alan and Martin-Lyons, Isabel and Truman, Paul J.},
year = {2024},
archiveprefix = {arXiv},
publisher = {arXiv},
copyright = {arXiv.org perpetual, non-exclusive license},
keywords = {arXiv},
doi = {2404.15929}
}
Skew braces provide an algebraic framework for studying bijective nondegenerate solutions of the set-theoretic Yang-Baxter equation. We show that left skew bracoids, recently introduced by two of the authors, can be used to obtain right nondegenerate solutions, and give a variety of examples arising from various methods of constructing left skew bracoids. We compare the solutions we obtain with the left nondegenerate solutions obtained via (left cancellative) left semibraces, and establish a correspondence between left semibraces and a class of left skew bracoids.
Colazzo, I., Okniński, J., & Van Antwerpen, A. (2024). Bijective solutions to the Pentagon Equation. arXiv.
@unpublished{240520406,
archiveprefix = {arXiv},
doi = {2405.20406},
author = {Colazzo, Ilaria and Okniński, Jan and Van Antwerpen, Arne},
title = {Bijective solutions to the Pentagon Equation},
year = {2024},
keywords = {arXiv},
publisher = {arXiv},
copyright = {arXiv.org perpetual, non-exclusive license}
}
A complete classification of all finite bijective set-theoretic solutions (S,s) to the Pentagon Equation is obtained. First, it is shown that every such a solution determines a semigroup structure on the set S that is the direct product E\times G of a semigroup of left zeros E and a group G. Next, we prove that this leads to a decomposition of the set S as a Cartesian product X\times A\times G, for some sets X,A and to a discovery of a hidden group structure on A. Then an unexpected structure of a matched product of groups A,G is found such that the solution (S,s) can be explicitly described as a lift of a solution determined on the set A\times G by this matched product of groups. Conversely, every matched product of groups leads to a family of solutions arising in this way. Moreover, a simple criterion for the isomorphism of two solutions is obtained. The results provide a far reaching extension of the results of Colazzo, Jespers and Kubat, dealing with the special case of the so called involutive solutions. Connections to the solutions to the Yang-Baxter equation and to the theory of skew braces are derived.
Colazzo, I., Jespers, E., Kubat, Ł., & Van Antwerpen, A. (2023). Structure algebras of finite set-theoretic solutions of the Yang–Baxter equation. arXiv.
@unpublished{230506023,
archiveprefix = {arXiv},
doi = {2305.06023},
author = {Colazzo, Ilaria and Jespers, Eric and Kubat, Łukasz and Van Antwerpen, Arne},
title = {Structure algebras of finite set-theoretic solutions of the {Y}ang--{B}axter equation},
year = {2023},
keywords = {arXiv},
publisher = {arXiv},
copyright = {arXiv.org perpetual, non-exclusive license}
}
Algebras related to finite bijective or idempotent left non-degenerate solutions (X,r) of the Yang–Baxter equation have been intensively studied. These are the monoid algebras K[M(X,r)] and K[A(X,r)], over a field K, of its structure monoid M(X,r) and left derived structure monoid A(X,r), which have quadratic defining relations.
In this paper we deal with arbitrary finite left non-degenerate solutions (X,r). Via divisibility by generators, i.e., the elements of X, we construct an ideal chain in M(X,r) that has very strong algebraic structural properties on its Rees factors. This allows to obtain characterizations of when the algebras K[M(X,r)] and K[A(X,r)] are left or right Noetherian. Intricate relationships between ring-theoretical and homological properties of these algebras and properties of the solution (X,r) are proven, which extends known results on bijective non-degenerate solutions. Furthermore, we describe the cancellative congruences of A(X,r) and M(X,r) as well as the prime spectrum of K[A(X,r)]. This then leads to an explicit formula for the Gelfand–Kirillov dimension of K[M(X,r)] in terms of the number of orbits in X under actions of certain finite monoids derived from (X,r). It is also shown that the former coincides with the classical Krull dimension of K[M(X,r)] in case the algebra K[M(X,r)] is left or right Noetherian. Finally, we obtain the first structural results for a class of finite degenerate solutions (X,r) of the form r(x,y)=(\lambda_x(y),ρ(y)) by showing that structure algebras of such solution are always right Noetherian.
Colazzo, I., Jespers, E., Kubat, Ł., & Van Antwerpen, A. (2023). Simple solutions of the Yang-Baxter equation. arXiv.
@unpublished{231209687,
archiveprefix = {arXiv},
doi = {2312.09687},
author = {Colazzo, Ilaria and Jespers, Eric and Kubat, Łukasz and Van Antwerpen, Arne},
title = {Simple solutions of the {Y}ang-{B}axter equation},
year = {2023},
keywords = {arXiv},
publisher = {arXiv},
copyright = {arXiv.org perpetual, non-exclusive license}
}
We present a characterization of simple finite non-degenerate bijective set-theoretic solutions of the Yang-Baxter equation in terms of the algebraic structure of the associated permutation skew left braces.
In particular, we prove that they need to have a unique minimal non-zero ideal and modulo this ideal one obtains a trivial skew left brace of cyclic type.
Accepted for publication
Colazzo, I., Ferrara, M., & Trombetti, M. (2022). On derived-indecomposable solutions of the Yang–Baxter equation. In accepted for publication in Publicacions Matemàtiques.
@unpublished{221008598,
doi = {2210.08598},
author = {Colazzo, Ilaria and Ferrara, Maria and Trombetti, Marco},
keywords = {accepted},
title = {On derived-indecomposable solutions of the Yang--Baxter equation},
journal = {accepted for publication in Publicacions Matemàtiques},
year = {2022},
copyright = {arXiv.org perpetual, non-exclusive license}
}
If (X,r) is a finite non-degenerate set-theoretic solution of the Yang–Baxter equation, the additive group of the structure skew brace G(X,r) is an FC-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an FC-group itself. If one additionally assumes that the derived solution of (X,r) is indecomposable, then for every element b of G(X,r) there are finitely many elements of the form b∗c and c∗b, with c∈G(X,r). This naturally leads to the study of a brace-theoretic analogue of the class of FC-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.
Published papers
Colazzo, I., Jespers, E., Kubat, Ł., Van Antwerpen, A., & Verwimp, C. (2023). Finite Idempotent Set-Theoretic Solutions of the Yang–Baxter Equation. International Mathematics Research Notices, 2024(7), 5458–5489. https://doi.org/10.1093/imrn/rnad183
@article{221207361,
author = {Colazzo, Ilaria and Jespers, Eric and Kubat, Łukasz and Van Antwerpen, Arne and Verwimp, Charlotte},
title = {{Finite Idempotent Set-Theoretic Solutions of the Yang–Baxter Equation}},
journal = {International Mathematics Research Notices},
volume = {2024},
number = {7},
pages = {5458-5489},
year = {2023},
month = aug,
issn = {1073-7928},
doi = {10.1093/imrn/rnad183},
url = {https://doi.org/10.1093/imrn/rnad183},
eprint = {https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnad183/51112327/rnad183.pdf},
arxiv = {2212.07361}
}
It is proven that finite idempotent left non-degenerate set-theoretic solutions (X,r) of the Yang–Baxter equation on a set X are determined by a left simple semigroup structure on X (in particular, a finite union of isomorphic copies of a group) and some maps q and \varphi_x on X, for x∈X. This structure turns out to be a group precisely when the associated Yang–Baxter monoid M(X,r) is cancellative and all the maps \varphi_x are equal to an automorphism of this group. Equivalently, the Yang–Baxter algebra K[M(X,r)] is right Noetherian, or in characteristic zero it has to be semiprime. The Yang–Baxter algebra is always a left Noetherian representable algebra of Gelfand–Kirillov dimension one. To prove these results, it is shown that the Yang–Baxter semigroup S(X,r) has a decomposition in finitely many cancellative semigroups S_u indexed by the diagonal, each S_u has a group of quotients G_u that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that X equals the diagonal is fully described by a single permutation on X.
Colazzo, I., Jespers, E., Van Antwerpen, A., & Verwimp, C. (2022). Left non-degenerate set-theoretic solutions of the
Yang-Baxter equation and semitrusses. J. Algebra, 610, 409–462. https://doi.org/10.1016/j.jalgebra.2022.07.019
@article{MR4466104,
author = {Colazzo, I. and Jespers, E. and Van Antwerpen, A. and Verwimp, C.},
title = {Left non-degenerate set-theoretic solutions of the
{Y}ang-{B}axter equation and semitrusses},
journal = {J. Algebra},
fjournal = {Journal of Algebra},
volume = {610},
year = {2022},
pages = {409--462},
issn = {0021-8693},
mrclass = {16T25 (16S36 16W22 16Y99 20M25)},
mrnumber = {4466104},
doi = {10.1016/j.jalgebra.2022.07.019},
url = {https://doi.org/10.1016/j.jalgebra.2022.07.019},
arxiv = {2109.04978}
}
To determine and analyze arbitrary left non-degenerate set-theoretic solutions of the Yang-Baxter equation (not necessarily bijective), we introduce an associative algebraic structure, called a YB-semitruss, that forms a subclass of the category of semitrusses as introduced by Brzeziński. Fundamental examples of YB-semitrusses are structure monoids of left non-degenerate set-theoretic solutions and (skew) left braces. Gateva-Ivanova and Van den Bergh introduced structure monoids and showed their importance (as well as that of the structure algebra) for studying involutive non-degenerate solutions. Skew left braces were introduced by Guarnieri, Vendramin and Rump to deal with bijective non-degenerate solutions. Hence, YB-semitrusses also yield a unified treatment of these different algebraic structures. The algebraic structure of YB-semitrusses is investigated, and as a consequence, it is proven, for example, that any finite left non-degenerate set-theoretic solution of the Yang-Baxter equation is right non-degenerate if and only if it is bijective. Furthermore, it is shown that some finite left non-degenerate solutions can be reduced to non-degenerate solutions of smaller size. The structure algebra of a finitely generated YB-semitruss is an algebra defined by homogeneous quadratic relations. We prove that it often is a left Noetherian algebra of finite Gelfand-Kirillov dimension that satisfies a polynomial identity, but in general, it is not right Noetherian.
Catino, F., Colazzo, I., & Stefanelli, P. (2021). Set-theoretic solutions to the Yang-Baxter equation and
generalized semi-braces. Forum Math., 33(3), 757–772. https://doi.org/10.1515/forum-2020-0082
@article{MR4250483,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {Set-theoretic solutions to the {Y}ang-{B}axter equation and
generalized semi-braces},
journal = {Forum Math.},
fjournal = {Forum Mathematicum},
volume = {33},
year = {2021},
number = {3},
pages = {757--772},
issn = {0933-7741},
mrclass = {16T25 (16N20 16Y99 81R50)},
mrnumber = {4250483},
mrreviewer = {Run-Qiang Jian},
doi = {10.1515/forum-2020-0082},
url = {https://doi.org/10.1515/forum-2020-0082},
arxiv = {2004.01606}
}
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 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.
Colazzo, I., & Van Antwerpen, A. (2021). The algebraic structure of left semi-trusses. J. Pure Appl. Algebra, 225(2), Paper No. 106467, 15. https://doi.org/10.1016/j.jpaa.2020.106467
@article{MR4125548,
author = {Colazzo, Ilaria and Van Antwerpen, Arne},
title = {The algebraic structure of left semi-trusses},
journal = {J. Pure Appl. Algebra},
fjournal = {Journal of Pure and Applied Algebra},
volume = {225},
year = {2021},
number = {2},
pages = {Paper No. 106467, 15},
issn = {0022-4049},
mrclass = {16Y99 (16T25 20M17)},
mrnumber = {4125548},
mrreviewer = {Abdelkader Ben Hassine},
doi = {10.1016/j.jpaa.2020.106467},
url = {https://doi.org/10.1016/j.jpaa.2020.106467},
arxiv = {1908.11744}
}
The distributive laws of ring theory are fundamental equalities in algebra. However, recently in the study of the Yang-Baxter equation, many algebraic structures with alternative "distributive" laws were defined. In an effort to study these "left distributive" laws and the interaction they entail on the algebraic structures, Brzeziński introduced skew left trusses and left semi-trusses. In particular the class of left semi-trusses is very wide, since it contains all rings, associative algebras and distributive lattices. In this paper, we investigate the subclass of left semi-trusses that behave like the algebraic structures that came up in the study of the Yang-Baxter equation. We study the interaction of the operations and what this interaction entails on their respective semigroups. In particular, we prove that in the finite case the additive structure is a completely regular semigroup. Secondly, we apply our results on a particular instance of a left semi-truss called an almost left semi-brace, introduced by Miccoli to study its algebraic structure. In particular, we show that one can associate a left semi-brace to any almost left semi-brace. Furthermore, we show that the set-theoretic solutions of the Yang-Baxter equation originating from almost left semi-braces arise from this correspondence.
Catino, F., Colazzo, I., & Stefanelli, P. (2020). The matched product of the solutions to the Yang-Baxter
equation of finite order. Mediterr. J. Math., 17(2), Paper No. 58, 22. https://doi.org/10.1007/s00009-020-1483-y
@article{MR4067191,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {The matched product of the solutions to the {Y}ang-{B}axter
equation of finite order},
journal = {Mediterr. J. Math.},
fjournal = {Mediterranean Journal of Mathematics},
volume = {17},
year = {2020},
number = {2},
pages = {Paper No. 58, 22},
issn = {1660-5446},
mrclass = {16T25 (16N20 16Y99 81R50)},
mrnumber = {4067191},
mrreviewer = {\L ukasz Kubat},
doi = {10.1007/s00009-020-1483-y},
url = {https://doi.org/10.1007/s00009-020-1483-y},
arxiv = {1904.07557}
}
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 r_S and r_T is of finite order if and only if r_S and r_T are. Furthermore, we show that with sufficient information on r_S and r_T 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
Colazzo, I., Jespers, E., & Kubat, Ł. (2020). Set-theoretic solutions of the pentagon equation. Comm. Math. Phys., 380(2), 1003–1024. https://doi.org/10.1007/s00220-020-03862-6
@article{MR4170296,
author = {Colazzo, Ilaria and Jespers, Eric and Kubat, \L ukasz},
title = {Set-theoretic solutions of the pentagon equation},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {380},
year = {2020},
number = {2},
pages = {1003--1024},
issn = {0010-3616},
mrclass = {20K10 (16T25)},
mrnumber = {4170296},
mrreviewer = {Jo\~{a}o Matheus Jury Giraldi},
doi = {10.1007/s00220-020-03862-6},
url = {https://doi.org/10.1007/s00220-020-03862-6},
arxiv = {2004.04028}
}
A set-theoretic solution of the Pentagon Equation on a non-empty set S is a map s:S^2\to S^2 such that s_23s_13s_12=s_12s_23, where s_12=s\times id, s_23=id\times s and s_13=(τ\times id)(id\times s)(τ\times id) are mappings from S^3 to itself and τ:S^2\to S^2 is the flip map, i.e., τ(x,y)=(y,x). We give a description of all involutive solutions, i.e., s^2=id. It is shown that such solutions are determined by a factorization of S as direct product X\times A\times G and a map σ:A\to Sym(X), where X is a non-empty set and A,G are elementary abelian 2-groups. Isomorphic solutions are determined by the cardinalities of A, G and X, i.e., the map σis irrelevant. In particular, if S is finite of cardinality 2^n(2m+1) for some n,m≥0 then, on S, there are precisely \binomn+22 non-isomorphic solutions of the Pentagon Equation.
Catino, F., Colazzo, I., & Stefanelli, P. (2020). The matched product of set-theoretical solutions of the
Yang-Baxter equation. J. Pure Appl. Algebra, 224(3), 1173–1194. https://doi.org/10.1016/j.jpaa.2019.07.012
@article{MR4009573,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {The matched product of set-theoretical solutions of the
{Y}ang-{B}axter equation},
journal = {J. Pure Appl. Algebra},
fjournal = {Journal of Pure and Applied Algebra},
volume = {224},
year = {2020},
number = {3},
pages = {1173--1194},
issn = {0022-4049},
mrclass = {16T25 (16N20 16Y99 81R50)},
mrnumber = {4009573},
mrreviewer = {Jo\~{a}o Matheus Jury Giraldi},
doi = {10.1016/j.jpaa.2019.07.012},
url = {https://doi.org/10.1016/j.jpaa.2019.07.012}
}
In this work, we develop a novel construction technique for set-theoretical solutions of the Yang-Baxter equation. Our technique, named the matched product, is an innovative tool to construct new classes of involutive solutions as the matched product of two involutive solutions is still involutive, and vice versa. This method produces new examples of idempotent solutions as the matched product of other idempotent ones. We translate the construction in the context of semi-braces, which are algebraic structures tightly linked with solutions that generalize the braces introduced by Rump. In addition, we show that the solution associated to the matched product of two semi-braces is indeed the matched product of the solutions associated to those two semi-braces.
Catino, F., Colazzo, I., & Stefanelli, P. (2019). Skew left braces with non-trivial annihilator. J. Algebra Appl., 18(2), 1950033, 23. https://doi.org/10.1142/S0219498819500336
@article{MR3917122,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {Skew left braces with non-trivial annihilator},
journal = {J. Algebra Appl.},
fjournal = {Journal of Algebra and its Applications},
volume = {18},
year = {2019},
number = {2},
pages = {1950033, 23},
issn = {0219-4988},
mrclass = {16T25 (16N20 16Y99 20B15 20G99 81R50)},
mrnumber = {3917122},
mrreviewer = {Loic Foissy},
doi = {10.1142/S0219498819500336},
url = {https://doi.org/10.1142/S0219498819500336}
}
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. Algebra222 (2018) 1670–1691].
Catino, F., Colazzo, I., & Stefanelli, P. (2017). Semi-braces and the Yang-Baxter equation. J. Algebra, 483, 163–187. https://doi.org/10.1016/j.jalgebra.2017.03.035
@article{MR3649817,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {Semi-braces and the {Y}ang-{B}axter equation},
journal = {J. Algebra},
fjournal = {Journal of Algebra},
volume = {483},
year = {2017},
pages = {163--187},
issn = {0021-8693},
mrclass = {16T25 (16N20 16Y99 81R50)},
mrnumber = {3649817},
mrreviewer = {Leandro Vendramin},
doi = {10.1016/j.jalgebra.2017.03.035},
url = {https://doi.org/10.1016/j.jalgebra.2017.03.035}
}
In this paper we obtain new solutions of the Yang–Baxter equation that are left non-degenerate through left semi-braces, a generalization of braces introduced by Rump. In order to provide new solutions we introduce the asymmetric product of left semi-braces, a generalization of the semidirect product of braces, that allows us to produce several examples of left semi-braces.
Catino, F., Colazzo, I., & Stefanelli, P. (2016). Regular subgroups of the affine group and asymmetric product
of radical braces. J. Algebra, 455, 164–182. https://doi.org/10.1016/j.jalgebra.2016.01.038
@article{MR3478858,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {Regular subgroups of the affine group and asymmetric product
of radical braces},
journal = {J. Algebra},
fjournal = {Journal of Algebra},
volume = {455},
year = {2016},
pages = {164--182},
issn = {0021-8693},
mrclass = {16Y99 (20B15)},
mrnumber = {3478858},
mrreviewer = {Leandro Vendramin},
doi = {10.1016/j.jalgebra.2016.01.038},
url = {https://doi.org/10.1016/j.jalgebra.2016.01.038}
}
In this paper we introduce the asymmetric product of radical braces, a construction which extends the semidirect product of radical braces. This new construction allows to obtain rather systematic constructions of regular subgroups of the affine group and, in particular, our approach allows to put in a more general context the regular subgroups constructed by Hegedűs (2000)
Catino, F., Colazzo, I., & Stefanelli, P. (2015). On regular subgroups of the affine group. Bull. Aust. Math. Soc., 91(1), 76–85. https://doi.org/10.1017/S000497271400077X
@article{MR3294261,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {On regular subgroups of the affine group},
journal = {Bull. Aust. Math. Soc.},
fjournal = {Bulletin of the Australian Mathematical Society},
volume = {91},
year = {2015},
number = {1},
pages = {76--85},
issn = {0004-9727},
mrclass = {20N99 (16Y30)},
mrnumber = {3294261},
mrreviewer = {Elena Zizioli},
doi = {10.1017/S000497271400077X},
url = {https://doi.org/10.1017/S000497271400077X}
}
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. J. Group Theory, 1 (2012), 17–23].
Other publications
Brzeziński, T., Colazzo, I., Doikou, A., & Vendramin, L. (2023). Mini-workshop: Skew Braces and the Yang-Baxter Equation. In Oberwolfach Reports (No.1; Vol. 20, Number 1, pp. 536–567).
@report{oberwolfach,
title = {Mini-workshop: Skew Braces and the Yang-Baxter Equation},
author = {Brzeziński, Tomasz and Colazzo, Ilaria and Doikou, Anastasia and Vendramin, Leandro},
journal = {Oberwolfach Reports},
volume = {20},
number = {1},
pages = {536--567},
year = {2023},
doi = {10.4171/OWR/2023/9}
}
The workshop was focused on the interplay between set-theoretic solutions to the Yang–Baxter equation and several algebraic structures used to construct and understand new solutions. In this vein, the YBE and properties of these algebraic structures are used as a source of inspiration to study other mathematical problems not directly related to the YBE.
Colazzo, I. (2022). Braces: between regular subgroups and solutions of the Yang-Baxter equation. In Algebra for cryptography. With a preface by Massimiliano Sala (pp. 35–38). Rome: Aracne Editrice.
@report{extended_abstract,
author = {Colazzo, Ilaria},
title = {Braces: between regular subgroups and solutions of the {Yang}-{Baxter} equation},
booktitle = {Algebra for cryptography. With a preface by Massimiliano Sala},
isbn = {979-12-5994-328-6; 979-12-5994-622-5},
pages = {35--38},
year = {2022},
publisher = {Rome: Aracne Editrice},
language = {English},
keywords = {16T25},
zbmath = {7623857}
}
Colazzo, I. (2017). Left semi-braces and the Yang-Baxter equation. Università del Salento; Ph.D. Thesis.
@thesis{Colazzo_thesis,
author = {Colazzo, I.},
title = {Left semi-braces and the Yang-Baxter equation},
year = {2017},
school = {Università del Salento},
note = {Ph.D. Thesis},
file = {thesis-colazzo.pdf}
}
This thesis focuses on the new algebraic structure: semi-brace. We study basic properties of this structure and we show that semi-braces are a generalization of braces. Moreover we introduce new constructions of semi-braces, the asymmetric product and the matched product, in order to obtain several examples of semi- braces. Finally, we prove that we may construct left non-degenerate solutions of the Yang-Baxter equation through left semi-braces.