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The algebraic structure of left semi-trusses. (English) Zbl 1461.16038

The study of a particular class of solutions of the Yang-Baxter equation, namely the non-degenerate involutive set-theoretical solutions, was initiated by P. Etingof et al. [Duke Math. J. 100, No. 2, 169–209 (1999; Zbl 0969.81030)] and T. Gateva-Ivanova and M. Van den Bergh [J. Algebra 206, No. 1, 97–112 (1998; Zbl 0944.20049)]. This led to the definition and study of the algebraic structure called a (right) brace by W. Rump [J. Algebra 307, No. 1, 153–170 (2007; Zbl 1115.16022)]. It was shown that this gives an algebraic way to describe all involutive, non-degenerate set-theoretic solutions. Subsequently many related algebraic structures were defined and studied, for example a skew left brace, skew left truss, left semi-truss and almost left semi-brace. Here the authors determine connections between some of these these structures and investigate their algebraic properties. In particular, they restrict themselves to the subclass of brace-like left semi-trusses, which includes all (skew) left braces, left semi-braces and almost left semi-braces.

MSC:

16T25 Yang-Baxter equations
16Y99 Generalizations
20M17 Regular semigroups
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References:

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