A note on Garside monoids and \({\mathcal{M}} \)-braces. (English) Zbl 1516.20139
Summary: We define an algebraic structure similar to that of a semiring, but without some of the requirements. As it is somehow also similar to the structure of left brace, we call it an \({\mathcal{M}} \)-brace. We present a connection between Garside monoids and more generally lcm-monoids with this algebraic structure. An lcm-monoid \(M\) is a left-cancellative monoid such that 1 is the unique invertible element in \(M\), and every pair of elements in \(M\) admit an lcm with respect to left-divisibility. The class of lcm-monoids contains the Gaussian, quasi-Garside and Garside monoids.
MSC:
| 20M75 | Generalizations of semigroups |
| 20M05 | Free semigroups, generators and relations, word problems |
| 16Y60 | Semirings |
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