Higher braid groups and regular semigroups from polyadic-binary correspondence

TL;DR

This paper introduces a novel algebraic framework connecting higher arity braid groups and regular semigroups through polyadic-binary correspondence, extending classical structures to higher dimensions and arities.

Contribution

It generalizes the construction of braid groups and regular semigroups to higher arity, introducing new algebraic structures linked with generalized polyadic braid equations and R-matrices.

Findings

Higher arity versions of braid groups and semigroups are constructed.

Connections between these higher structures and generalized polyadic braid equations are established.

Higher degree Coxeter and symmetry groups are defined and analyzed.

Abstract

In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and $R$-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, $n$-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.