  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.
