Green's Relations on the Monoid of Regular Hypersubstitutions

Abstract

The theory of hyperidentities and hypervarieties is based on the fact that the set Hyp (τ) of all hypersubstitutions of a fixed type τ forms a monoid, with a Galois connection between submonoids of this monoid and complete sublattices of the lattice of all varieties of type τ. For this reason, there is interest in studying the semigroup or monoid properties of Hyp (τ) and its submonoids. One approach is to study the five relations known as Green's relations definable on any semigroup. In this paper, we consider the type τ = (n) with one n-ary operation symbol for n≥ 1, and the submonoid Reg (n) of regular hypersubstitutions. We characterize Green's relations on every subsemigroup of Reg (n); then using this characterization we describe which subsemigroups of Reg (n) are 𝒢-subsemigroups of Reg (n) defined by Levi.