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Semilattices of Stratified Semigroups

James Renshaw, William Warhurst

Abstract

In 1995 Grillet introduced the concept of a stratified semigroup as a kind of generalisation of finite nilsemigroups. We extend these ideas here by allowing a more general Base and describe them in terms of extensions of semigroups by stratified semigroups. We consider semillatices of certain types of group-bound semigroups and also semillatices of Clifford semigroups and show how to describe them as semilattices of these stratified extensions and provide a number of interesting examples.

Semilattices of Stratified Semigroups

Abstract

In 1995 Grillet introduced the concept of a stratified semigroup as a kind of generalisation of finite nilsemigroups. We extend these ideas here by allowing a more general Base and describe them in terms of extensions of semigroups by stratified semigroups. We consider semillatices of certain types of group-bound semigroups and also semillatices of Clifford semigroups and show how to describe them as semilattices of these stratified extensions and provide a number of interesting examples.
Paper Structure (4 sections, 33 theorems, 7 equations)

This paper contains 4 sections, 33 theorems, 7 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Stratified Extensions
  3. Semilattices of group bound semigroups
  4. Strict extensions of Clifford Semigroups

Key Result

Proposition 1.1

Every extension of $S$ is strict if and only if $S$ has an identity.

Theorems & Definitions (33)

  • Proposition 1.1: grillet-petrich-68
  • Theorem 1.2: grillet-petrich-68
  • Lemma 1.3
  • Lemma 1.4
  • Theorem 1.5: shevrin-95
  • Lemma 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 23 more