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Partial actions and proper extensions of two-sided restriction semigroups

Mikhailo Dokuchaev, Mykola Khrypchenko, Ganna Kudryavtseva

Abstract

We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We introduce and study several classes of partial actions of two-sided restriction semigroups that generalize partial actions of monoids and of inverse semigroups. We establish an adjunction between the category ${\mathcal{P}}(S)$ of proper extensions of a restriction semigroup (or, in particular, an inverse semigroup) $S$ and a category ${\mathcal{A}}(S)$ of partial actions of $S$ subject to certain conditions going back to the work of O'Carroll. In the category ${\mathcal{A}}(S)$, we specify two isomorphic subcategories, one being reflective and the other one coreflective, each of which is equivalent to the category ${\mathcal{P}}(S)$.

Partial actions and proper extensions of two-sided restriction semigroups

Abstract

We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We introduce and study several classes of partial actions of two-sided restriction semigroups that generalize partial actions of monoids and of inverse semigroups. We establish an adjunction between the category of proper extensions of a restriction semigroup (or, in particular, an inverse semigroup) and a category of partial actions of subject to certain conditions going back to the work of O'Carroll. In the category , we specify two isomorphic subcategories, one being reflective and the other one coreflective, each of which is equivalent to the category .

Paper Structure

This paper contains 21 sections, 38 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Restriction semigroups
  4. Premorphisms and partial actions of restriction semigroups
  5. Premorphisms and their corresponding partial actions
  6. Strong and order-preserving premorphisms.
  7. Locally strong premorphisms
  8. Locally multiplicative premorphisms
  9. Structure of proper extensions of restriction semigroups
  10. Proper morphisms and proper extensions of restriction semigroups.
  11. The partial action product of $Y$ by $S$
  12. The two underlying premorphisms of a proper extension
  13. Decomposition of a proper extension into a partial action product
  14. Classes of proper extensions
  15. Order-proper extensions
  16. ...and 6 more sections

Key Result

Lemma 2.2

Let $S$ be a restriction semigroup and $s,t\in S$. Then:

Theorems & Definitions (81)

  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 71 more