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Categorical representation of DRC-semigroups

James East, Matthias Fresacher, P. A. Azeef Muhammed, Timothy Stokes

Abstract

DRC-semigroups model associative systems with domain and range operations, and contain many important classes, such as inverse, restriction, Ehresmann, regular $*$-, and $*$-regular semigroups. In this paper we show that the category of DRC-semigroups is isomorphic to a category of certain biordered categories whose object sets are projection algebras in the sense of Jones. This extends the recent groupoid approach to regular $*$-semigroups of the first and third authors. We also establish the existence of free DRC-semigroups by constructing a left adjoint to the forgetful functor into the category of projection algebras.

Categorical representation of DRC-semigroups

Abstract

DRC-semigroups model associative systems with domain and range operations, and contain many important classes, such as inverse, restriction, Ehresmann, regular -, and -regular semigroups. In this paper we show that the category of DRC-semigroups is isomorphic to a category of certain biordered categories whose object sets are projection algebras in the sense of Jones. This extends the recent groupoid approach to regular -semigroups of the first and third authors. We also establish the existence of free DRC-semigroups by constructing a left adjoint to the forgetful functor into the category of projection algebras.

Paper Structure

This paper contains 30 sections, 64 theorems, 238 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on biordered categories
  3. DRC-semigroups
  4. Definitions and basic properties
  5. From DRC-semigroups to biordered categories
  6. Projection algebras
  7. Definitions and basic properties
  8. The category of projection algebras
  9. The chain category of a projection algebra
  10. Chained projection categories
  11. Weak projection categories
  12. Projection categories
  13. Weak chained projection categories
  14. Chained projection categories
  15. From chained projection categories to DRC-semigroups
  16. ...and 15 more sections

Key Result

Lemma 2.2

Suppose $\mathcal{C}$ is a small category for which the following two conditions hold: Then $(\mathcal{C},\leq_\ell,\leq_r)$ is a biordered category with orders given by Moreover, any biordered category has the above form. ∎

Figures (3)

  • Figure 1: Certain (sub)classes of DRC-semigroups, and the containments among them.
  • Figure 2: The projections and morphisms associated to $b\in\mathcal{C}(q,r)$ and $p,s\in P$; see Definitions \ref{['defn:ef']}, \ref{['defn:lamrho']} and \ref{['defn:CPC']}, and Lemma \ref{['lem:ef']}. Dashed lines indicate $\leq$ relationships. Axiom \ref{['C2']} says that the hexagon at the bottom of the diagram commutes.
  • Figure 3: Formation of the product $a\bullet b$; see Definition \ref{['defn:pr']}. Dashed lines indicate $\leq$ relationships.

Theorems & Definitions (154)

  • Lemma 2.2
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3: cf. Stokes2015
  • proof
  • Lemma 3.4
  • proof
  • Corollary 3.6
  • proof
  • Remark 3.7
  • ...and 144 more