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A pretorsion theory for right groups

Alberto Facchini, Carmelo Antonio Finocchiaro

Abstract

Let $S$ be a right group. Then there exist two congruences $\sim$ and $\equiv$ on $S$ such that $S$ is the product of its quotient semigroups $S/{\sim}$ and $S/{\equiv}$, where $S/{\sim}$ is a group and $S/{\equiv}$ is a right zero semigroup. If $E$ is the set of all idempotents of $S$ and we fix an element $e_0\in E$, then the pointed right group $(S,e_0)$ is the coproduct of its pointed subsemigroups $(Se_0,e_0)$ and $(E,e_0)$ in the category of pointed right groups. In general, there is a pretorsion theory in the category of right groups in which the torsion objects are right zero semigroups and the torsion-free objects are groups.

A pretorsion theory for right groups

Abstract

Let be a right group. Then there exist two congruences and on such that is the product of its quotient semigroups and , where is a group and is a right zero semigroup. If is the set of all idempotents of and we fix an element , then the pointed right group is the coproduct of its pointed subsemigroups and in the category of pointed right groups. In general, there is a pretorsion theory in the category of right groups in which the torsion objects are right zero semigroups and the torsion-free objects are groups.
Paper Structure (8 sections, 16 theorems, 21 equations)

This paper contains 8 sections, 16 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions and terminology
  3. Direct-product decompositions of semigroups
  4. Right groups
  5. Categories of right groups
  6. Homomorphisms of right groups
  7. Right inverses $\varphi\colon S/{\sim}\to S$ to the canonical projection $\pi\colon S\to S/{\sim}$.
  8. Pretorsion theory for right groups

Key Result

Lemma 2.2

Let $S$ be a semigroup and $\sim,\ \equiv$ be two congruences on $S$. If $\sim$ and $\equiv$ are permutable (that is, $\sim\circ\equiv$ and $\equiv\circ\sim$ coincide), then the least upper bound $\sim\vee\equiv$ of $\sim$ and $\equiv$ in the lattice of all congruences of $S$ coincides with ${\sim}\

Theorems & Definitions (35)

  • Remark 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Example 2.8
  • Proposition 3.1
  • ...and 25 more