Several locality semigroups, path semigroups and partial semigroups

TL;DR

This paper investigates the structure of locality semigroups, path semigroups, and partial semigroups, establishing their relationships and properties, including the characterization of free objects and subclasses within these algebraic structures.

Contribution

It introduces the concept of free refined locality semigroups, characterizes the path locality semigroup of a quiver, and clarifies the hierarchical relationships among various locality and partial semigroups.

Findings

Path locality semigroup of a quiver is the free refined locality semigroup.

Path locality semigroup is a proper subclass of the intersection of locality and partial semigroups.

Refined locality semigroups form a proper subclass of strong locality semigroups.

Abstract

Locality semigroups were proposed recently as one of the basic locality algebraic structures, which are studied in mathematics and physics. Path semigroups and partial semigroups were also developed by many authors in the literature. In this paper, we study free objects in the category of refined locality semigroups. It turns out that the path locality semigroup of a quiver is the free refined locality semigroup on a locality set. We also explore the relationships among locality semigroups, partial semigroups and path locality semigroups, concluding that the path locality semigroup is a proper subclass of the intersection of locality semigroups and partial semigroups. In particular, the class of refined locality semigroups is a proper subclass of strong locality semigroups. Furthermore, we show that, when a partial semigroup is a refined locality semigroup, one can extend it a strong semigroup with zero.