Trace- and pseudo-products: restriction-like semigroups with a band of projections

TL;DR

This paper explores the structure of restriction-like semigroups with projections forming a band, generalizing the ESN Theorem to a self-dual context with relaxed conditions, and examines related algebraic structures and special cases.

Contribution

It introduces a self-dual framework for restriction-like semigroups with band projections, extending the ESN Theorem and analyzing one-sided reducts and groupoid cases.

Findings

Conditions for pseudo-products and trace products are established.

The band condition on projections is shown to be natural and significant.

Connections to D-semigroups and groupoids are analyzed.

Abstract

We ascertain conditions and structures on categories and semigroups which admit the construction of pseudo-products and trace products respectively, making their connection as precise as possible. This topic is modelled on the ESN Theorem and its generalization to ample semigroups. Unlike some other variants of ESN, it is self-dual (two-sided), and the condition of commuting projections is relaxed. The condition that projections form a band (are closed under multiplication) is shown to be a very natural one. One-sided reducts are considered, and compared to (generalized) D-semigroups. Finally the special case when the category is a groupoid is examined.