On the zero-classes of monoid semi-congruences

TL;DR

This paper explores the structure of zero-classes of monoid semi-congruences, introducing a syntactic relation to characterize these classes and establishing a hierarchy of conditions for submonoids to understand their properties.

Contribution

It introduces a new syntactic relation that characterizes zero-classes of semi-congruences and develops a hierarchy of conditions on submonoids to analyze their properties.

Findings

A syntactic relation characterizes zero-classes of semi-congruences.

Hierarchy of conditions on submonoids ensures compatibility and relational properties.

Illustrative cases include positive cones and normal submonoids.

Abstract

This paper studies the zero-classes of monoid semi-congruences, understood as internal reflexive relations on a monoid. Classical examples include normal submonoids, which arise as zero-classes of congruences, and positive cones, which are the zero-classes of preorders; both admit well-known syntactic characterizations via the Eilenberg syntactic equivalence and the syntactic preorder introduced by Pin, respectively. Beyond these cases, however, no general notion of a syntactic object characterizing zero-classes of semi-congruences, also called clots, has been established. We address this gap by introducing a syntactic relation that is reflexive and characterizes clots whenever it is compatible with the monoid operation, a property that is not automatic in contrast to the congruence and preorder settings. We further develop a hierarchy of conditions on submonoids of a given monoid that induce syntactic relations with progressively stronger properties: first ensuring compatibility with the monoid operation, and subsequently enforcing additional relational properties such as transitivity and symmetry. Several illustrative situations are discussed, including the cases of positive cones and normal submonoids.