Axiomatisability problems for S-posets

TL;DR

This paper investigates the conditions under which classes of S-posets over a pomonoid are axiomatisable in first-order logic, focusing on free, projective, and flatness-related classes, extending prior research.

Contribution

It provides new criteria and strategies for determining axiomatisability of various classes of S-posets, including some not previously studied.

Findings

Established general strategies for axiomatisability

Extended results to new classes of S-posets

Connected findings to previous work by Pervukhin and Stepanova

Abstract

Let C be a class of algebras of a given fixed type t. Associated with the type is a first order language L_t. One can then ask the question, when is the class C axiomatisable by sentences of L_t. In this paper we will be considering axiomatisability problems for classes of left S-posets over a pomonoid S (that is, a monoid S equipped with a partial order compatible with the binary operation). We aim to determine the pomonoids S such that certain categorically defined classes are axiomatisable. The classes we consider are the free S-posets, the projective S-posets and classes arising from flatness properties. Some of these cases have been studied in a recent article by Pervukhin and Stepanova. We present some general strategies to determine axiomatisability, from which their results for the classes of weakly po-flat and po-flat S-posets will follow. We also consider a number of classes not previously examined.