Regular S-acts with primitive normal and antiadditive theories

TL;DR

This paper characterizes when classes of regular S-acts over commutative monoids are both primitive normal and antiadditive, linking these properties to the linearity of an associated semigroup order.

Contribution

It establishes the equivalence between primitive normality, antiadditivity, and the linear order of a semigroup in the context of regular S-acts over commutative monoids.

Findings

Primitive normality is equivalent to antiadditivity for these classes.

Linearity of the semigroup order characterizes the properties of the S-acts.

The results unify algebraic and model-theoretic properties of regular S-acts.

Abstract

In this work, we investigate the commutative monoids over which the axiomatizable class of regular S-acts is primitive normal and antiadditive. We prove that the primitive normality of an axiomatizable class of regular S-acts over the commutative monoid S is equivalent to the antiadditivity of this class and it is equivalent to the linearity of the order on a semigroup R such that an S-act SR is a maximal (under the inclusion) regular subact of the S-act SS.