Directly Indecomposables in Semidegenerate Varieties of Connected po-Groupoids

TL;DR

This paper investigates semidegenerate varieties of connected po-groupoids, establishing a first-order axiomatization for directly indecomposable structures and analyzing the complexity of these definitions.

Contribution

It provides a first-order axiomatization for directly indecomposables in semidegenerate varieties of connected po-groupoids, extending previous results to this specific class.

Findings

Axiomatization of directly indecomposables is possible for these varieties.

Quantifier complexity of the axioms is bounded.

Connectedness and semidegeneracy are key conditions for the results.

Abstract

We study varieties with a term-definable poset structure, "po-groupoids". It is known that connected posets have the "strict refinement property" (SRP). In [arXiv:0808.1860v1 [math.LO]] it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegenerate varieties of connected po-groupoids and show its quantifier complexity is bounded in general.