On some algebraic properties of Plonka sums and regularized varieties

TL;DR

This paper explores algebraic properties of Plonka sums within regularized varieties, extending known results to types with constants and analyzing their structural and congruence properties.

Contribution

It extends existing results on Plonka sums to algebraic types with constants and characterizes their congruences and subvariety lattice splittings.

Findings

Complete characterization of congruences of Plonka sums

Preservation of surjective epimorphisms and injective monomorphisms

Insights into subvariety lattice splittings

Abstract

P\l onka sums consist of a general construction that provides structural description for algebras in regularized varieties, whose examples range from Clifford semigroups to many algebras of logic including involutive bisemilattices, Bochvar algebras and certain residuated structures. While properties such as subdirectly irreducible algebras, subvariety lattices, and free algebras are well-understood for plural types without constants, the general case involving nullary operations remains largely unexplored. In this paper, we extend these results to algebraic types with constants and provide new insights into splittings within the lattice of subvarieties of a regularized variety. Furthermore, we offer a complete characterization of the congruences of a P\l onka sum and establish that the construction preserves surjective epimorphisms and injective monomorphisms.