On some algebraic properties of Plonka sums and regularized varieties
S. Bonzio, G. Zecchini
TL;DR
The paper develops a comprehensive framework for Plonka sums in regularized varieties, extending known results to languages with constants and analyzing how subvarieties split, how free algebras are structured, and how congruences behave in this setting. Central contributions include a complete congruence description for Plonka sums over semilattice direct systems, a robust SI/classification extended to constants, and a demonstration that epimorphism-surjectivity and monomorphism-injectivity are preserved under regularization. It also clarifies the interaction between a strongly irregular variety $\mathcal{V}$ and its regularization $R(\mathcal{V})$, showing that key structural properties transfer and that splitting identities can be regularized. The results unify and extend the theory for Clifford semigroups, involutive bisemilattices, and dual weak braces, providing practical tools for analyzing varieties arising from logic and non-classical algebraic structures.
Abstract
Płonka 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 Płonka sum and establish that the construction preserves surjective epimorphisms and injective monomorphisms.