On some algebraic properties of Plonka sums and regularized varieties

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.

💡 Research Summary

The paper investigates the algebraic properties of Płonka sums and regularized varieties, extending the theory to languages that include nullary operations (constants). Historically, the structural description of algebras in regularized varieties via Płonka sums has been well‑developed for plural signatures without constants, covering subdirectly irreducible algebras, subvariety lattices, and free algebras. However, the presence of constants left a gap in the literature. The authors close this gap by presenting four major contributions.

First, they generalize the Płonka decomposition theorem to signatures with constants. By adding a condition (PF6) to the definition of a partition function, they ensure that each constant is interpreted in the distinguished minimal component of the semilattice direct system. Consequently, the construction of the Płonka sum remains well‑defined and retains its universal property even when constants are present.

Second, they study the relationship between the lattice of subvarieties L(V) of a strongly irregular variety V and the lattice L(R(V)) of its regularization. They define a map h: W↦R(W) and prove that it is an embedding of L(V) into L(R(V)). This embedding works uniformly for signatures with or without constants, because the minimal element of the underlying semilattice guarantees that regular identities are preserved while irregular ones are eliminated in the regularization process.

Third, the authors give a complete description of the congruences of a Płonka sum. They show that any congruence Θ on the sum can be uniquely assembled from a family {θ_i} of congruences on the component algebras A_i together with a congruence θ_I on the indexing semilattice I, subject to natural compatibility conditions (the transition homomorphisms must respect the component congruences). This result resolves an open problem from earlier work and yields explicit characterizations of generated and factor congruences, again valid when constants are present.

Fourth, they prove that the Płonka construction preserves two fundamental categorical properties: surjective epimorphisms and injective monomorphisms. If a homomorphism between algebras in the original variety V is onto (resp. one‑to‑one), then the induced homomorphism between their Płonka sums in R(V) is also onto (resp. one‑to‑one). This preservation ensures that free algebras, generators, and relational identities survive the regularization process unchanged.

The paper is organized as follows. Section 2 collects preliminaries on Płonka sums, partition functions, and the adjustments needed for constants. Section 3 extends the classical results on subdirectly irreducible algebras and the subvariety lattice to the constant‑inclusive setting, using a semilattice‑of‑subalgebras viewpoint. Section 4 investigates splitting pairs in L(R(V)) and shows how splittings in L(V) are reflected in the regularized lattice. Section 5 provides a comprehensive description of free algebras in regularized varieties, adapting the construction of Romano‑Skowroński to accommodate constants. Section 6 contains the main congruence theorem, together with applications to generated and factor congruences. Finally, Section 7 establishes the preservation of surjective epimorphisms and injective monomorphisms under the Płonka construction.

Through these results, the authors demonstrate that the core algebraic features of Płonka sums—decomposition, congruence structure, subvariety lattice, and categorical behavior—remain robust in the presence of constants. This broadens the applicability of the theory to a wide range of algebraic structures arising in non‑classical logics (e.g., involutive bisemilattices, Bochvar algebras), residuated structures, skew braces, and dual weak braces. The work thus provides a unified framework for analyzing regularized varieties across many logical and algebraic contexts, opening new avenues for research in universal algebra, categorical algebra, and the algebraic semantics of logics.