Coproducts of distributive lattice-based algebras.
The study of coproducts in different varieties of algebras can be found in mathematics' literature for a long time, the characterization of coproducts often being motivated by the interest in amalgation properties, determination of free objects, axiomatizations or colimits. The main aim of this paper is to present a uniform theory, based on duality, of coproducts in classes of algebras that admit a bounded distributive lattice reduct. Let \(\mathcal D\) denote the variety of bounded distributive lattices. A class \(\mathcal A\) of algebras with language \(\mathcal L\) is a \(\mathcal D\)-based class if the algebras of \(\mathcal A\) have reduct in \(\mathcal D\). In this paper the authors study coproducts in \(\mathcal D\)-based categories of algebras by relating them to coproducts in \(\mathcal D\) (it is well known the fact that Priestley duality for (bounded) distributive lattices establishes a dual equivalence between the category \(\mathcal D\) and the category \(\mathcal P\) of Priestley spaces). For every \(\mathcal D\)-based class of algebras, \(\mathcal A\), we have a (forgetful) functor \(U_{\mathcal A}\) from \(\mathcal A\) into \(\mathcal D\) and one of the main problems of this paper is to determine when \(U_{\mathcal A}\) preserves coproducts. They also investigate the possible behaviours of this functor as regards coproducts in \(\mathcal A\) under weaker assumptions. The authors restrict their attention to the situation in which \(\mathcal A\) is a finitely generated quasivariety (quasivarieties ensure the existence of coproducts for each set of algebras in the class). The principal results of this paper are presented in Section 3 and Theorem 3.6 (Coproducts Preservation Theorem) is the most important of them. It establishes equivalent conditions for the forgetful functor \(U_{\mathcal A}\) to preserve coproducts. In the final part of this section the authors give a short description of the free product and present the connections between the properties of coproducts in a finitely generated \(\mathcal D\)-based quasivariety and the type of natural duality that it admits. In Section 4, the authors ``venture beyond the cofines of finitely generated \(\mathcal D\)-based quasivarieties, with the aim of revealing how far certain results in Section 3 hold in greater generality''. Finally, in Section 5, they apply their ``results and techniques to particular well-known classes of finitely generated varieties'' (De Morgan algebras, Kleene algebras, pseudocomplemented distributive lattices, quasivarieties of Heyting algebras generated by finite chains, MV-algebras, \(\mathcal Q\)-lattices, \(n\)-valued pre-Moisil and pre-Łukasiewicz-Moisil algebras and Moisil and Łukasiewicz-Moisil algebras). The paper ``unifies pre-existing description of coproducts, both as regards the descriptions and as regards the methodology for obtaining them''.
scientific article; zbMATH DE number 6369493
scientific article