Abstract In “Monk Algebras and Ramsey Theory,” J. Log. Algebr. Methods Program. (2022), Kramer and Maddux prove various representability results in furtherance of the goal of finding the smallest weakly representable but not representable relation algebra. They also pose many open problems. In the present paper, we address problems and issues raised by Kramer and Maddux. In particular, we prove that the algebra obtained by splitting the atoms of an n atom Monk algebra is representable for $$n=32$$ n = 32 and $$n=116$$ n = 116 , and hence Proposition 7 in Kramer-Maddux does not generalize. We answer Problem 1(3) in the negative: relation algebra $$1311_{1316}$$ 1311 1316 is not representable. Thus $$1311_{1316}$$ 1311 1316 is a good candidate for the smallest weakly representable but not representable relation algebra.

Abstract In this paper, we prove the following characterization: an abelian Mal’cev variety is finitely based if and only it has finite type, its ring of idempotent binary terms is finitely presented, and its module of unary terms is finitely presented.

Abstract In this note, we determine, by a disjunctive normal form theorem, which functions on the standard n -nuanced Łukasiewicz-Moisil algebra are representable by formulas and we show how this result may help in establishing the structure of the free algebras in this class.

Abstract This paper investigates the existence and structure of square roots in hoops, with a particular focus on cancellative, Wajsberg, and basic hoops that are not necessarily bounded. We introduce novel characterizations and decomposition theorems in this context. Specifically, we show that a cancellative hoop $$\textbf{N}(G)$$ N ( G ) , arising from the negative cone of an Abelian $$\ell $$ ℓ -group $$\textbf{G}$$ G , admits a square root $$\mu $$ μ if and only if $$\textbf{G}$$ G is two-divisible. For Wajsberg hoops with square roots, we introduce the notion of strong strict square roots and prove that any such hoop admits a unique decomposition into a direct product of a strong strict hoop and an idempotent hoop (which is, equivalently, a generalized Boolean algebra), by employing the framework of a nested family of hoops. We also investigate square roots on ordinal sums of hoops and define the ordinal sum of a family of square roots. This approach enables us to characterize linearly ordered hoops admitting square roots. Finally, we extend these results to basic hoops with square roots and identify generators for certain special subvarieties within this class.

Abstract Płonka sums are a powerful technique for the representation of algebras in regular varieties. However, certain representations of algebras in irregular varieties—like Polin’s variety or the variety of pseudocomplemented semilattices—bear striking similarities to Płonka sums, although they differ from them in some important respects. We aim at finding a convenient umbrella under which these constructions, as well as other ones of a similar kind, can be subsumed. Inspired by Grätzer and Sichler’s work on Agassiz sums , we appropriately enrich the structure of semilattice direct systems and we modify the attendant definition of a sum, while still encompassing Płonka sums as a special case. We prove that the above-mentioned representations of Polin algebras and pseudocomplemented semilattices can be recast in terms of this new framework. Finally, we investigate the problem as to which identities are preserved by the construction.