We consider some frame-theoretic properties of the hull-kernel and the inverse topologies on the set of minimal prime ideals of an algebraic frame with the finite intersection property on its compact elements. Denote by Alg_do the subcategory of Frm consisting of such frames together with dense onto coherent maps. We construct a functor $${{\sf T} : {\bf Alg}_{\rm do} \rightarrow {\bf Frm}}$$ and a natural transformation $${\tau : {\sf E} \rightarrow {\sf T}}$$ , where E is the inclusion functor from Alg_do to Frm.

Although there have been repeated attempts to define the concept of an Archimedean algebra for individual classes of residuated lattices, there is no all-purpose definition that suits the general case. We suggest as a possible candidate the notion of a normal-valued and e-cyclic residuated lattice that has the zero radical compact property—namely, a normal-valued and e-cyclic residuated lattice in which every principal convex subuniverse has a trivial radical (understood as the intersection of all its maximal convex subuniverses). We characterize the Archimedean members in the variety of e-cyclic residuated lattices, as well as in various special cases of interest. A theorem to the effect that each Archimedean and prelinear GBL-algebra is commutative, subsuming as corollaries several analogous results from the recent literature, is grist to the mill of our proposal’s adequacy. Finally, we revisit the concept of a hyper-Archimedean residuated lattice, another notion with which researchers have engaged from disparate angles, and investigate some of its properties.

Arch denotes the category of archimedean ℓ-groups and ℓ-homomorphisms. Tych denotes the category of Tychonoff spaces with continuous maps, and α denotes an infinite cardinal or ∞. This work introduces the concept of an αcc-disconnected space and demonstrates that the class of αcc-disconnected spaces forms a covering class in Tych. On the algebraic side, we introduce the concept of an αcc-projectable ℓ-group and demonstrate that the class of αcc-projectable ℓ-groups forms a hull class in Arch. In addition, we characterize the αcc-projectable objects in W—the category of Arch-objects with designated weak unit and ℓ-homomorphisms that preserve the weak unit—and construct the αcc-hull for G in W. Lastly, we apply our results to negatively answer the question of whether every hull class (resp., covering class) is epireflective (resp., monocoreflective) in the category of W-objects with complete ℓ-homomorphisms (resp., the category of compact Hausdorff spaces with skeletal maps).

For a given variety $${\mathcal {V}}$$ of algebras, we define a class relation to be a binary relation $$R\subseteq S^2$$ which is of the form $$R=S^2\cap K$$ for some congruence class K on $$A^2$$, where A is an algebra in $$ {\mathcal {V}}$$ such that $$S\subseteq A$$. In this paper we study the following property of $${\mathcal {V}}$$: every reflexive class relation is an equivalence relation. In particular, we obtain equivalent characterizations of this property analogous to well-known equivalent characterizations of congruence-permutable varieties. This property determines a Mal’tsev condition on the variety and in a suitable sense, it is a join of Chajda’s egg-box property as well as Duda’s direct decomposability of congruence classes.

Let A be a finite algebra in a congruence permutable variety. We assume that for every subdirectly irreducible homomorphic image of A the centralizer of the monolith is n-supernilpotent. Then the clone of polynomial functions on A is determined by relations of arity |A|ⁿ⁺¹. As consequences we obtain finite implicit descriptions of the polynomial functions on finite local rings with 1 and on finite groups G such that in every subdirectly irreducible quotient of G the centralizer of the monolith is a p-group.

In this paper we study certain actions of a pomonoid S on a complete lattice, which we call S-quantales. Our aim is to characterize epimorphisms in the category of S-quantales. For this purpose we show that this category is a monadic construct and has the amalgamation property.

We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine vector spaces over a division ring.