The problem of determining (up to lattice isomorphism) the lattices that are sublattices of free lattices is in general an extremely difficult and an unsolved problem. A notable result towards solving this problem was established by Galvin and Jónsson when they classified (up to lattice isomorphism) all of the distributive sublattices of free lattices in 1959. In this paper, we weaken the requirement that a sublattice of a free lattice be distributive to requiring that a such a lattice belongs in the variety of lattices generated by the pentagon $$N_5$$ . Specifically, we use McKenzie’s list of join-irreducible covers of the variety generated by $$N_5$$ to extend Galvin and Jónsson’s results by proving that all sublattices of a free lattice that belong to the variety generated by $$N_5$$ satisfy three structural properties. Afterwards, we explain how the results in this paper can be partially extended to lattices from seven known infinite sequences of semidistributive lattice varieties.

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.