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By Dube, Themba

1 Citations

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.

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By Ledda, Antonio; Paoli, Francesco; Tsinakis, Constantine

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.

By Carrera, Ricardo E.

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).

By Ahmed, Tarek Sayed

4 Citations

This is a survey article on the concept of neat reducts. An old venerable idea in algebraic logic, in this paper we show why it is regaining momentum.

By Mayr, Peter

5 Citations

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.

By Xiao, Zhijiang; Zhang, Xia

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.

By Olšák, Miroslav

We prove that a weakest non-trivial strong Maltsev condition given by a single identity of the form $$t(\text {variables}) = t(\text {variables})$$ is the existence of a term s satisfying $$s(x,y,y,z,z,x)=s(y,x,z,y,x,z)$$.

By Dormán, Miklós

In this paper we investigate a class of inverse transformation monoids constructed from finite lattices, and we describe a necessary and sufficient condition for such a transformation monoid to be collapsing.

By Kearnes, Keith A.; Kiss, Emil W.; Szendrei, Ágnes

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.

By Zhuk, Dmitriy N.

7 Citations

All minimal clones on three elements were found by B. Csákány. In this paper, for each minimal clone the cardinality of the set of all clones containing this clone is found.

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