Applied Categorical Structures – Official Blog

by Belluce, Lawrence P.; Nola, Antonio; Gerla, Brunella

In this paper, after recounting the basic properties of perfect MV-algebras, we explore the role of such algebras in localization issues. Further, we analyze some logics that are based on Łukasiewicz connectives and are complete with respect to linearly ordered perfect MV-algebras.

DOI: 10.1007/s10485-007-9069-4
Online Date: 2/21/2007
Print publication date: 4/1/2007
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by Olmen, Christophe; Verwulgen, Stijn

It has never been systematically studied under which algebraic actions lower semi-continuity is a stable property. This is despite the fact that this continuity variant is commonly used, in different branches of mathematics. In this note we propose an abstraction of the lower semi-continuous function frame on a topological space and describe this as the Eilenberg–Moore algebras of a Hom-functor monad. We also point out the connection with approach frames and with classical frame theory. Thereby we obtain several different characterizations of the introduced category.

DOI: 10.1007/s10485-007-9064-9
Online Date: 2/14/2007
Print publication date: 4/1/2007
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by Zenk, Eric Richard

Functorial nearness structures and coreflections mesh tightly together. After presenting a new construction closely related to the completion of a nearness frame, this article shows a bijection exists between monocoreflections and strong, locally fine, completion stable, functorial nearness structures. This bijection gives a simple description of the monocoreflective subcategory generated by any class of frames.

DOI: 10.1007/s10485-007-9065-8
Online Date: 2/13/2007
Print publication date: 4/1/2007
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by Hager, Anthony W.; Martínez, Jorge

This article considers coherent frame homomorphisms h : L → M between coherent frames, which induce an isomorphism between the boolen frames of polars, with M projectable, and such that M is generated by h(L) and certain complemented elements of M. This abstracts the passage from a semiprime commutative ring with identity to its projectable hull. The frame theoretic setting is investigated thoroughly, first without any assumptions beyond the Zermelo–Fraenkel axioms of set theory, and, subsequently, assuming that algebraic frames are spatial. The culmination of this effort is the result that the spectrum of d-elements of M is obtained from that of L by refining the given hull–kernel topology to the patch topology. The second part of the article relates the projectable hull to the (von Neumann) regular hull, in a variety of contexts, including that of f-rings. For a uniformly complete f-algebra A, it is shown that the maximal ℓ-ideals of A that are traces of real maximal ideals of the regular hull HA are precisely the almost P-points of the space of maximal ℓ-ideals of A.

DOI: 10.1007/s10485-007-9062-y
Online Date: 2/13/2007
Print publication date: 4/1/2007
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