Panagis Karazeris

When is flatness coherent

ABSTRACT We characterize those small categories with the property that flat (contravariant) functors on them are coherently axiomatized in the language of presheaves on them. They are exactly the categories with the property that every finite diagram into them has a finite set of (weakly) initial cocones.

Coupling an MV-algebra with a Boolean algebra

Abstract In this article we study Boolean powers of MV-algebras. We show that the property of semisimplicity is preserved by Boolean powers over separable complete Boolean algebras or by bounded Boolean powers over any Boolean algebra. In addition we give an explicit method of construction, based on Boolean powers, of an MV-algebra with given center. This last construction allows one to combine a qualitative (idempotent) structure like the Boolean algebra with a quantitative (non-idempotemt) structure like the MV-algebra.

Final Coalgebras in Accessible Categories

We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in this case. On the other hand, there are interesting examples of final coalgebras beyond the realm of l.f.p. categories to which our results apply. We rely on ideas developed by Tom Leinster for the study of self-similar objects in topology.

A note on representing and interpreting MV-algebras

We try to make a distinction between the idea of representing and that of interpreting a mathematical structure. We present a slight generalization of Di Nolas Representation Theorem as to incorporate this point of view. Furthermore, we examine some preservation and functorial aspects of the Boolean power construction.

Gabriel Filters and the Spectrum of an MV-Algebra

In this work we treat monoidal lattices (or multiplicative semilattices, in some people’s terminology, but we stick to the terminology in [Sun, 1994]) and monoidal closed (=residuated) lattices and MV-algebras (as special cases) from the viewpoint of the theory of quantales. We use the machinery of Gabriel topologies on coherent quantales in order to describe (the locale of opens of) the prime spectrum of an MV-algebra. For that we rely heavily on results obtained in [Karazeris, 1998]. Let us introduce the types of structures we study, in a hierarchy of increasing structural complexity:

Gabriel topologies on coherent quantales

Abstract The set of Gabriel topologies on a coherent quantale ordered under inclusion is a frame (studied by Rosenthal, Simmons and others). The set of all those Gabriel topologies that are inaccessible by directed joins (we call such topologies compact) is a subframe of it. When the quantale under consideration is commutative the frame of compact topologies is coherent. Several notions of spectra in ring theory appear as instances of this construction. When the quantale is non-commutative and coherent and its finite elements are closed under (right) implication then the frame of compact topologies is locally compact and compact. We present an interpretation of the notion of compact Gabriel topology on a coherent quantale in terms of deductively closed sets of formulae for a system of prepositional logic without the contraction and possibly the exchange rule (but admitting weakening). Our local compactness (and the subsequent spatiality) results for the frame of compact topologies correspond to a completeness theorem for such a system.