Quantale algebras as a generalization of lattice-valued frames

Year 2016, Volume: 45 Issue: 3, 781 - 809, 01.06.2016

Sergey A. Solovyov

Abstract

Recently, I. Stubbe constructed an isomorphism between the categories

of right Q-modules and cocomplete skeletal Q-categories for a given

unital quantale Q. Employing his results, we obtain an isomorphism

between the categories of Q-algebras and Q-quantales, where Q is additionally

assumed to be commutative. As a consequence, we provide

a common framework for two concepts of lattice-valued frame, which

are currently available in the literature. Moreover, we obtain a convenient

setting for lattice-valued extensions of the famous equivalence

between the categories of sober topological spaces and spatial locales,

as well as for answering the question on its relationships to the notion

of stratification of lattice-valued topological spaces.

Keywords

(Cocomplete) (skeletal) Q-category, lattice-valued frame, latticevalued partially ordered set, quantale, quantale algebra, quantale module, sober topological space, spatial locale, stratification degree, stratified topological space

References

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