Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V . The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V...

The purpose of this paper is to construct a weak hyper semi-quantale as a generalization of the concept of semi-quantale and used it as an appropriate hyperlattice-theoretic basis to formulate new lattice-valued topological theories. Based on such weak hyper semi-quantale, we aim to construct the notion of a weak hypervalued-topology as a generalized form of the so-called lattice-valued t...

In fuzzy set theory non-idempotency arises when the conjunction is interpreted by arbitrary t-norms. There are many instances in mathematics where set theory ought to be non-commutative and/or non-idempotent. The purpose of this paper is to combine both ideas and to present a theory of non-commutative and non-idempotent quantale sets (among other things, standard concepts like fuzzy preorders a...

This survey paper, specifically targeted at a readership of fuzzy logicians and fuzzy set theorists, aims to provide a gentle introduction to the basic notions of quantaloidenriched category theory. We discuss at length the definitions of quantaloid, quantaloidenriched category, distributor and functor, always giving several examples that – hopefully – appeal to the intended readership. To indi...

This paper considers some extensions of the notion filter to quantale-valued context, including saturated prefilter, ?-filter and bounded prefilter. The question is whether these constructions give rise monads on category sets. It shown that answer depends structure quantale. Specifically, if quantale unit interval equipped with a continuous t-norm, then only implication operator corresponding ...

For a quantaloid Q, considered as a bicategory, Walters introduced categories enriched in Q. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad T over the discrete presheaf monad of the small quantaloid Q the primary data of the theory, rather than the l...

For a monad S on a categoryK whose Kleisli category is a quantaloid, we introduce the notion of modularity, in such a way that morphisms in the Kleisli category may be regarded as V(bi)modules (= profunctors, distributors), for some quantale V . The assignment S // V is shown to belong to a global adjunction which, in the opposite direction, associates with every (commutative, unital) quantale ...

The notion of sobriety is extended to the realm topological spaces valued in a commutative and unital quantale, via an adjunction between category quantale modules quantale-valued spaces. Relations such sober domains based on flat ideals are investigated. In particular, it shown that if unit element coincides with its top element, then every domain (relative ideals) Scott topology space.

In this paper, the category of L-FINT is topological category over Set. We investigate the functorial relations between L-FTOP and LFINT where L is a strictly two-sided, commutative quantale lattice. Mathematics Subject Classification: 03E72, 54A40, 54B10

The ‘the’ in the title hides a subtlety. A metric space induces not one but four topologies - by means of open sets, closed closure, and interior they just so happen to coincide. agreement between these structures arising from function d:X×X→[0,∞] is due combination axioms lattice structure [0,∞]. Further motivation materializes Lawvere's observation 1973 effect that (slightly generalized) cate...