In fairly elementary terms this paper presents how the theory of preordered fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy sets, is established under the guidance of enriched category theory. Motivated by several key results from the theory of quantaloid-enriched categories, this paper develops all needed ingredients purely in order-theoretic languages for the rea...

The topic of sets with fuzzy order relations valuated in complete lattices with additional structure has been quite active in the recent decade, and a number of papers have been published (see [3, 5, 6] among many others). Based on a quantale-valued order relation and subset membership, counterparts to common order-theoretic notions can be defined, like monotone mappings, adjunctions, joins and...

The aim of this paper is to extend the truth value table oflattice-valued convergence spaces to a more general case andthen to use it to introduce and study the quantale-valued fuzzy Scotttopology in fuzzy domain theory. Let $(L,*,varepsilon)$ be acommutative unital quantale and let $otimes$ be a binary operationon $L$ which is distributive over nonempty subsets. The quadruple$(L,*,otimes,varep...

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...

The concepts of fuzzy source and fuzzy successor operators for an L-fuzzy automaton (L is a latticeordered monoid) are introduced, which turn out to be L-fuzzy closure operators. When L is a quantale, these operators introduce L-fuzzy topologies. These observations are then used to give topological characterization of the separatedness and connectedness properties of an L-fuzzy automaton. c ©20...

For a completely distributive quantale L, L-fuzzy strongest postcondition predicate transformers are introduced, and it is shown that, under reasonable assumptions, they are linear or affine continuous mappings between continuous L-idempotent semimodules of L-fuzzy monotonic predicates. © 2012 Elsevier B.V. All rights reserved.

In this paper, we study the relations (L, )-fuzzy topologies and (L, )-filters on strictly two-sided, commutative quantale lattices (L, ) and (L, ∗). Mathematics Subject Classification: 03E72, 54A40, 54B10

In this paper, we study (L, )-fuzzy topologies induced by operations and (L, )-filters on strictly two-sided, commutative quantale lattices (L, ) and (L, ∗). Mathematics Subject Classification: 03E72, 54A40, 54B10

This paper introduces a new approach to the theory of Ω-categories enriched by a frame. The approach combines ideas from various areas such as generalized ultrametric domains, Ω-categories, constructive analysis, and fuzzy mathematics. As the basic framework, we use the Wagner’s Ω-category [18,19] with a frame instead of a quantale with unit. The objects and morphisms in the category will be ca...

Each divisible and unital quantale is associated with two quantaloids. Categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders. This paper deals with the relationship between these two kinds of enriched categories. This is a special case of the change-base issue in enriched category theory...