Sup-lattice 2-forms and quantales

A 2-form between two sup-lattices L and R is defined to be a sup-lattice bimorphism L x R -> 2. Such 2-forms are equivalent to Galois connections, and we study them and their relation to quantales, involutive quantales and quantale modules. As examples we describe applications to C*-algebras.Comment: 30 pages. Contains more detailed background section and corrections of several typos and mistake

Fuzzy Galois connections on fuzzy sets

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 readership of fuzzy set theorists, with particular attention paid to fuzzy Galois connections between preordered fuzzy sets.Comment: 30 pages, final versio

Convolution, Separation and Concurrency

A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales. A number of examples underpin the universality of these constructions, the most prominent ones being separation logics, where convolution is separating conjunction in an assertion quantale; interval logics, where convolution is the chop operation; and stream interval functions, where convolution is used for analysing the trajectories of dynamical or real-time systems. A Hoare logic is constructed in a generic fashion on the power series quantale, which applies to each of these examples. In many cases, commutative notions of convolution have natural interpretations as concurrency operations.Comment: 39 page

The Coproduct of Unital Quantales

In this paper, the definition of the saturated element in quantale is given, Based on the coproduct of monoids, the concrete forms of the coproduct of unital quantales is obatined. Also, some properties of their are discussed. KeyWords: Quantale; Monoid; Saturated element; Coproduct; Categor

Enriched lower separation axioms and the principle of enriched continuous extension

[EN] This paper presents a version of the lower separation axioms and the principle of enriched continuous extension for quantale-enriched topological spaces. As a remarkable result, among other things, we point out that in the case of commutative Girard quantales the principle of continuous extension holds for projective modules in Sup.The authors acknowledge support from the Basque Government (grant IT1483-22). The first named author also acknowledges support from a postdoctoral fellowship of the Basque Government (grant POS-2022-1-0015)

A theory of quantale-enriched dcpos and their topologization

There have been developed several approaches to a quantale-valued quantitative domain theory. If the quantale Q is integral and commutative, then Q-valued domains are Q-enriched, and every Q-enriched domain is sober in its Scott Q-valued topology, where the topological «intersection axiom» is expressed in terms of the binary meet of Q (cf. D. Zhang, G. Zhang, Fuzzy Sets and Systems (2022)). In this paper, we provide a framework for the development of Q-enriched dcpos and Q-enriched domains in the general setting of unital quantales (not necessarily commutative or integral). This is achieved by introducing and applying right subdistributive quasi-magmas on Q in the sense of the category Cat(Q). It is important to point out that our quasi-magmas on Q are in tune with the «intersection axiom» of Q-enriched topologies. When Q is involutive, each Q-enriched domain becomes sober in its Q-enriched Scott topology. This paper also offers a perspective to apply Q-enriched dcpos to quantale computatio