A Representation Theorem for Quantale Valued sup-algebras

Abstract

With this paper we hope to contribute to the theory of quantales and quantale-like structures. It considers the notion of Q-sup-algebra and shows a representation theorem for such structures generalizing the well-known representation theorems for quantales and sup-algebras. In addition, we present some important properties of the category of Q-sup-algebras.

A categorical equivalence between logical quantale modules and quantum B‐modules

“…As the key notion for the treatment of process semantics, quantale module was first introduced by Abramsky and Vickers [1]. Paseka reintroduced the notion of quantale module from the viewpoint of ring algebra; it replaces ring by quantale and abelian group by complete lattice [11]. Since the notion of quantale module was introduced, it has attracted many scholars' eyes.…”

Quantum B‐modules

“…Moreover, there is a correspondence between quantale modules and lattice‐valued sup‐lattices, established in [17] and developed further in [15, 16], which can be viewed as a fuzzification of the well‐known isomorphism between the categories of 2 ‐modules and $$\textstyle \bigvee$$‐semilattices (cf. [9, 14] for quantale‐valued sup‐algebras). The pioneering work on lattice‐valued sup‐lattices goes back to [20], and also to [18, 19], where L ‐frames were introduced and where it was shown that the category of L ‐frames is isomorphic to the category of L ‐fuzzy frames of Zhang and Liu.…”