Some Contributions to Rough Fuzzy Quantales and Quantale Modules

Abstract

The theory of quantale was rst introduced by Mulvey [57]. With algebraic struc- tures and lattice-ordered structures, Quantale introduces a lattice setting of the study of non-commutative C?- algebra and an initiation of the study of quantum mechanics. A connection between quantale theory and linear logic was introduced by Yetter in 1990, in which he established a complete class of models for linear intuitionistic logic [102]. Quantales may be utilized in many interesting research topics like algebraic the- ory [44], rough set theory ([49, 67, 68, 70, 91, 96]), topological theory [30], theoretical computer science [77] and linear logic [28]. The idea of quantale module was introduced by Abramsky and Vickers [1]. The quantale module has attracted many scholars eyes. The idea of quantale module was motivated by the thought of module over a ring [5]. It replaces rings by quantales and abelian groups by complete lattices. The concept of quantale module showed up out of the blue for the rst time as the key notion in the uni ed treatment of process semantics done by Abramsky and Vickers. A family of models of full linear logic is provided by modules over a commutative unital quantaleas as shown by Rosenthal [80]. Fuzzy set theory, at rst proposed by Zadeh [105], has given an important scienti c and mathematical tool to the description of those frameworks which are unreason- ably perplexing or uncertain. Moreover, those conditions including vulnerabilities or ambiguities even more solidly, the unit interval [0; 1] is replaced with a lattice and L-fuzzy sets were proposed by Goguen [29]. Gradually by applying fuzzy sets to the lattice-ordered environment, an important branch, has attracted consideration of re- searchers [114, 115], recently since fuzzy lattices have been extensively used as a part of designing, software engineering, topology, logic etc [64, 65]. Further, fuzzy algebra has furthermore transformed into a promising subject, since fuzzy algebraic structures have been viably associated with a wide range of elds [49, 67]. The idea of fuzziness is generally utilized in the theory of formal languages and automata. Numerous scientists utilized this idea to generalize notions of algebra. Rosenfeld de ned fuzzy subgroups. Ahsan et al. proposed fuzzy semirings [2]. There are several authors who applied the theory of fuzzy sets to quantale, for instance, Luo and Wang [49] applied the fuzzy set theory to quantales. They de ned fuzzy prime, fuzzy semi-prime and fuzzy primary ideals of quantales.