Roughness in quantales

Primeness in Quantales

“…The same may occur in quantales. The definition of prime ideals proposed in [1] is based on elements of a quantale and we ponder it is geared to commutative environment. When we move from commutative to the noncommutative setting, elementwise should be replaced by an approach based on ideals.…”

“…The main aim of this paper is to study the notion of primeness in the following perspective: we rename prime ideal defined in [1] to completely prime ideal and define a new concept of prime ideal for general quantales. Then we translate an important result in ring theory for quantales environment (theorem 2) to prove that these two concepts coincide in the commutative setting, but are no longer valid in the noncommutative setting (see Proposition 3).…”

Generalized Rough Fuzzy Ideals in Quantales

“…In many cases, as pointed out by numerous researchers, the implementation of theory of rough set becomes restrictive if we use the condition of the equivalence relation in the model of Pawlak rough set. To get control of this issue, several authors generalized the classical rough set theory by using more general binary relations [16][17][18][19][20] or by employing coverings [21,22]. Besides, theory of rough sets can also be generalized to the fuzzy environment by employing the notion of fuzzy sets of Zadeh [23], and the resulting notions are called fuzzy rough sets [24][25][26][27].…”

L-subquantales and L-filters of quantales based on quasi-coincidence of fuzzy points with parameters

“…In 1990, Yetter applied quantale theory to linear logic and provided a sound and complete class of models for linear intuitionstic logic [42]. Henceforth, quantales have invoked many interesting research topics in theoretical computer science [33], algebraic theory [15,18,51], rough set theory [24,41], groupoid theory [33], linear logic (see [11,34]), topological theory [14], etc. Based on the aforementioned analysis, the study combining fuzzy sets and quantales may become a promising topic that deserves further investigation.…”