A new so-called fuzzifying measurable theory that generalizes the classical measurable theory is established and the structures of such new theory are discussed very detailed. In the last, we study the product of two fuzzifying measures and consider a problem, which is like the third of the open problems in fuzzy measure presented by Z. Wang. We have solved this problem satisfactorily in the ne...

The generalization of binary operation in the classical algebra to fuzzy is an important development field algebra. paper proposes a new vector spaces over field, which called M-hazy field. Some fundamental properties spaces, and subspaces are studied, some results also proved. Furthermore, linear transformation studied their Finally, it shown that M-fuzzifying convex induced by subspace space.

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...

Based on a complete Heyting algebra, we modify the definition oflattice-valued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesian-closed category, calledthe category of $L-$ordered fuzzifying convergence spaces, in whichthe category of $L-$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called the...

In this paper, characterizations of the degree to which a mapping $mathcal{T} : L^{X}longrightarrow M$ is an $(L, M)$-fuzzy topology are studied in detail.What is more, the degree to which an $L$-subset is an $L$-open set with respect to $mathcal{T}$ is introduced.Based on that, the degrees to which a mapping $f: Xlongrightarrow Y$ is continuous,open, closed or a quotient mapping with respect t...

Let G be a group with identity e. Let R be a G-graded commutative ring and let M be a graded R-module. The graded classical prime spectrum Cl.Specg(M) is defined to be the set of all graded classical prime submodule of M. The Zariski topology on Cl.Specg(M); denoted by ϱg. In this paper we establish necessary and sufficient conditions for Cl.Specg(M) with the Zariski topology to be a Noetherian...

this paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. it incorporates the most important settings of lattice-valued topology, including poslat topology of s.~e.~rodabaugh, $(l,m)$-fuzzy topology of t.~kubiak and a.~v{s}ostak, and $m$-fuzzy topology on $l$-fuzzy sets of c.~guido. moreover, its respe...

By fuzzifying the number of occurrences of an element of a multiset, we obtain a new fuzzy structure; similarly, by fuzzifying the number of occurrences of an element of a hybrid set, we obtain another new fuzzy structure. The aim of the present work is twofold: to provide a concise definition of these new fuzzy structures and their properties and to apply them in natural computing. More specif...

‎‎Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎We define the primary spectrum of $M$‎, ‎denoted by $mathcal{PS}(M)$‎, ‎to be the set of all primary submodules $Q$ of $M$ such that $(operatorname{rad}Q:M)=sqrt{(Q:M)}$‎. ‎In this paper‎, ‎we topologize $mathcal{PS}(M)$ with a topology having the Zariski topology on the prime spectrum $operatorname{Spec}(M)$ as a sub...