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...

A new dominance relation is defined in set-valued information system based on similarity grade of attribute value sets the problem of attribute reduction and the judgment in set-valued information system are studied, from which a new approach to attribute reductions is provided in set-valued information system, and the influence of similarity level on attribute reduction are discussed..

We present an extension of GLukG, a logic that was introduced in [6] as a three-valued logic under the name of G3. GLukG is a paraconsistent logic defined in terms of 15 axioms, which serves as the formalism to define the p-stable semantics of logic programming. We introduce a new axiomatic system, N-GLukG, a paraconsistent logic that possesses strong negation. We use the 5-valued logic N ′ 5, ...

Zooming reasoning systems is a mechanism for reasoning using granular computing. The key concept of the zooming reasoning system is focus, which represents sentences we use in the current step of reasoning. Murai et al. has proposed a three-valued valuation based on focus. On the other hand, the authors have proposed another concept of granularity, called visibility, and constructed a four-valu...

in this paper, we study the existence of generalized solutions for the infinite dimensional nonlinear stochastic differential inclusions $dx(t) in f(t,x(t))dt +g(t,x(t))dw_t$ in which the multifunction $f$ is semimonotone and hemicontinuous and the operator-valued multifunction $g$ satisfies a lipschitz condition. we define the it^{o} stochastic integral of operator set-valued stochastic pr...

Closed classes of three-valued logic generated by periodic symmetric funtions that equal 1 in tuples from {1, 2} and equal 0 on the rest tuples are considered. Criteria for bases existence and finite bases existence for these classes is obtained. The problem of the bases existence for some families of closed classes of the three-valued logic functions is considered in the paper. E. Post [1] (se...

Partial functions are abundant in mathematics and program specifications. Unfortunately, their importance has been mostly ignored in automated theorem proving. In this paper, we develop a theorem proving strategy for Partial Classical Logic (PCL). Proof search takes place in Kleene Logic. We show that PCL theories can be translated into equivalent sets of formulas in Kleene logic. For proof sea...

Łukasiewicz three-valued logic Ł3 is often understood as the set of 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using D...

In this article, we introduce a perturbed system of generalized mixed quasi-equilibrium-like problems involving multi-valued mappings in Hilbert spaces. To calculate the approximate solutions of the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems, firstly we develop a perturbed system of auxiliary generalized multi-valued mixed quasi-equilibrium-like problems,...

in this paper, the notions of $(t,s)$-composition matrix and$(t,s)$-interval-valued intuitionistic fuzzy equivalence matrix areintroduced where $(t,s)$ is a dual pair of triangular module. theyare the generalization of composition matrix and interval-valuedintuitionistic fuzzy equivalence matrix. furthermore, theirproperties and characterizations are presented. then a new methodbased on $tilde{...