A T-partial order obtained from T-norms

summary:A partial order on a bounded lattice $L$ is called t-order if it is defined by means of the t-norm on $L$. It is obtained that for a t-norm on a bounded lattice $L$ the relation $a\preceq_{T}b$ iff $a=T(x,b)$ for some $x\in L$ is a partial order. The goal of the paper is to determine some conditions such that the new partial order induces a bounded lattice on the subset of all idempotent elements of $L$ and a complete lattice on the subset $A$ of all elements of $L$ which are the supremum of a subset of atoms

On the property of T-distributivity

WOS: 000320947500001In this paper, we introduce the notion of T-distributivity for any t-norm on a bounded lattice. We determine a relation between the t-norms T and T', where T' is a T-distributive t-norm. Also, for an arbitrary t-norm T, we give a necessary and sufficient condition for T-D to be T-distributive and for T to be T-boolean AND-distributive. Moreover, we investigate the relation between the T-distributivity and the concepts of the T-partial order, the divisibility of t-norms. We also determine that the T-distributivity is preserved under the isomorphism. Finally, we construct a family of t-norms which are not distributive over each other with the help of incomparable elements in a bounded lattice

Resolution of finite fuzzy relation equations based on strong pseudo-t-norms

AbstractThis work studies the problem of solving a sup-T composite finite fuzzy relation equation, where T is an infinitely distributive strong pseudo-t-norm. A criterion for the equation to have a solution is given. It is proved that if the equation is solvable then its solution set is determined by the greatest solution and a finite number of minimal solutions. A necessary and sufficient condition for the equation to have a unique solution is obtained. Also an algorithm for finding the solution set of the equation is presented

An extension of the ordering based on nullnorms

summary:In this paper, we generally study an order induced by nullnorms on bounded lattices. We investigate monotonicity property of nullnorms on bounded lattices with respect to the $F$-partial order. Also, we introduce the set of incomparable elements with respect to the F-partial order for any nullnorm on a bounded lattice. Finally, we investigate the relationship between the order induced by a nullnorm and the distributivity property for nullnorms

Logics without the contraction rule and residuated lattices

In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Lukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework

Fourier transformation of Sato's hyperfunctions

A new generalized function space in which all Gelfand-Shilov classes $S^{\prime 0}_\alpha$ ($\alpha>1$) of analytic functionals are embedded is introduced. This space of {\it ultrafunctionals} does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on $R^k$ is constructed that extends the Fourier transformation of Roumieu-type ultradistributions and is naturally interpreted as the Fourier transformation of hyperfunctions. The notion of carrier cone that replaces the notion of support of a generalized function for ultrafunctionals is proposed. A Paley-Wiener-Schwartz-type theorem describing the Laplace transformation of ultrafunctionals carried by proper convex closed cones is obtained and the connection between the Laplace and Fourier transformation is established.Comment: 34 pages, final version, accepted for publication in Adv. Mat