As an extension of interval-valued pseudo t-norms, pseudo-overlap functions (IPOFs) play a vital role in solving multi-attribute decision making problems. However, their corresponding algebraic structure has not been studied yet. On the other hand, with development non-commutative (non-associative) fuzzy logic, study residuated lattice theory is gradually deepening. Due to conditions operators ...

In this paper we study systems of fuzzy relation inequalities and equations of the formU ◦Vi 6 Vi ◦U (i ∈ I), where U is an unknown and Vi (i ∈ I) are given fuzzy relations, the dual systems Vi ◦ U 6 U ◦ Vi (i ∈ I), their conjunctions, the systems of the formU◦Vi = Vi◦U (i ∈ I), and certain special types of these systems. We call themweakly linear systems. For each weakly linear system, with a ...

The study of the Gentzen system Gew determined by the well known sequent calculus FLew [Ono98, Ono03c] is interesting for the study of the substructural aspects of t-norm based logics [Háj98, EG01]. In [BGV05] we studied the 〈∨, ∗,¬, 0, 1〉 and the 〈∨,∧, ∗,¬, 0, 1〉-fragments of this Gentzen system and the same fragments of the logic of residuated lattices IPC∗\c. In this paper we continue the st...

This paper presents basic notions about fuzzy measures over algebras of fuzzy subsets of a fuzzy set. It also presents basic ideas on fuzzy integrals defined using these fuzzy measures. Definitions of new types of fuzzy measures and integrals are motivated by our research on generalized quantifiers. Several useful properties of fuzzy measures and fuzzy integrals are stated and proved. Definitio...

Extensions of monoidal t-norm logic MTL and related fuzzy logics with truth stresser modalities such as globalization and “very true” are presented here both algebraically in the framework of residuated lattices and proof-theoretically as hypersequent calculi. Completeness with respect to standard algebras based on t-norms, embeddings between logics, decidability, and the finite embedding prope...

The commutative residuated lattices were first introduced by M. Ward and R.P. Dilworth as generalization of ideal lattices of rings. Non-commutative residuated lattices, called sometimes pseudo-residuated lattices, biresiduated lattices or generalized residuated lattices are algebraic counterpart of substructural logics, that is, logics which lack some of the three structural rules, namely cont...

In many applications we have a set of objects together with their properties. Since the available information is often incomplete and/or imprecise, the true knowledge about subsets of objects can be determined approximately only. In this paper we present a fuzzy generalisation of two relation–based operations suitable for fuzzy set approximations. Main properties of these operations are present...

In Section 2 replace the definition of ∗◦Q in Definition 1 by x∗◦Q y = inf{u ∗◦ v | u > x, v > y}. It is defined only if the infimum exists. Proposition 1 remains unchanged. Theorem 0. Let (X, ∗◦,→∗◦,≤) be a commutative residuated semigroup on a complete chain equipped with the order topology. Let a, b, c ∈ X be such that a = b→∗◦ c. Let (x, y) ∈ X × X be such that 1. neither x nor y equals the...

We introduce the notion of (L, e)-filters with fuzzy partially order e on complete residuated lattice L. We investigate (L, e)-filters induced by the family of (L, e)-filters and functions. In fact, we study the initial and final structures for the family of (L, e)-filters and functions. From this result, we define the product and co-product for the family of (L, e)-filters and functions.

Fuzzy description logics can be used to model vague knowledge in application domains. This paper analyses the consistency and satisfiability problems in the description logic SHI with semantics based on a complete residuated De Morgan lattice. The problems are undecidable in the general case, but can be decided by a tableau algorithm when restricted to finite lattices. For some sublogics of SHI...