Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

Visual characterization of associative quasitrivial nondecreasing operations on finite chains

In this paper we provide visual characterization of associative quasitrivial nondecreasing operations on finite chains. We also provide a characterization of bisymmetric quasitrivial nondecreasing binary operations on finite chains. Finally, we estimate the number of functions belonging to the previous classes.Comment: 25 pages, 18 Figure

Characterizations of quasitrivial symmetric nondecreasing associative operations

We provide a description of the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We also prove that associativity can be replaced with bisymmetry in the definition of this class. Finally we investigate the special situation where the chain is finite

Relating Kleene algebras with pseudo uninorms

This paper explores a strict relation between two core notions of the semantics of programs and of fuzzy logics: Kleene Algebras and (pseudo) uninorms. It shows that every Kleene algebra induces a pseudo uninorm, and that some pseudo uninorms induce Kleene algebras. This connection establishes a new perspective on the theory of Kleene algebras and provides a way to build (new) Kleene algebras. The latter aspect is potentially useful as a source of formalism to capture and model programs acting with fuzzy behaviours and domains.publishe

On the reinforcement of uninorms and absorbing norms

DUKE_HCERES2020Aggregation operators Reinforcement ... We propose a n-ary extension of absorbing norms, defined with the help of generative functions, and its relationship with additive generating functions of uninorms. In this paper, we also present new aggregation operators, namely the k-uninorms and k-absorbing norms. These operators are a generalization of usual uninorms and absorbing norms for which a set combination of inputs is introduced. Their main ability is to provide reinforcement for contradictory inputs, as nullnorms and as opposed to uninorms. On the other hand it still provides full reinforcement for agreeing inputs, as uninorms and as opposed to nullnorms. Numerous examples are given in order to illustrate the behavior of the proposed operators

On triangular norms and uninorms definable in ŁΠ12

AbstractIn this paper, we investigate the definability of classes of t-norms and uninorms in the logic ŁΠ12. In particular we provide a complete characterization of definable continuous t-norms, weak nilpotent minimum t-norms, conjunctive uninorms continuous on [0,1), and idempotent conjunctive uninorms, and give both positive and negative results concerning definability of left-continuous t-norms (and uninorms). We show that the class of definable uninorms is closed under construction methods as annihilation, rotation and rotation–annihilation. Moreover, we prove that every logic based on a definable uninorm is in PSPACE, and that any finitely axiomatizable logic based on a class of definable uninorms is decidable. Finally we show that the Uninorm Mingle Logic (UML) and the Basic Uninorm Logic (BUL) are finitely strongly standard complete w.r.t. the related class of definable left-continuous conjunctive uninorms

A principal topology obtained from uninorms

summary:We obtain a principal topology and some related results. We also give some hints of possible applications. Some mathematical systems are both lattice and topological space. We show that a topology defined on the any bounded lattice is definable in terms of uninorms. Also, we see that these topologies satisfy the condition of the principal topology. These topologies can not be metrizable except for the discrete metric case. We show an equivalence relation on the class of uninorms on a bounded lattice based on equality of the topologies induced by uninorms