Narrow Results

To cover almost all known Lebesgue type integrals constructed by means of pseudo-operations, the (S, U)-integral based on a t-conorm S, a uninorm (or t-norm) U and an S-measure (decomposable measure) is introduced and its properties are studied. Also its relationship to aggregation operators is discussed.

This paper presents a characterization of a new family of operators. Namely, all non-decreasing, associative binary operators U: [0, 1]² → [0, 1] with a left (or right) neutral element e ∈ [0, 1], and such that they satisfy an additional hypothesis on continuity which is called here left (or right) pseudocontinuity.

Paper deals with binary operations in unit interval. We investigate connections between families of triangular norms, triangular conorms and uninorms. Certain structures of uninorms admits only the idempotent case.

A characterization of all idempotent uninorms satisfying the distributive property is given. The special cases of left-continuous and right-continuous idempotent uninorms are presented separately and it is also proved that all idempotent uninorms are autodistributive. Moreover, all distributive pairs of idempotent uninorms (pairs U₁, U₂ such that U_l is distributive over U₂ and U₂ is distributive over U₁) are also characterized.

We consider the interval ]-1, 1[ and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t. 0 are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.

In this paper, a new kind of fuzzy relational equations (FREs for short) A ∘^R_*x = b is first introduced, and then the problem of solving solution to the FREs is discussed, where A is an m × n matrix, x and b are an n and an m dimensional column vectors, respectively. More specifically, their solvability and unique solvability are investigated, the corresponding necessary and sufficient conditions are presented, the complete solution set is obtained. It is worth noting the method to construct the complete solution set.

Uninorms, as binary operations on the unit interval, have been widely applied in fuzzy set theory. In this paper, we study uninorms with nilpotent underlying t-norm and t-conorm. We prove that such a uninorm belongs to or . Moreover, some construction methods of uninorms from given t-norm and t-conorm are discussed.

The α-migrative property for uninorms with different neutral elements is presented, and some general results are given. The case for idempotent uninorms is studied, characterizing those uninorms (from any of the main classes of uninorms), which are α-migrative over an idempotent uninorm. The solutions obtained generalize the results where both uninorms have the same neutral elements.

Distributivity between two operations is a property posed many years ago — that is especially interesting in the framework of logical connectives because of its applications to fuzzy logic and approximate reasoning as their applications. Since semi-uninorms have been used in these topics, the study of the distributivity between two semi-uninorms becomes of particular interest that calls for thorough studies. The distributivity between two semi-uninorms, which are non-commutative and non-associative uninorms, has been developed only in the cases when both semi-uninorms are examples of very special classes of semi-uninorms. On the other hand, in general, the distributivity does not rely on the commutativity and associativity. The objective of this work is twofold. The first one is to show new solutions to distributivity equations for semi-uninorms. The second one is to check whether the results concerning the distributivity between two uninorms are valid for semi-uninorms. We investigate the distributivity involving two semi-uninorms when only one semi-uninrom lies in the most studied classes of semi-uninorms, achieving the above two objectives simultaneously.

The paper attempts to bridge the gap between widely accepted models of biological systems based on the Tsetlin automata acting in random environment and traditional artificial neural networks that consist of the McCalloch and Pitts neurons. Using recently developed algebra with uninorm and absorbing norm aggregators, we consider the neurons as extended Tsetlin automata that implement multi-valued not - xor operator applied to the aggregated inputs and internal states, and then construct the network using these neurons. The inputs of the neurons are specified by the synapses that implement multi-valued joined and and or operations. We demonstrate that for favorable (in the sense of learning) states the suggested neurons act similarly to the traditional neurons, while for unfavorable states they immediately change their activity to the reverse one. Such properties of the neurons both results in the correct activity of the network and demonstrates better correspondence with the logics of natural neural networks.

This paper investigates uninorms that are neither conjunctive nor disjunctive on bounded lattices. New methods are introduced for construction of such uninorms, where some restrictions on the identity and the annihilator are considered. In particular, new types of idempotent uninorms on bounded lattices are obtained. Furthermore, some specific examples are provided to illustrate that these constructions differ from the existing ones.

In this paper, we study uni-nullnorms and null-uninorms on bounded lattices, which are generalizations of uninorms, nullnorms, t-norms and t-conorms. We construct two uni-nullnorms and two null-uninorms on a bounded lattice $L$. We determine the smallest-greatest uni-nullnorms with 2-neutral element ${\{e,1\}}_a$ and the smallest-greatest null-uninorms with 2-neutral element ${\{0,e\}}_a$ by using these construction methods.

In this paper, we are mainly to solve the functional equations given by the modularity condition. Modularity condition between disjunctive (resp. conjunctive) uni-nullnorms and some most studied classes of binary aggregation operators (i.e., t-norms, t-conorms, uninorms and semi-t-operators) are discussed. Both positive and negative results of modularity condition for uni-nullnorms are obtained. Since uni-nullnorms are an extension of nullnorms and uninorms, previous results about modularity condition for nullnorms and uninorms in ${{𝒰}}_{max}$ and ${{𝒰}}_{min}$ can be obtained as simple corollaries.

Uninorms, as important generalizations of triangular norms and conorms, let the identity $e$ exist anywhere on a bounded lattice. In this paper, we focus on new characterizations of uninorms allowed to act on more general bounded lattices. In particular, we present several necessary and sufficient conditions to verify the construction approaches introduced by (Çaylı and Karaçal, Kybernetika 53 (2017) 394–417) and (Çaylı, Fuzzy Sets Syst. 395 (2020) 107–129), yielding a uninorm on bounded lattices.

In this paper, the concept of $U$-compatible congruence and its related properties are studied. We also focus on homomorphism theorems for uninorms. Decomposition of uninorms is investigated via factor congruences, at first. Subsequently, we investigate decomposition of conjunctive and disjunctive uninorms, respectively. After all, we determine necessary and sufficient conditions for decomposition of some uninorms that are neither conjunctive nor disjunctive. In addition to all these, we shed light on the application of our results with the examples.

Uninorms combining t-conorms and t-norms on bounded lattices have lately drawn extensive interest. In this article, we propose two ways for constructing uninorms on a bounded lattice with an identity element. They benefit from the appearance of the t-norm (resp. t-conorm) and the closure operator (resp. interior operator) on a bounded lattice. Additionally, we include some illustrative examples to highlight that our procedures differ from others in the literature.

This paper looks over a class of operators introduced in ([2]), called t–operators. Introduced in order to be applied to fuzzy preorders, their properties lead them to be also appropriate in some fields like aggregation problems and expert systems. We characterize these operators as a special combination of a t-norm and a t-conorm on [0, 1] in a similar way of uninorms in ([5]). We study duality and self duality on t–operators with respect to a strong negation N. We also give a classification of continuous t–operators through ordinal sums. Finally, we obtain from some t–operators (those idempotent) a special kind of E.A.F. by extending them to E=∪_n≥1[0,1]ⁿ.

In this paper, we show that the set of all left semi-uninorms and the set of all implications on a complete lattice are all complete lattices and obtain the formulas for calculating the smallest left semi-uninorm, the largest left semi-uninorm, the largest implication and the smallest implication generated by a binary operation.