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Uninorms on bounded lattices have recently attracted widespread attention. In this study, we first propose the necessary structure for uninorms with Archimedean underlying t-norms and t-conorms on bounded lattices. We also discuss the characterization of more general classes of uninorms. We then study the necessity for a particular structure of uninorms that is required by many existing methods for constructing uninorms on bounded lattices. Finally, we propose a construction approach as an example.

In this paper, we first generalize some constructions of t-norms on bounded lattices by using order-preserving functions and propose the necessary and sufficient conditions for this kind of construction. Then, we simplify these conditions for practical use. Based on these results, we propose a new type of ordinal sum construction for t-norms. It can be regarded as a generalization of h-ordinal sum. Some examples and comparisons are also provided.

Overlap and grouping functions as two kinds of special continuous, commutative aggregation functions have wide applications in image processing and classification. Modularity conditions between the class of overlap (grouping) functions satisfying O ( 1 , e ) = e ( G ( 0 , e ) = e ) and special classes of uninorms, and those of another special class of overlap (grouping) functions over nullnorms have been studied in [49] . In this paper we extend these results to any overlap (grouping) function. Moreover, we study the modularity conditions between overlap (grouping) functions and uni-nullnorms (null-uninorms), which are the generalization of uninorms (nullnorms).

The uninorm has been used successfully in many different fields as a generalization of both the triangular norm and triangular conorm. Different classes of uninorms have been discussed in previous studies. In particular, the class of representable uninorms has been characterized from different viewpoints. In the present study, we provide a characterization of representable uninorms using a functional equation

In this paper we study the additive generators of t-norms and t-conorms on bounded lattices. They are not only good tools to construct but also crucial to studying the representations of t-norms and t-conorms. First we extend the classical additive generators theorem to the partially ordered cases by adding one more condition. Then we discuss some properties of these constructed t-norms such as the Archimedean property, nilpotent elements, strict monotonicity, et cetera. Besides, we give a more convenient modification of our theorem on finite bounded lattices and several examples for a further step on the characteristics of the additive generators.

The focus of this paper is the modularity equation including semi-t-operators and Mayor's aggregation operators. We discuss the modularity equation in three cases, namely Mayor's aggregation operators over Mayor's aggregation operators, Mayor's aggregation operators over semi-t-operators, and semi-t-operators over Mayor's aggregation operators, respectively. Necessary and sufficient conditions are established for these cases. Meanwhile, we obtain that Mayor's aggregation operators satisfying the modularity equation are mostly ordinal sums of semi-t-norms or semi-t-conorms.

Recently nullnorms on bounded lattices have been frequently investigated by many researchers. In this paper, we give the full characterization for quite a general class of nullnorms on bounded lattices and represent them by their underlying t-norms and t-conorms. An example is given to show the relationship between each block of the nullnorms. We also fully determine the values of arbitrary nullnorms when only one of the two variables is comparable with the absorbing elements. For the rest cases, we give some necessary conditions.

In this study, we show some construction methods to obtain a semi-t-operator from a given pseudo-t-conorm and pseudo-t-norm on bounded lattices. Using these methods, we prove that the known methods of constructing semi-t-operators on bounded lattices proposed by Fang and Hu are incorrect, and present the correct forms at the same time. Furthermore, some other construction methods for semi-t-operators on bounded lattices are also added.

In this paper, we study uni-nullnorms with both neutral elements and zero elements on bounded lattices. In order to illustrate the existence of uni-nullnorms on bounded lattices, we present two construction methods for obtaining such a uni-nullnorm. Also, we show two additional methods for the construction of idempotent nullnorms on bounded lattices, which are generalizations of the one proposed by Cayli and Karacal. Furthermore, we discuss the presence of idempotent uni-nullnorms on bounded lattices, and propose two construction methods to obtain an idempotent uni-nullnorm on any bounded lattice.

Inspired by the paving construction method, we construct a new operation on the unit interval from a given operation defined on the unit interval and a discrete operation defined on an index set. In particular, we construct a new t-norm from a t-norm on the unit interval and a discrete t-norm; a t-conorm from a t-norm and a discrete t-superconorm; a proper uninorm from a t-norm and a discrete t-conorm and a uninorm from a t-norm and a discrete uninorm. Dually, we also construct some uninorms (including t-norms and t-conorms) from a t-conorm and a discrete t-conorm, t-subnorm, t-norm or uninorm.

In this paper, the study of the functional equation of modularity for 2-uninorms recently published by Fechner, Rak and Zedam is revisited. A result from Corollary 6.3 in the mentioned paper is corrected and it is now given in a correct form. Furthermore, we characterize more general cases of modularity for 2-uninorms.

It is well known that under some conditions, a convex combination of the drastic product T D and the strict triangular norm is still a triangular norm. Uninorms, as a special kind of generalization of both triangular norms and triangular conorms, have been widely used in fuzzy set theory. In this paper, the convex combination of uninorms is studied. Some properties of the class of uninorms with the underlying triangular norm T D are presented firstly. Then the convex combination of this class of uninorms is discussed and it is shown that only in some trivial cases, the convex combination is still a uninorm.

This paper is mainly devoted to investigating the modularity condition for overlap and grouping functions, which may have neither neutral elements nor associativity. At present, the modularity equation has been discussed in some families of certain associative operators, such as t-norms, t-conorms, uninorms and nullnorms. In this study we explore the modularity equation between overlap (grouping) functions and overlap functions, grouping functions, uninorms and nullnorms. The results that we obtained are complete and generalize the known ones about modularity for certain associative operators.

In this paper, we describe the structure of 2-uninorms defined on the unit interval. We divide the class of 2-uninorms into five mutually exclusive subclasses based on their boundary behaviour, as determined by the absorbing element and the conjunctive/disjunctive nature of the underlying uninorms, which every 2-uninorm is known to have. For each of these subclasses, we fully characterize the structure of its members, albeit for two subclasses only under an additional continuity assumption, a common practice in the study of associative operations .

The aim of this paper is mainly to solve the functional equations given by the modularity condition. Several years ago, the modularity equations were discussed in families of certain commutative operations, such as t-norms, t-conorms, uninorms, 2-uninorms and t-operators. In this study, we continue the investigation of this same topic by focusing on semi-t-operators, which are generalization of t-operators by omitting commutativity. Our results are not only the full characterization of this modularity condition for semi-t-operators with continuous underlying operators, but also generalize the case for t-operators under certain conditions.

The distributivity equation has been widely studied involving different classes of aggregation functions from t-norms and t-conorms to uninorms, nullnorms and generalizations of them. It is important in the framework of logical connectives because of its applications in fuzzy logic and approximate reasoning as well as in image processing. Since uninorms have been used in these topics, the study of the distributivity between two uninorms becomes specially interesting. In a recent paper by the same authors the already known solutions were compiled and completed when the first uninorm is in any of the most studied classes of uninorms and the second uninorm is anyone. In this paper we want to achieve this study by focusing on the reverse direction, that is, for the cases when the second uninorm lies in any of the most studied classes of uninorms and the first one is any uninorm. We show along the paper that this new point of view leads to many new solutions.

The aim of this paper is mainly to solve the functional equations given by the modularity condition. Several years ago, the modularity equations for t-norms, t-conorms, uninorms and t-operators, which are commutative and associative, have been studied. Our investigations are motivated by modularity condition for generalizations of these operators by removing associativity or commutativity. In this work, the following main conclusions are proved: (1) a continuous t-norm with respect to a continuous semicopula is modular if and only if they are equal. The case for a semicopula with respect to a strict t-norm is also the same. A semicopula with respect to a co-semicopula is modular if and only if the semicopula is min and the co-semicopula is max. The modularity condition does not hold for a co-semicopula with respect to a semicopula. (2) Necessary and sufficient conditions are given for a semi-t-operator with respect to a semi-uninorm, a pseudo-uninorm with respect to a semi-t-operator to satisfy the modularity condition equation. New solutions to the modularity condition equations of the Case (1) are characterized.

In this paper, the study of the distributivity equation involving uninorms and t-conorms given in Li and Liu (2016) [3] is revised. Some errors in the proof of theorems in the mentioned reference are pointed out and their right versions are offered. Furthermore, the class of uninorms with continuous underlying operators which are (conditionally) distributive over a continuous t-conorm is characterized.

Recently, a class of uninorms locally internal on the boundary was proposed by Mas et al. during the study of the migrativity property for uninorms. In this paper, some properties of this class of uninorms are presented and the characterizations are obtained for different cases.

In this paper, an algebraic construction–called rotation–is introduced, which produces a fuzzy implication from a fuzzy implication. This construction method is similar to the rotation construction for triangular norms. An infinite number of new families of such fuzzy implications can be constructed in this way which provides a broad spectrum of choices for e.g. fuzzy connectives in fuzzy set theory. A preservation of the logical properties of the initial implication in the final one is investigated.

Coimplications are one of the important connectives used in fuzzy logic and fuzzy inference because they are a generalization of binary coimplications existing in classical logic. In this work, we further study two classes of coimplications derived from pseudo-uninorms on a complete lattice. Firstly, we present some characterizations of ( U , N ) -coimplications derived from a pseudo-uninorm and a strong negation. Then, we investigate residual coimplications and ( U , N ) -coimplications jointly rather than separately.

The migrativity equation with interesting applications in decision making and image processing has been extensively discussed involving different kinds of aggregation functions from t-norms and t-conorms to uninorms, nullnorms and some generalizations of them. In recent papers, the already known results concerning the migrativity of two uninorms are based on the assumption that both uninorms belong to one of the most studied classes of uninorms. In this paper we will explore the migrativity equation involving uninorms in a most general setting. Specifically, we will study the migrativity between two uninorms in the cases when the second uninorm lies in any of the most studied classes of uninorms, but the first one is any uninorm with no further assumptions. We will show along the paper that many new solutions appear from this new point of view that were not included in the previous approaches.

Mas et al. have investigated a special kind of commutative and associative aggregation operators with annihilator on the unit interval [ 0 , 1 ] , which includes S-uninorms, S-nullnorms, T-uninorms, T-nullnorms, bi-uninorms and bi-nullnorms. This paper is devoted to extending this kind of operators from the unit interval [ 0 , 1 ] to a finite chain. A characterization of this kind of operators on a finite chain is given, including some examples. This class of operators can be viewed as a generalization of both uninorms and nullnorms on a finite chain.

The aim of this paper is to complete some results in the paper "Migrative uninorms and nullnorms over t-norms and t-conorms" 1. That paper studied the alpha-migrativity of uninorms over t-norms. Here, migrativity in the other direction is investigated.

The α-migrative uninorms over a fixed uninorm with different neutral elements are presented. All cases when both uninorms lay in any one of the most usual classes of uninorms are analyzed, characterizing all solutions of the migrativity equation for some possible combinations. The solutions obtained generalize the results where both uninorms have the same neutral elements.

This paper deals with the distributivity and conditional distributivity of uninorms with continuous underlying operators over continuous triangular conorms. In particular, the involved triangular conorm is either maximum operator or an ordinal sum with only one summand in which the corresponding triangular conorm is strict. From the obtained results, it is deduced that distributivity and conditional distributivity are equivalent. Moreover, we obtain the full characterization of some cases of this class of uninorms of which either the underlying triangular norm or triangular conorm is strict.

Recently the distributivity equation was discussed in families of certain operations (e.g. triangular norms, conorms, uninorms and nullnorms (or called t-operators)). In this paper we describe the solutions of distributivity for semi t-operators over uninorms. Previous results about distributivity for nullnorms over uninorms can be obtained as a corollary.

Distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives. For this reason, the distributivity property has been extensively studied for several families of operations like triangular norms and conorms, some kinds of uninorms and nullnorms (also called t-operators) and even for some generalizations of them. In this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms, leading to many new solutions.

In this paper, quasi-homogeneous t-norms are generalized and then pseudo-homogeneous t-norms are introduced and characterized. It is shown that only some particular strict t-norms and the minimum t-norm T M are pseudo-homogeneous. The concepts of pseudo-homogeneous t-conorms and proper uninorms are given as well. Different from t-norms, only the maximum t-conorm S M is pseudo-homogeneous and any proper uninorm is not pseudo-homogeneous.

In this paper, the notions of α-migrative uninorms over semi t-operators and α-migrative semi t-operators over uninorms are introduced and investigated. All solutions of the migrativity equations for all possible combinations of semi t-operators and uninorms are characterized.

Conditional distributivity of nullnorm over uninorm with continuous operators is discussed.The structure of uninorms with idempotent operators is discussed. The problem of conditional distributivity for different two aggregation operators plays an important role in many different fields such as utility theory and integration theory. This paper deals with the conditional distributivity of a continuous nullnorm with respect to a uninorm with the continuous underlying t-norm and t-conorm.

The distributivity and conditional distributivity of a uninorm and a continuous t-conorm present an open problem recalled by Klement in the Linz2000 closing session. In 2006, Ruiz and Torrens 14 solved this problem for the most usual known classes of uninorms. Recently, Rak 13 solved the same problem for several classes of semi-uninorms, i.e., a special class of conjunctive semi-uninorms, a special class of disjunctive semi-uninorms and the class of idempotent semi-uninorms. In this work we continue to investigate the same topic as the above by focusing on two other classes of semi-uninorms, i.e., representable semi-uninorms and continuous semi-uninorms. The obtained results are different from the case of representable uninorms and also different from the cases of other known classes of semi-uninorms. Moreover, the dual case of distributivity and conditional distributivity of the above two classes of semi-uninorms over continuous t-norms is also investigated and similar results are obtained.

Recently, ( α , U 0 ) -migrative uninorms U were introduced and discussed, where U and U 0 are two uninorms with neutral element e. In this paper, we will further study ( α , U 2 ) -migrative uninorms U 1 , where U 1 and U 2 are two uninorms with different neutral elements e 1 and e 2 , respectively. All cases when the uninorm U 1 and U 2 are in different families out of the most usual classes of uninorms are analyzed, characterizing all solutions of the migrativity equation for some possible combinations of U 1 and U 2 .

A new class of fuzzy implications, called ordinal sum implications, is introduced by means of the ordinal sum of a family of given implications, which is similar to the ordinal sum of t -norms (or t -conorms). Basic properties of ordinal sum implications are discussed. It is shown that the ordinal sum implication is really a new class, which is different from the known ( S , N ) -, R -, QL - and Yager’s f - and g -implications.

Pseudo-uninorms are a generalization of uninorms by removing the commutativity from the axioms of the uninorms. In this paper, we further study coimplications generated from pseudo-uninorms on a complete lattice. Firstly, we further discuss the properties of residual coimplications generated from pseudo-uninorms on a complete lattice. Then, we recall the induced operators by coimplications on a complete lattice and give conditions such that they are pseudo-uninorms. Finally, we give out some characterizations of the residual coimplications generated from (right) infinitely ?-distributive disjunctive pseudo-uninorms on a complete lattice.

This paper shows by an example that Proposition 15(iv)(c) in Mas et al. (2013) about migrative property for uninorms in U"c"o"s","m"i"n is false, and corrects these propositions.

The aim of this paper is to correct and improve some results obtained in the paper “Quantitative logic” [Information Sciences 179 (2009) 226–247].

In this paper, Yager’s g -implications are generalized, and a new class of implications, called ( g , u ) -implications, is introduced. It is shown that ( g , u ) -implications are not only the generalization of Yager’s g -implications, but also the generalization of ( S , N ) -, R -, QL -implications. Basic properties and characterization of these implications are discussed. Furthermore, the distributive equations of these implications are investigated.

First, we show that the class of left-continuous triangular norms whose associated negations are discontinuous only at one point is not generally continuous except for the trivial case. Second, we find the strongest left-continuous triangular norm with a given associated negation and shows that there is no weakest left-continuous triangular norm with a given associated negation.

A new class of fuzzy implications, called (h,min)-implications, is introduced by means of generalized h-generators. Basic properties of these implications are discussed. It is shown that the (h,min)-implications are really a new class different from the known (S,N)-, R-, QL- and Yager's f- and g-implications. Generalizations of three classical logic tautologies with implications, viz., law of importation, contraction law and distributivity over triangular norms (t-norms) and triangular conorms (t-conorms) are investigated. A series of necessary and sufficient conditions are established, under which the corresponding functional equations are satisfied.

This paper investigates the functional equation of distributivity of uninorms over nullnorms. We consider the case where the uninorm is continuous in (0,1)^2 or is representable. It has been proved that the absorbing element of the nullnorm is an idempotent element of the uninorm if the distributive equation holds. And then combining the structures of the uninorm and the nullnorm, we give the characterization of the pair of the functions of the uninorm and the nullnorm. Moreover, when the nullnorm is continuous, we obtain the sufficient and necessary conditions under which the distributive equation holds.

Since Yager introduced f-generated implications in 2004, this class of fuzzy implications has been extensively investigated. In this paper, we generalize f-generated implications and get a new class of fuzzy implications called (f,g)-implications, which is different from the usual known classes of fuzzy implications. We discuss the basic algebraic properties of (f,g)-implications and study some classical logic tautologies (i.e., law of importation, contrapositive symmetry and distributivity over t-norms or t-conorms) for (f,g)-implications. Characterization of solutions to the corresponding fuzzy functional equations is obtained.

In this paper, we investigate the distributive functional equation I(x,S"1(y,z))=S"2(I(x,y),I(x,z)), where I:[0,1]^2->[0,1] is an unknown function, S"2 a continuous Archimedean t-conorm and S"1 a continuous t-conorm given as an ordinal sum. First, based on the special case with one summand in the ordinal sum of S"1, all the sufficient and necessary conditions of solutions to the distributive equation above are given and the characterization of its continuous solutions is derived. It is shown that the distributive equation does not have continuous fuzzy implication solutions. Subsequently, we characterize its non-continuous fuzzy implication solutions. Finally, it is pointed out that the case with finite summands in the ordinal sum of S"1 is equivalent to the one with one summand.

An extension of a uninorm called a semi-uninorm is introduced and discussed in this paper. First, we introduce the concept of semi-uninorms on a complete lattice. Then, we discuss two kinds of residual operators of semi-uninorms and give conditions such that the operators are implications. We also give equivalent conditions for infinitely @?-distributive left- and right-conjunctive semi-uninorms. Furthermore, we define two classes of induced operators by implications on a complete lattice and give conditions such that they are semi-uninorms. We also provide the equivalent conditions for the infinitely @?-distributive implications in their second variables.

This paper investigates an iterative Boolean-like law with fuzzy implications derived from uninorms. More precisely, we characterize the solutions to the functional equation I(x,y)=I(x,I(x,y)) that involve RU-, (U,N)- and QLU-implications generated by the most usual classes of uninorms.

Two kinds of extensions of triangular norms (t-norms) are proposed, and the relations between these extensions and fuzzy implications are discussed in this paper. First, two classes of pseudo-t-norms (ps-t-norms), called type-A and type-B ps-t-norms, and their respective residual operators are defined. Then, we prove that these residual operators are fuzzy implications and satisfy the left neutral property. For these two classes of pseudo-t-norms, we give a series of equivalent conditions of left-continuity with respect to their first or second variable. By combining the axioms of the two classes of pseudo-t-norms, we simply get the definition of the triangular seminorms. Furthermore, we define two classes of induced operators from fuzzy implications and give the conditions under which they are type-A ps-t-norms, type-B ps-t-norms and t-seminorms. For a fuzzy implication, a series of equivalent conditions of right-continuity with respect to its second variable are established. Finally, another method inducing type-A ps-t-norms, type-B ps-t-norms and t-seminorms by implications is proposed.

In this paper we present new methods for solving multi-criteria decision-making problem in an intuitionistic fuzzy environment. First, we define an evaluation function for the decision-making problem to measure the degrees to which alternatives satisfy and do not satisfy the decision-maker’s requirement. Then, we introduce and discuss the concept of intuitionistic fuzzy point operators. By using the intuitionistic fuzzy point operators, we can reduce the degree of uncertainty of the elements in a universe corresponding to an intuitionistic fuzzy set. Furthermore, a series of new score functions are defined for multi-criteria decision-making problem based on the intuitionistic fuzzy point operators and the evaluation function and their effectiveness and advantage are illustrated by examples.