Representable Uninorms Research Articles

Generating methods of some classes of uninorms and associated structures

Uninorms are an important class of fuzzy logic connectives that have, so far, neither generating methods nor algebraic structures. In this work, we attempt to propose some novel generating methods of some well established classes of uninorms (with a fixed but arbitrary neutral element e) and investigate the algebraic structures that these generating methods can impose on those specific classes of uninorms. Towards this, we show that the operations point-wise maximum (∨) and the point-wise minimum (∧) on the set of idempotent uninorms are closed and make it a distributive lattice. Also, we show that the operations ∧ and ∨ on some sub-classes of idempotent uninorms with other classes of uninorms generate uninorms. Since the operations ∨ and ∧ on the set of uninorms are not closed in general, we propose some novel generating methods of uninorms that come from various classes such as representable uninorms, pseudo-continuous uninorms, uninorms continuous in the open unit square, uninorms with underlying continuous Archimedean t-norms and t-conorms and for each class of these uninorms, we investigate the algebraic structure imposed on it by the proposed generating methods with the help of results obtained recently in Vemuri et al. (2022) [51].

Deriving priorities based on representable uninorms from fuzzy preference relations

Deriving priorities of objects from preference relations is an important research topic for decision making with preference information. In this paper, this issue is solved for fuzzy preference relations using representable uninorms, with which several alternative definitions of representable uninorm-based consistency (⊗-consistency) are provided for complete and incomplete fuzzy preference relations. In particular, a procedure to evaluate incomplete fuzzy preference relations with arbitrary n−1 independent preference values is provided. Furthermore, a novel acceptable ⊗-consistency is defined by the proposed local priority vectors, and then a method to check and reach the ⊗-consistency of fuzzy preference relations by reducing the distances between the optimal priority vector and local ones is proposed. Specific examples are illustrated to show how the models work, and comparisons with the existing methods are also offered to show the advantages of the proposed method.

The distributivity of extended uninorms over extended overlap functions on the membership functions of type-2 fuzzy sets

This paper mainly investigates the distributivity of an extended uninorm over an extended overlap function when the uninorm is continuous in (0,1)2. It first studies the distributivity of the extended representable uninorms over the extended overlap functions. Then, it presents the distributivity of the extended uninorms over the extended overlap functions when the uninorms belong to one of the usual classes Ucos,min, Ucos,max.

The generalized sigmoid function and its connection with logical operators

In this study, we present the operator-dependent sigmoid function, which is derived from a universal unary operator called the kappa function. Here, we describe how the generalized sigmoid function is related to representable uninorms (i.e., Dombi's aggregative operator). Namely, we show that the inverse of a generalized sigmoid function is an additive generator of the aggregative operator. We provide the necessary and sufficient conditions for the form of the function that transforms the aggregative operator into a conjunctive or disjunctive logical operator. This transformation is also based on the generalized sigmoid function. Here, we show how conjunctive and disjunctive operators, which form a De Morgan system with a negation, can be derived from the aggregative operator. We point out that, under certain conditions, a set of generalized sigmoid functions is closed under the negation and modifier operators. Lastly, we demonstrate an important connection between the weighted aggregative operator and the generalized sigmoid function. Based on this connection, we provide a new interpretation of the feed-forward neural networks. We show that a perceptron-based neural network can be modeled using the aggregative operator and the generalized sigmoid function.

The n-uninorms with continuous underlying t-norms and t-conorms

ABSTRACT The n-uninorms with continuous underlying t-norms and t-conorms are studied. We show that each 2-uninorm with continuous underlying functions can be expressed as an ordinal sum of a uninorm with continuous underlying functions (possibly also of a countable number of semigroups related to representable uninorms, continuous Archimedean t-norms, continuous Archimedean t-conorms and internal uninorms) and a 2-uninorm with continuous underlying functions such that (possibly restricted to open or half-open unit square ). Similar results are shown for n-uninorms with continuous underlying functions, where

Some new results on the migrativity of uninorms over overlap and grouping functions

In 2018, Qiao and Hu [48] studied the α-migrativity of uninorms over overlap and grouping functions when the uninorm U belongs to one certain class (e.g., Umin, Umax, the family of idempotent uninorms, representable uninorms or uninorms continuous on ]0,1[2). In addition, they obtained some equivalent characterizations of them. This paper will continue to consider the characterizations of this kind of migrativity equations by means of the ordinal sum of overlap and grouping functions. We give the necessary and sufficient conditions for the solutions of the (α,O)-migrativity equation when the uninorm U becomes a t-norm or a conjunctive uninorm locally internal on the boundary and the (α,G)-migrativity equation when the uninorm U becomes a t-conorm or a disjunctive uninorm locally internal on the boundary, respectively.

Convex combinations of uninorms and triangular subnorms

The problem whether a non-trivial convex combination of two continuous triangular norms can be a t-norm was extensively studied in the last decades, bringing the negative answer in all examined special cases. On the other hand, in the field of non-continuous t-norms (including left-continuous t-norms) several examples were found, including mainly a convex combination of an α-migrative t-norm and the drastic product t-norm, or a convex combination of ordinal sums of product related t-subnorms. Recently, convex combinations of uninorms were discussed in [18]. Since any uninorm with the neutral element e∈[0,1] restricted to [0,e]2 is isomorphic to a t-norm, this investigation extends the original problem of a convex combination of t-norms given by Alsina, Frank and Schweizer. In this work we will study convex combinations of uninorms further, covering uninorms with continuous underlying functions, uninorms with different neutral elements and representable uninorms. We will also show that convex combinations of triangular subnorms play a major role in this investigation.

Migrative uninorms and nullnorms over t-norms and t-conorms revisited

In 2015, Mas et al. investigated the migrativity of uninorms and nullnorms over any fixed t-norm and t-conorm. First, they introduced the concept of migrative uninorm over any fixed t-norm. In addition, they showed equivalent characterizations of the migrativity equation when the uninorm belongs to one certain class (e.g., Umin, Umax, the family of idempotent uninorms, representable uninorms or uninorms continuous on ]0,1[2). And then, they gave the notion of migrative uninorms over any fixed t-conorm and proposed the migrativity equation using an analogous method. Finally, they discussed the migrativity of nullnorms over any fixed t-norm and t-conorm and obtained equivalent characterizations of the related migrativity equations, respectively. However, they did not discuss the migrativity property of a conjunctive (resp. disjunctive) uninorm non locally internal on the boundary over any fixed t-norm (resp. t-conorm) and the migrativity property of a conjunctive (resp. disjunctive) uninorm locally internal on the boundary over any continuous t-norm (resp. t-conorm). To fill the gap, in this paper, we explore some new results on the migrativity property of a conjunctive (resp. disjunctive) uninorm over any fixed t-norm (resp. t-conorm) and give some new characterizations of the migrativity property of nullnorms over any fixed t-norm and t-conorm, respectively.

A short note on the migrativity properties of overlap functions over uninorms

Recently, as an addendum to Qiao and Hu (2018) [21], Zhu and Hu (2020) [25] studied the α-migrativity of overlap and grouping functions over uninorms and nullnorms. Later on, Zhou and Yan (2019) [24] further investigated the α-migrativity properties of overlap functions over uninorms. They showed equivalent characterizations of the (α,U)-migrativity equation when the uninorm U belongs to one certain class (e.g., Umin, Umax, the family of idempotent uninorms, representable uninorms or uninorms continuous on ]0,1[2). However, we find that Propositions 3.5, 3.8 and Remark 3.5 (2) presented by Zhu and Hu (2020) [25] are incorrect. In this note, at first, we establish the updated results. And then, we study the α-migrativity of overlap functions over t-norms when t-norms are continuous and give their characterizations. In particular, we point out that the relationship between the α-migrativity of overlap functions over uninorms and the α-migrativity of uninorms over overlap functions. Finally, we discuss the (α,U)-migrativity equation for grouping functions using an analogous method.

New Results on the Distributive Laws of Uninorms Over Overlap Functions

Recently, Qiao studied the distributive laws of uninorms over overlap functions when the uninorms were one of the usual classes $\mathcal {U}_{\rm min}$ and $\mathcal {U}_{\rm max}$ , the family of representable uninorms or uninorms continuous in $(0,1)^{2}$ and obtained some partially characterizations. This article will continue to consider the characterizations of this kind of distributivity equations when the uninorms lie in the family of uninorms continuous in $(0,1)^{2}$ . First of all, we investigate the distributive laws of continuous t -norms over overlap functions and give their full characterizations. Then, we give the necessary and sufficient conditions for the solutions of the distributivity equations when uninorms are continuous in $(0,1)^{2}$ .

On a characterization of representable uninorms

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

On Distributive Laws of Uninorms Over Overlap and Grouping Functions

Overlap and grouping functions, as two new classes of special binary aggregation functions, have been investigated by several authors for applications in image processing, decision making, classification, and fuzzy community detection problems. On the other hand, after Aczel studied the distributive law between two operations, the distributive laws among particular binary aggregation functions become an interesting and natural research area. In this paper, we continue to discuss this topic for uninorms and overlap and grouping functions. First, we give some basic properties for the distributive law of uninorms over overlap functions. Second, we study this distributive law when the uninorms belong to one of the usual classes (e.g., $\mathcal {U}_{\min }$ , $\mathcal {U}_{\max }$ , the family of idempotent uninorms, representable uninorms or uninorms continuous on $]0,1[^2$ ). Finally, we investigate the distributive law of uninorms over grouping functions by an analogous way.

Migrativity properties of overlap functions over uninorms

The migrativity of aggregation functions is an interesting property from both theoretic and applied points of view in fuzzy set theory, and many profound results on the migrative functional equation for different combinations of some specific subclasses of aggregation functions have been achieved in the last decades. The present paper chooses to study the migrativity properties of overlap functions over uninorms, where overlap functions are a newly-born subclass of nonassociative aggregation functions for modeling overlap problems in decision making and image processing. The main results are characterizations of solutions to some migrative functional equations for overlap functions over uninorms in the usual classes such as uninorms continuous on the boundary except at their neutral elements, representable uninorms, idempotent uninorms and uninorms continuous in the open unit square, together with plenty of supporting examples for positive solutions and illustrating remarks for negative results.

On a class of left-continuous uninorms constructed from the representable uninorm

On a class of left-continuous uninorms constructed from the representable uninorm

On the migrativity of uninorms and nullnorms over overlap and grouping functions

In this paper, we investigate the migrativity of uninorms and nullnorms over any fixed overlap function or grouping function. First, we introduce the concept of (α,O)-migrative uninorms over any fixed overlap function O. In addition, we show equivalent characterizations of the (α,O)-migrativity equation when the uninorm U belongs to one certain class (e.g. Umin, Umax, the family of idempotent uninorms, representable uninorms or uninorms continuous on ]0,1[2). And then, we give the notion of (α,G)-migrative uninorms over any fixed grouping function G and propose the (α,G)-migrativity equation using an analogous method. Finally, we discuss the (α,O)-migrative and (α,G)-migrative nullnorms and obtain equivalent characterizations of the related (α,O)-migrativity and (α,G)-migrativity equations, respectively.

On Extended Representable Uninorms and Their Extended Fuzzy Implications (Coimplications)

In this work, by Zadeh’s extension principle, we extend representable uninorms and their fuzzy implications (coimplications) to type-2 fuzzy sets. Emphatically, we investigate in which algebras of fuzzy truth values the extended operations are type-2 uninorms and type-2 fuzzy implications (coimplications), respectively.

Characterization of uninorms with continuous underlying t-norm and t-conorm by means of the ordinal sum construction

The uninorms with continuous underlying t-norm and t-conorm are characterized via the ordinal sum construction of Clifford. Using the previous results of the author, where each uninorm with continuous underlying operations was characterized by properties of its set of discontinuity points, it is shown that each such a uninorm can be decomposed into an ordinal sum of a countable number of semigroups related to representable uninorms, continuous Archimedean t-norms, continuous Archimedean t-conorms and internal uninorms (including the min and the max operator).

The cross-migrative property for uninorms

This paper aims to study the cross-migrative property for uninorms. We only consider the most usual classes of uninorms as follows: uninorms in \({\mathcal{U}_{min}}\) and \({\mathcal{U}_{max}}\), representable uninorms, idempotent uninorms and uninorms continuous in the open unit square, and limit the research to those uninorms which have the same neutral element. This study shows that there is no cross-migrativity between representable uninorms and other classes of uninorms. The relationship is the same between conjunctive uninorms and disjunctive uninorms. We give the sufficient and necessary conditions for the cross-migrativity equation to hold for all of the other possible combinations of uninorms.

Ordinal sums of representable uninorms

We investigate properties of an ordinal sum of uninorms in the case that the summands are proper representable uninorms. We show sufficient and necessary conditions for a uninorm to be an ordinal sum of representable uninorms. An example is also included.

On almost equitable uninorms

Uninorms, as binary operations on the unit interval, have been widely applied in information aggregation. The class of almost equitable uninorms appears when the contradictory information is aggregated. It is proved that among various uninorms of which either underlying t-norm or t-conorm is continuous, only the representable uninorms belong to the class of almost equitable uninorms. As a byproduct, a characterization for the class of representable uninorms is obtained.