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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.

We report acute antibody responses to SARS-CoV-2 in 285 patients with COVID-19. Within 19 days after symptom onset, 100% of patients tested positive for antiviral immunoglobulin-G (IgG). Seroconversion for IgG and IgM occurred simultaneously or sequentially. Both IgG and IgM titers plateaued within 6 days after seroconversion. Serological testing may be helpful for the diagnosis of suspected patients with negative RT-PCR results and for the identification of asymptomatic infections.

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.

Recently, the α-migrativity has been discussed in families of certain operations (e.g., triangular norms, conorms, uninorms, and nullnorms). In this paper, the migrative property for semi-t-operators is studied. All solutions of the migrativity equation for all possible combinations of semi-t-operators are analyzed and characterized.

This paper is mainly devoted to investigating the migrativity equations involving nullnorms and 2-uninorms. Depending on whether the absorbing elements of 2-uninorms and unllnorms are same or not, all solutions of the migrativity equations for all possible combinations of the three defined subclasses of 2-uninorms and nullnorms are analyzed and characterized respectively. And for such equations, there are new solutions which extend the known ones about the migrativity for uninorms and nullnorms.

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.

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.

It is known that bisymmetry generalizes the simultaneous commutativity and associativity in the framework of the unit interval. In this work, we will completely characterize two classes of bisymmetric aggregation operators: one with a neutral element and the other with the vertical and horizontal sections of the idempotent elements being smooth on a finite chain, but not necessarily smooth and commutative. Thus, the previous results, based on the smoothness that is known as a very restrictive condition, are improved. For example, there is only one smooth Archimedean t-norm on a finite chain. In this paper, the discrete bisymmetric aggregation operators are explored without the limit of the smoothness. As a by-product, it is deduced that for smooth aggregation operators on a finite chain, the bisymmetry is equivalent to the commutativity and associativity, which improves the conclusion obtained by Mas et al. that associativity and bisymmetry are equivalent for commutative smooth aggregation operators on a finite chain.

In this study, silicon oxynitride (SiOxNy) films and three-paired organosilicon/SiOxNy multilayered structures prepared by the plasma-enhanced chemical vapor deposition were respectively deposited onto the polyethylene terephthalate and silicon substrate with overcut and undercut patterned photoresist to mimic the packaging of the patterned organic light emitting diodes (OLEDs) and surface contaminants. The step coverage at the bottom and sidewall surface was conducted from the cross-sectional scanning electron microscopy observations. A methodology for investigating the position-dependent coverage at the concave region of the sidewall surface on the undercut patterns was developed. The increase in the water vapor transmission when these barrier structures were deposited onto polyethylene terephthalate substrate was linked to the decrease in their coverage at the sidewall surface, which was ascribed to the reduction in the diffusion path for water vapor permeation. It also was demonstrated that the degradation in the barrier property was significant as those structures coated onto the undercut patterns originating from the precursors hardly arrived at the concave region. However, although the film growth at the concave region of the sidewall surface was limited, paired organosilicon/SiOxNy structures still showed a superior barrier property to that of single SiOxNy film as the appearance of the organosilicon film in the paired structure was favorable to prolonging the diffusion length for water vapor permeation. The investigation into the barrier structures coated onto the patterned surface was beneficial for better understanding the package reliability on the patterned OLED devices.

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.

Commuting is an important property in any two-step information aggregation process where the results should not depend on the order in which the single steps are carried out. In case of aggregation functions with the neutral elements, Saminger et al. (IEEEs Transactions on Fuzzy Systems, 2007, 15, 1032–1045) already reduced characterization of commuting functions to resolving the functional equation of a unary function. In this paper, we will investigate the commuting functions with the annihilator elements, especially, the nullnorms. A characterization of the commuting aggregation functions with the annihilator elements will be obtained in this case.

By weakening the neutral element condition of semiuninorms, we introduce a new concept called weak-neutral semiuninorms (shortly, wn-semiuninorms). After analyzing their structure, several classes of wn-semiuninorms are presented and discussed. Particularly, based on a kind of monotone unary functions which are not necessarily continuous and strictly monotone, we introduce representable wn-semiuninorms and discuss some of their properties in detail. We show that there is no idempotent proper wn-semiuninorm. Each representable wn-semiuninorm is Archimidean but not strictly monotone, and its additive generator is unique up to a positive multiplicative constant under some conditions. In the discussion about the representable wn-semiuninorms, we also characterize the solutions to a class of Cauchy functional equations on a restricted domain.

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.

Subcancellative triangular subnorms have been taken as aggregation functions and applied in multicriteria decision making. In this paper, the class of conditionally subcancellative triangular subnorms is introduced and its full characterization is given through a functional equation. Furthermore, we give the sufficient and necessary conditions under which such triangular subnorms become triangular norms.

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.

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.

Recently, Su and Liu have introduced a new class of fuzzy implications, called ordinal sum implications, and discussed some of their desirable properties, such as neutrality property, consequent boundary, exchange principle, etc. In this paper, we explore the class of ordinal sum implications with respect to distributivity. Necessary and sufficient conditions, under which ordinal sum implications are distributive over t-norms and t-conorms are given.