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Following a historical overview of the development of ordinal sums, a presentation of the most relevant results for ordinal sums of triangular norms and copulas is given (including gluing of copulas, orthogonal grid constructions and patchwork operators). The ordinal sums of copulas considered here are constructed not only by means of the comonotonic copula, but also by using the lower Fréchet-Hoeffding bound and the independence copula. We provide alternative proofs to some results on ordinal sums, elaborate properties common to all or just some of the ordinal sums discussed. Also included are a discussion of the relationship between ordinal sums of copulas and the Markov product and an overview of ordinal sums of multivariate copulas, illustrating aspects to be considered when extending concepts for ordinal sums of bivariate copulas to the multivariate case. [ABSTRACT FROM AUTHOR]

We study the properties of z -ordinal sum construction and show that it can always be expressed in the reduced basic form. This means that it is enough to assume only trivial semigroups in the branching set and we can always remove semigroups with duplicate carriers. We also investigate the cardinality of the minimal branching set corresponding to monotone (and non-monotone) functions defined on the unit interval constructed via z -ordinal sum construction. [ABSTRACT FROM AUTHOR]

Similar to the construction of ordinal sums of overlap functions, Baczyński et al. introduced two new kinds of ordinal sums of fuzzy implications without additional restrictions on summands in 2017. In this paper, based on the ordinal sum of fuzzy implications with the first method, we discuss its basic properties, such as iterative Boolean law, right ordering property and strong boundary condition. Meanwhile, we give characterizations of such ordinal sum of fuzzy implications that is a QL -implication constructed from tuples (O , G , N ⊤) or a D -implication derived from grouping function G , and show some conclusions about its relations with (G , N) - and R O -implications. Moreover, we study the distributivity of such ordinal sum of fuzzy implications over a class of overlap and grouping functions. More specifically, we give necessary and sufficient conditions under which this ordinal sum of fuzzy implications is distributive over overlap and grouping functions satisfying some conditions. [ABSTRACT FROM AUTHOR]

This paper introduces two fundamental types of ordinal sum constructions for partially ordered monoids that are determined by two specific partial orderings on the disjoint union of the partially ordered monoids. Both ordinal sums of partially ordered monoids are generalized with the help of operators on posets, which combine, in some sense, the properties of interior and closure operators on posets. The proposed approach provides a unified view on several known constructions of ordinal sums of t-norms and t-conorms on posets (lattices) and introduces generalized ordinal sums of uninorms on posets (lattices). [ABSTRACT FROM AUTHOR]

Recently, Qiao gave the concrete form of the continuity of overlap and grouping functions on complete lattices by using meet- and join-preserving properties. By eliminating the property join-preserving (resp. meet-preserving), one gets right (resp. left) continuous functions, namely, C R - (resp. C L -) overlap and grouping functions. In this paper, we extend the notion of ordinal sums of overlap functions from the unit interval to complete lattices directly and indicate it does not necessarily lead to an overlap function. We find that the ordinal sums of finitely many overlap functions can create an overlap function on a frame that can be partitioned into a chain of subintervals. We also investigate ordinal sums of C R - and C L -overlap functions on complete lattices, where the endpoints of summand carriers constitute a chain. Moreover, we have an analogous discussion on grouping functions on complete lattices. [ABSTRACT FROM AUTHOR]

First, we present construction methods for interior operators on a meet semilattice. Second, under the assumption that the underlying meet semilattices constitute the range of an interior operator, we prove an ordinal sum theorem for countably many (finite or countably infinite) triangular norms on bounded meet semilattices, which unifies and generalizes two recent results: one by Dvořák and Holčapek and the other by some of the present authors. We also characterize triangular norms that are representable as the ordinal sum of countably many triangular norms on given bounded meet semilattices. [ABSTRACT FROM AUTHOR]

The structure of idempotent n -uninorms is studied. We show that each idempotent 2-uninorm can be expressed as an ordinal sum of an idempotent uninorm (possibly also of a countable number of idempotent semigroups with operations min and max) and a 2-uninorm from Class 1 (possibly restricted to open or half-open unit square). Similar results are shown also for idempotent n -uninorms. Further, it is shown that idempotent n -uninorms are in one-to-one correspondence with special lower semi-lattices defined on the unit interval. The z -ordinal sum construction for partially ordered semigroups is also defined. [ABSTRACT FROM AUTHOR]

The commutative, associative and monotone (CAM) functions on a horizontal sum of bounded chains are studied. Depending on their range, these functions are divided into two main classes and properties of CAM functions from each of these classes are described. Several necessary and sufficient conditions under which a CAM function defined on a horizontal sum of bounded chains can be expressed as a non-trivial (z)-ordinal sum of semigroups are also introduced. Special classes of CAM functions are discussed and several examples are included. Various construction methods for special CAM functions defined on a horizontal sum of bounded chains, such as t-norms, t-conorms, uninorms and nullnorms, are also presented. [ABSTRACT FROM AUTHOR]

The ordinal sum construction provides a very effective way to generate a new triangular norm on the real unit interval from existing ones. One of the most prominent theorems concerning the ordinal sum of triangular norms on the real unit interval states that a triangular norm is continuous if and only if it is uniquely representable as an ordinal sum of continuous Archimedean triangular norms. However, the ordinal sum of triangular norms on subintervals of a bounded lattice is not always a triangular norm (even if only one summand is involved), if one just extends the ordinal sum construction to a bounded lattice in a naïve way. In the present paper, appropriately dealing with those elements that are incomparable with the endpoints of the given subintervals, we propose an alternative definition of ordinal sum of countably many (finite or countably infinite) triangular norms on subintervals of a complete lattice, where the endpoints of the subintervals constitute a chain. The completeness requirement for the lattice is not needed when considering finitely many triangular norms. The newly proposed ordinal sum is shown to be always a triangular norm. Several illustrative examples are given. [ABSTRACT FROM AUTHOR]

The concept of ordinal sums in the sense of Clifford have long been blamed for their limitations in constructing new t-norms including inability to cope with general bounded lattices. Motivated by this observation, and based on the lattice-based sum of lattices that has been recently introduced by El-Zekey et al., we propose a new sum-type construction of t-norms, called a lattice-based sum of t-norms , for building new t-norms on bounded lattices from given ones. The proposed sum is generalizing the well-known ordinal and horizontal sum constructions of t-norms by allowing for lattice ordered index sets. We demonstrate that, like the ordinal sum of t-norms, the lattice-based sum of t-norms can be generalized using as summands so-called t-subnorms, still leading to a t-norm. Subsequently, we apply the results for constructing several new families of t-norms and t-subnorms on bounded lattices. In the same spirit, by the duality, we will also introduce lattice-based sums of t-conorms and t-subconorms. [ABSTRACT FROM AUTHOR]

Recently, Baczyński, Drygaś, Król and Mesiar have proposed two constructions of ordinal sums of fuzzy implications based on the construction of ordinal sum of overlap functions. In this paper, we study the class of ordinal sum implications constructed using the first method, and show that this new class of fuzzy implications is different from the known (S , N) -, R -, QL -, f - and g -implications. Furthermore, we explore this new class of ordinal sum implications with respect to distributivity. [ABSTRACT FROM AUTHOR]

Our starting point are several general classes of real functions defined on the unit square satisfying some basic properties such as a boundary condition or several types of monotonicity and continuity. Applying to these functions some parameterized transformations and other constructions such as the transpose and flipping (which describe different aspects of symmetry) and truncation, we ask for conditions yielding (again) a bivariate copula. Some of these transformations are involutive (on one or more classes of functions), others are not even injective, and occasionally they induce additional properties, yielding, e.g., a (quasi-)copula. For several typical scenarios we identify the (not necessarily convex) sets of parameters leading to a copula and conditions imposing a minimal set of parameters. [ABSTRACT FROM AUTHOR]

The non-commutative, idempotent, associative functions are studied. It is well known that each commutative, idempotent (internal), associative function can be represented by a partial (linear) order. In this work it is shown that each non-commutative, idempotent (internal), associative function can be represented by a (linear) pair-order. Moreover, each internal, associative function can be expressed as an ordinal sum of trivial semigroups and semigroups, where the corresponding semigroup operation is the projection to one of the coordinates. Vice versa, each linear pair-order induces a unique internal, associative function and a condition under which each (non-linear) pair-order defined on a chain induces a unique monotone, idempotent, associative function is also introduced. Several examples of non-commutative, idempotent, associative functions and related pair-orders are shown. [ABSTRACT FROM AUTHOR]

Although Fang provided a systematic characterization of the α -cross-migrativity property of t-conorms over fuzzy implications, there are some faults in the results obtained by Fang, such as insufficient conditions, redundant conditions, false proofs and faulty conclusions. Thus we rectify those conclusions to remedy those defects in this paper. For the convenience of readers, we also present a table to show the correspondences between those faulty results obtained by Fang and the corrected conclusions. [ABSTRACT FROM AUTHOR]

Abstract Overlap functions, as a new special class of binary aggregation functions, have been investigated in many recent works for applications in image processing, classification problems and decision making. In addition, α -migrativity as a vital property of binary functions on unit interval has been discussed in the literature. In particular, recently, an interesting and natural research topic for α -migrativity is the investigation of the generalized forms of α -migrativity for one peculiar binary aggregation functions over any fixed special binary aggregation function (see, e.g., the α -migrativity of t-norms over any fixed t-norm, t-conorms over any fixed t-conorm, uninorms over any fixed uninorm, and uninorms and nullnorms, respectively, over any fixed t-norm or t-conorm.). Thus, in this paper, we continue to study this research topic for overlap functions. At first, we generalize the α -migrativity of any overlap function O from the usual formula O (α x , y) = O (x , α y) to the so-called (α , O ⁎ , O †) -migrativity O (O ⁎ (α , x) , y) = O (x , O † (α , y)) , where O ⁎ and O † are two fixed overlap functions. And then, we investigate the (α , O ⁎ , O †) -migrativity of an overlap function by taking O ⁎ and O † as the minimum overlap function and give an equivalent characterization of it by the ordinal sum of overlap functions. In addition, we propose the (α , O ⁎ , O †) -migrativity of an overlap function by taking O ⁎ and O † as the p -product overlap function and show an equivalent characterization of it by its additive generator pair. In particular, we obtain an equivalent characterization of the usual α -migrativity of an overlap function by its additive generator pair. Finally, we discuss the (α , O ⁎ , O †) -migrativity of an overlap function by taking O ⁎ and O † as the 1-product overlap function and p -product overlap function, respectively, and give two characterizations of it by its additive generator pair. [ABSTRACT FROM AUTHOR]

We characterize all pseudo-uninorms with continuous Archimedean underlying functions. By discussing all possible combinations of strict and nilpotent underlying functions we show that a pseudo-uninorm with continuous Archimedean underlying functions is not a uninorm only in the case when both underlying functions are strict. Moreover, we show that in the case of pseudo-uninorms with continuous Archimedean underlying functions the set of points where the commutativity could be violated reduces to { (0 , 1) , (1 , 0) }. [ABSTRACT FROM AUTHOR]

In this study, we discuss the weak dominance relation between two binary operations which is closely related with the dominance relation. First, we present some basic properties of the weak dominance and prove that there exists no t-norm weakly dominating a t-conorm. Then we provide the necessary and sufficient conditions of a t-conorm weakly dominating a t-norm for different cases: both are Archimedean or ordinal sums. Lastly, the weak dominance between t-norms is discussed. [ABSTRACT FROM AUTHOR]

In an ordinal sum of two combinatorial games G and H , denoted by G : H , a player may move in either G (base) or H (subordinate), with the additional constraint that any move on G completely annihilates the component H . It is well-known that the ordinal sum does not depend on the form of its subordinate, but depends on the form of its base. In this work, we analyze G ( G : H ) where G and H are impartial forms, observing that the G -values are related to the concept of minimum excluded value of order k . As a case study, we introduce the ruleset oak , a generalization of green hackenbush . By defining the operation gin sum , it is possible to determine the literal forms of the bases in polynomial time. [ABSTRACT FROM AUTHOR]

The ordinal sum of triangular norms on unit interval has been used to construct other triangular norms. But, in general, ordinal sum construction of triangular norms and triangular conorms may not work on an arbitrary bounded lattice. In this study, we focus on ordinal sums of triangular norms and triangular conorms on an arbitrary bounded lattice. Moreover, we show a complete generalization of our ordinal sums of triangular norms and triangular conorms on an arbitrary bounded lattice. And some illustrative examples are added for clarity. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The monotonicity of functions defined on the unit interval, constructed via (z)-ordinal sum is discussed. In Part I of this two-part paper we characterize all non-decreasing functions defined on the unit interval which are constructed by a non-trivial ordinal sum of semigroups. We also give necessary and sufficient conditions for a function constructed via ordinal sum to be monotone. In Part II we describe the structure of a monotone function defined on the unit interval which is constructed via z -ordinal sum construction with respect to a finite branching set and we give necessary and sufficient conditions for a function constructed via z -ordinal sum to be monotone in the case when intermediate condition is fulfilled. The case when intermediate condition is not fulfilled is discussed as well. [ABSTRACT FROM AUTHOR]

This is the second part of a two-part paper which discusses monotonicity of functions defined on the unit interval, constructed via (z)-ordinal. In Part I we characterized all non-decreasing functions defined on the unit interval which are constructed by a non-trivial ordinal sum of semigroups and gave necessary and sufficient conditions for a function constructed via ordinal sum to be monotone. In the present Part II we describe the structure of a monotone function defined on the unit interval which is constructed via z -ordinal sum construction with respect to a finite branching set and we give necessary and sufficient conditions for a function constructed via z -ordinal sum to be monotone in the case when the intermediate condition is fulfilled. The case when the intermediate condition is not fulfilled is discussed as well. [ABSTRACT FROM AUTHOR]

As a weaker form of the classical commuting equation, α -cross-migrativity properties between conjunctive connectives including t-norms, uninorms, and overlap functions have been extensively investigated. However, there has been no comparative analysis of the α -cross-migrativity between disjunctive connectives and fuzzy implications. The work is dedicated to the study of α -cross-migrativity involving t-conorms and fuzzy implications. The investigation is presented in two separate parts: the first part focuses on the case that fuzzy implication satisfies some property, especially, the order property. The second one deals with the situation where fuzzy implication belongs to some special classes, i.e., (S , N) -implications, R -implications, and Yager's implications (i.e., f - and g -generated implications). Some interesting results are obtained: full characterizations of α -cross-migrativity for continuous t-conorms over fuzzy implications satisfying the order property, (S , N) -implications, R -implications induced by border continuous t-norms and Yager's implications. As a by-product, some constructions of ordinal sum t-conorms and ordinal sum fuzzy implications satisfying α -cross-migrativity are obtained. Moreover, some supporting examples for solutions are given. [ABSTRACT FROM AUTHOR]

In this paper, we present several necessary and sufficient conditions to ensure an ordinal sum of t-norms is still a t-norm on bounded lattices. For a fixed or an arbitrary bounded sublattice of a bounded lattice, we provide a direct characterization theorem, respectively. Meanwhile, we extend these results to the ordinal sum for a family of pairwise disjoint bounded sublattices. Furthermore, we investigate ordinal sum of t-norms on product lattices and reveal that there exist ordinal sum t-norms on non-trivial bounded sublattices of product lattices, which is different from the previous results of ordinal sum of t-norms defined on subintervals. [ABSTRACT FROM AUTHOR]

In this paper, the ordinal sum of a family of given triangular norms on subintervals in a bounded meet semi-lattice is constructed when the subintervals are pairwise non-overlapped and all the endpoints of the subintervals constitute a chain. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The continuous extensions of a t-norm know only on some subregion of the unit interval to the whole unit interval are studied. The conditions which a t-norm T 1 should satisfy on [ a , b ] 2 necessary for the existence of a continuous t-norm T 2 that coincides with T 1 on [ a , b ] 2 are given. Construction of such a continuous t-norm T 2 is described. [ABSTRACT FROM AUTHOR]

Triangular norms and conorms with continuous diagonals are discussed. In literature we can find examples of non-continuous t-norms with continuous diagonals, however, their deeper study is still missing. In this paper we introduce a sufficient condition under which a t-norm with continuous diagonal is continuous. Moreover, we show that an Archimedean t-norm continuous on the boundary is continuous. Several illustrative examples are also included. [ABSTRACT FROM AUTHOR]

The idempotent uninorms are characterized by means of the ordinal sum of Clifford. It is shown that idempotent uninorms are in one-to-one correspondence with special linear orders on [ 0 , 1 ] . A connection between respective linear order on [ 0 , 1 ] and the characterizing multi-function of the uninorm is also investigated. [ABSTRACT FROM AUTHOR]

Copulas are nothing else but supermodular functions with absorbing element 0 and neutral element 1. Although a copula C cannot be modular on the unit square itself, it is effectively so on every rectangle with zero C -volume. In this paper, we show that an appropriate modular function on the unit square can be transformed into a copula by performing a simple truncation operation (either from below or above to ensure the compatibility with the Fréchet–Hoeffding bounds). Modular functions on the unit square admit an additive representation in terms of two univariate functions. We specifically focus on cases where these univariate functions are sections (horizontal, vertical, diagonal) of a given copula. Ample illustrations are provided. [ABSTRACT FROM AUTHOR]

Overlap functions are a particular type of aggregation functions, given by increasing continuous commutative bivariate functions defined over the unit square, satisfying appropriate boundary conditions. Overlap functions are applied mainly in classification problems, image processing and in some problems of decision making based on some kind of fuzzy preference relations, in which the associativity property is not strongly required. Moreover, the class of overlap functions is reacher than the class of t-norms, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap functions. This flexibility of overlap functions increases their applicability. The aim of this papers is to introduce the concept of Archimedean overlap functions, presenting a study about the cancellation, idempotency and limiting properties, and providing a characterization of such class of functions. The concept of ordinal sum of overlap functions is also introduced, providing constructing/representing methods of certain classes of overlap functions related to idempotency, cancellation, limiting and Archimedean properties. [ABSTRACT FROM AUTHOR]