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We discuss the structure of idempotent uninorms defined on bounded lattices such that the set of all points incomparable with the neutral element contains at most one point. We show that the structure of idempotent uninorms on lattices with one point incomparable with the neutral element is much more complex than the structure of those defined on lattices where all points are comparable with the neutral element. In Part I of this two-part paper, we completely characterize idempotent uninorms defined on bounded lattices such that all points are comparable with the neutral element. Moreover, we show some basic properties and observations on idempotent uninorms on bounded lattices with one point incomparable with the neutral element and we completely characterize the class of internal uninorms on such lattices. In Part II, we will characterize two special classes of idempotent uninorms defined on bounded lattices with one point incomparable with the neutral element, including idempotent uninorms with annihilator incomparable with the neutral element, and by the composition of these special cases, we will show the complete characterization of idempotent uninorms on such lattices. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of conditional monotonicity and other related relaxed monotonicities within the framework of intervals equipped with admissible orders. It generalizes the work of Sesma-Sara et al., who introduced weak/directional monotonicity on intervals endowed with the Kulisch–Miranker order, and the work of Santiago et al., who introduced the notion of g-weak monotonicity in the fuzzy setting. The paper also explores properties of conditional monotonicities, introduce the notion of ordinal sum for a family of functions and examines the connections between conditional monotonicity, ordinal sums and implications. [ABSTRACT FROM AUTHOR]

Aggregation operators have the important application in any fields where the fusion of information is processed. The dominance relation between two aggregation operators is linked to the fusion of fuzzy relations, indistinguishability operators and so on. In this paper, we deal with the weak dominance relation between two aggregation operators which is closely related with the dominance relation. Weak domination of isomorphic aggregation operators and ordinal sum of conjunctors is presented. More attention is paid to the weak dominance relation between ordered weighted averaging operators and Lukasiewicz t-norm. Furthermore, the relationships between weak dominance and some functional inequalities of aggregation operators are discussed. [ABSTRACT FROM AUTHOR]

In this paper, the class of S-generalized distances such that the involved t-conorms S are ordinal sums is discussed. It is shown that these S-generalized distances can be thought of as families of generalized distances with respect to some Archimedean t-conorms. We also deal with the S-generalized distance aggregations, which merge a family of Si-generalized distances into a new S-generalized distance. [ABSTRACT FROM AUTHOR]

We define several sorts of mappings on a poset like monotone, strictly monotone, upper cone preserving and variants of these. Our aim is to study in which posets some of these mappings coincide. We define special mappings determined by two elements and investigate when these are strictly monotone or upper cone preserving. If the considered poset is a semilattice then its monotone mappings coincide with semilattice homomorphisms if and only if the poset is a chain. Similarly, we study posets which need not be semilattices but whose upper cones have a minimal element. We extend this investigation to posets that are direct products of chains or an ordinal sum of an antichain and a finite chain. We characterize equivalence relations induced by strongly monotone mappings and show that the quotient set of a poset by such an equivalence relation is a poset again.

We introduce and characterize various gluing constructions for residuated lattices that intersect on a common subreduct, and which are subalgebras, or appropriate subreducts, of the resulting structure. Starting from the 1-sum construction (also known as ordinal sum for residuated structures), where algebras that intersect only in the top element are glued together, we first consider the gluing on a congruence filter, and then add a lattice ideal as well. We characterize such constructions in terms of (possibly partial) operators acting on (possibly partial) residuated structures. As particular examples of gluing constructions, we obtain the non-commutative version of some rotation constructions, and an interesting variety of semilinear residuated lattices that are 2-potent. This study also serves as a first attempt toward the study of amalgamation of non-commutative residuated lattices, by constructing an amalgam in the special case where the common subalgebra in the V-formation is either a special (congruence) filter or the union of a filter and an ideal. [ABSTRACT FROM AUTHOR]

In this paper, the ordinal sum of quantales is provided. Based on this, the ordinal sums of many familiar ordered algebraic structures can be obtained, such as complete residuated lattice, spatial quantale, etc. Then we investigate the relationship between the quantale completion of partially ordered semigroups and the ordinal sum, and prove that for any family of two-side posemigroups, the ordinal sum of their quantale completions coincides with the quantale completion of their ordinal sum. [ABSTRACT FROM AUTHOR]

The notion of a pre-period of an algebra A is defined by means of the notion of the pre-period λ(f) of a monounary algebra hA; fi: it is determined by sup{λ(f) | f is an endomorphism of A}. In this paper we focus on the pre-period of a finite modular lattice. The main result is that the pre-period of any finite modular lattice is less than or equal to the length of the lattice; also, necessary and sufficient conditions under which the pre-period of the glued sum is equal to the length of the lattice, are shown. Moreover, we show the triangle inequality of the pre-period of the glued sum. [ABSTRACT FROM AUTHOR]

The ordinal sum construction is an important way to generate new triangular norms (t-norms for short) on real unit interval from existing ones. Saminger extended the ordinal sum construction of t-norms to bounded lattices in a direct way and proved that, except for very special cases the ordinal sum of t-norms on bounded lattices may be not a t-norm. To ensure the ordinal sum of t-norms is always a t-norm, several modified ordinal sum of t-norms have been developed. This note reviews several such constructions. As a byproduct, we point out that some recent results concerning ordinal sum of t-norms by A¸sici can be seen as corollaries of these construction. [ABSTRACT FROM AUTHOR]

As a proper generalization of the ordinal sum t-norm construction on bounded lattices proposed in [E. Aşıcı, R. Mesiar, New constructions of triangular norms and triangular conorms on an arbitrary bounded lattice, International Journal of General Systems, 49(2) (2020), 143-160], the present paper studies a new extension of a triangular norm on a subinterval [0, α] via an interior operator to the underlying entire bounded lattice, where the necessary and sufficient conditions under which the constructed operation is again a t-norm are given. By comparing the graphic structures of two t-norms on a common bounded lattice which are constructed in different ways, it is shown that the new method in this paper is essentially different from the ones existing in the literature. As an end, this new construction is generalized to construct ordinal sums of finitely many t-norms by recursion on bounded lattices. The dual results for ordinal sum construction of t-conorms via closure operators on bounded lattices are also presented. [ABSTRACT FROM AUTHOR]

This paper focuses on the topic of ordinal sums of semigroups in the sense of A. H. Clifford – a method for constructing a new semigroup from a given system of semigroups indexed by a linearly ordered index set. We completely describe the linearly ordered index set for an ordinal sum of semigroups yielding a uninorm. [ABSTRACT FROM AUTHOR]

We define several sorts of mappings on a poset like monotone, strictly monotone, upper cone preserving and variants of these. Our aim is to study in which posets some of these mappings coincide. We define special mappings determined by two elements and investigate when these are strictly monotone or upper cone preserving. If the considered poset is a semilattice then its monotone mappings coincide with semilattice homomorphisms if and only if the poset is a chain. Similarly, we study posets which need not be semilattices but whose upper cones have a minimal element. We extend this investigation to posets that are direct products of chains or an ordinal sum of an antichain and a finite chain. We characterize equivalence relations induced by strongly monotone mappings and show that the quotient set of a poset by such an equivalence relation is a poset again. [ABSTRACT FROM AUTHOR]

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]

A prominent example of a perturbation of the bivariate product copula (which characterizes stochastic independence) is the parametric family of Eyraud-Farlie-Gumbel-Morgenstern copulas which allows small dependencies to be modeled. We introduce and discuss several perturbations, some of them perturbing the product copula, while others perturb general copulas. A particularly interesting case is the perturbation of the product based on two functions in one variable where we highlight several special phenomena, e.g., extremal perturbed copulas. The constructions of the perturbations in this paper include three different types of ordinal sums as well as flippings and the survival copula. Some particular relationships to the Markov product and several dependence parameters for the perturbed copulas considered here are also given.

This paper is a contribution to the topic of ordinal sums of semigroups in the sense of A. H. Clifford. The goal of this article is twofold. Firstly, we present sufficient and necessary conditions for obtaining uninorms as an ordinal sum of semigroups, of which the underlying sets are non-empty subintervals of the unit interval [0, 1]. Secondly, we show that each uninorm locally internal on A(e) can be decomposed into an ordinal sum of such semigroups. [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 ordinal sum of t-norms on a bounded lattice has been used to construct other t-norms. However, an ordinal sum of binary operations (not necessarily t-norms) defined on the fixed subintervals of a bounded lattice may not be a t-norm. Some necessary and sufficient conditions are presented in this article for ensuring that an ordinal sum on a bounded lattice of two binary operations is, in fact, a t-norm. In particular, the results presented here provide an answer to an open problem put forward by Ertuğrul and Yeşilyurt, ordinal sums of triangular norms on bounded lattices, Inf. Sci., vol. 517, (2020) 198–216. [ABSTRACT FROM AUTHOR]

Recently, the topic of construction methods for triangular norms (triangular conorms), uninorms, nullnorms, etc. has been studied widely. In this paper, we propose construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices by using interior and closure operators, respectively. Thus, we obtain some proposed methods given by Ertuğrul, Karacal, Mesiar [15] and Çaylı [8] as results. Also, we give some illustrative examples. Finally, we conclude that the introduced construction methods can not be generalized by induction to a modified ordinal sum for t-norms and t-conorms on bounded lattices. [ABSTRACT FROM AUTHOR]

We characterize all functions that can be obtained as a z-ordinal sum of semigroups related to continuous t-norms, t-conorms, representable uninorms and idempotent semigroups. We show that this class of functions is bigger than the class of n-uninorms with continuous underlying functions. Vice versa, we show that the characterization of n-uninorms with continuous underlying functions via z-ordinal sum can be extended to any commutative, associative and nondecreasing binary function on the unit interval, which has continuous Archimedean components and is continuous on the diagonal. [ABSTRACT FROM AUTHOR]

Overlap functions, as one kind of special binary aggregation functions, have been continuously concerned and studied by many scholars and widely applied in classification, decision making, image processing, and fuzzy community detection problems. Meanwhile, the migrativity as a vital, and particularly, interesting property for binary aggregation functions, has been studied in the literature since it was proposed. This article continues to consider this research topic and mainly focuses on the $\alpha$ -cross-migrativity for overlap functions. First, we introduce the concept of the $\alpha$ -cross-migrativity for overlap functions and show some vital properties of $\alpha$ -cross-migrative overlap functions. Second, we study the overlap functions that are $\alpha$ -cross-migrative with respect to the minimum overlap function $O_M$ and obtain an equivalent characterization of them by the ordinal sum of overlap functions. Third, we prove that an overlap function is $\alpha$ -cross-migrative with respect to the $p$ -product overlap function $O_p$ only if $p$ is equal to 1 and give an equivalent characterization of the overlap functions that satisfy the $\alpha$ -cross-migrativity with respect to the 1-product overlap function $O_1$ also by the ordinal sum of overlap functions. Finally, we extend the $\alpha$ -cross-migrativity of overlap functions to the 0-overlap functions status and get two equivalent characterizations of the 0-overlap functions that are $\alpha$ -cross-migrative with respect to $O_M$ and an equivalent characterization of the 0-overlap functions that satisfy the $\alpha$ -cross-migrativity with respect to $O_1$ , respectively. [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]

We discuss ordinal sums as one of powerful tools in the aggregation theory serving, depending on the context, both as a construction method and as a representation, respectively. Up to recalling of several classical results dealing with ordinal sums, in particular dealing, e.g., with continuous t-norms, copulas, or recent results, e.g., concerning uninorms with continuous underlying functions, we present also several new results, such as the uniqueness of the link between t-norms or t-conorms, and related Archimedean components, problems dealing with the cardinality of the considered index sets in ordinal sums, or infinite ordinal sums of aggregation functions covering by one type of ordinal sums both t-norms and t-conorms ordinal sums. [ABSTRACT FROM AUTHOR]

In 2015, a new class of fuzzy implications, called ordinal sum implications, was proposed by Su et al. They then discussed the distributivity of such ordinal sum implications with respect to t-norms and t-conorms. In this paper, we continue the study of distributivity of such ordinal sum implications over two newly-born classes of aggregation operators, namely overlap and grouping functions, respectively. The main results of this paper are characterizations of the overlap and/or grouping function solutions to the four usual distributive equations of ordinal sum fuzzy implications. And then sufficient and necessary conditions for ordinal sum implications distributing over overlap and grouping functions are given. [ABSTRACT FROM AUTHOR]

In this paper, we give construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on appropriate bounded lattices. Then, we compare our methods and well-known methods proposed in [2, 8, 19]. Finally, we give different construction methods for t-norms and t-conorms on an appropriate bounded lattice by using recursion. Also, we provide some examples to discuss introduced methods. [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 present article proposes two construction ways to study the general forms of ordinal sums of fuzzy implications with the intent of unifying the ordinal sums existing in the literature. The first ordinal sum construction way, which we call “Implication Complementing,” is to study how to complement a specific fuzzy implication to the linear transformations of given fuzzy implications defined on respective disjoint subsquares whose principal diagonals are segments of the principal diagonal of the unit square, in order that the resulting ordinal sum on the unit square is a fuzzy implication. The second way, which we call “Implication Reconstructing,” is to study how to reconstruct an initial fuzzy implication through replacing its some values on given rectangular regions of the unit square with the linear transformations of respective given fuzzy implications such that the redefined function is a new fuzzy implication. In both ways, necessary and sufficient conditions for the final reconstructed functions to be fuzzy implications are given and several new constructions of ordinal sums of fuzzy implications are obtained, which would generalize the existing ordinal sums from several aspects. In particular, by adopting the idea behind the second way, the generalized ordinal sums of fuzzy implications are proposed, in which the regions where the linear transformations of given fuzzy implications are defined are neither necessarily subsquares nor necessarily along the principal or minor diagonal of the unit square. [ABSTRACT FROM AUTHOR]

Recently, the topic related to the construction of triangular norms and triangular conorms on bounded lattices using ordinal sums has been extensively studied. In this paper, we introduce a new ordinal sum construction of triangular norms and triangular conorms on an appropriate bounded lattice. Also, we give some illustrative examples for clarity. Then, we show that a new construction method can be generalized by induction to a modified ordinal sum for triangular norms and triangular conorms on an appropriate bounded lattice, respectively. And we provide some illustrative examples. [ABSTRACT FROM AUTHOR]

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 U 2 (0 , 1) = z 1 (possibly restricted to open or half-open unit square [ 0 , 1 [ 2 , ] 0 , 1 ] 2 ). Similar results are shown for n-uninorms with continuous underlying functions, where n ∈ N , n > 2. [ABSTRACT FROM AUTHOR]

In this paper, we study t-norms and t-conorms on bounded lattices. We propose new methods for generating these operators, applicable on any bounded lattice M by use of the presence of a t-norm on [0M, k] and a t-conorm on [k, 1M] for an element k ∊ M\{0M, 1M}. In addition, some corresponding examples are provided for well understanding the structure of new t-norms and t-conorms. [ABSTRACT FROM AUTHOR]