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We present a complete representation theorem for nullnorms on bounded lattices. Explicitly, any nullnorm on a bounded lattice can be represented in terms of two order-preserving maps, a triangular conorm, a triangular norm and a conditionally associative function. As a particular case, we retrieve the known representation of nullnorms only taking values that are comparable with the annihilator (in terms of two order-preserving maps, a triangular conorm and a triangular norm). As an illustration of the representation theorem, we generate construction methods for nullnorms, especially those taking at least one value that is incomparable with the annihilator. [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]

Internal uninorms on a bounded lattice always output one of the two input values and are nothing else but idempotent uninorms when the lattice is a chain. In this paper, we study the existence and representation of internal uninorms with a given neutral element e on a bounded lattice. We obtain a necessary condition for the existence of such an internal uninorm. In the case that those elements that are incomparable with the neutral element (collected in the set I e) constitute a chain, we prove that this necessary condition also becomes a sufficient one and allows to represent any internal uninorm on a bounded lattice in terms of an internal uninorm on the bounded chain obtained by deleting I e from the lattice and an internal quasi-uninorm on I e connected via a family of lower sets and a family of upper sets indexed by I e. [ABSTRACT FROM AUTHOR]

We introduce three broad classes of nullnorms on a bounded lattice and lay bare the structure of their members. For that purpose, we introduce particular subsets of a bounded lattice, called upper (resp. lower) beams, and appropriate associative operations on them, called beam (resp. dual beam) operations, which conveniently generalize triangular norms (resp. triangular conorms). It is shown that nullnorms in the first (resp. second) class are characterized by a triangular conorm (resp. triangular norm) and a beam operation (resp. dual beam operation), while nullnorms in the third class are characterized by a triangular conorm and a triangular norm. We also discuss the relationships among these three classes. [ABSTRACT FROM AUTHOR]

As generalizations of the classes Umin and Umax of uninorms on the real unit interval, we introduce the classes Umin and Umax of uninorms on a bounded lattice and lay bare the structure of their members. It is shown that uninorms in Umin (resp. Umax) are characterized by a triangular conorm (resp. triangular norm) and a triangular subnorm (resp. triangular superconorm). The characterization theorems unify all the existing construction methods for these two classes of uninorms known in the literature. We also introduce four new related classes of uninorms on a bounded lattice and present dedicated construction methods using triangular subnorms and triangular superconorms. Finally, connections among the introduced classes of uninorms are discussed. [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]

Recently, Wang et al. proposed the notion of a uni-nullnorm on a bounded lattice and provided two construction methods. In this paper, we point out that the first construction method is invalid since it does not always lead to a uni-nullnorm. Then we present a method to construct a uni-nullnorm on a bounded lattice via a disjunctive uninorm and a triangular norm. Making use of existing construction methods for uninorms on a bounded lattice, we derive several construction methods for uni-nullnorms, including the second construction method by Wang et al. Dually, we present a method to construct a null-uninorm on a bounded lattice via a conjunctive uninorm and a triangular conorm, as well as several derived construction methods for null-uninorms. [ABSTRACT FROM AUTHOR]

In this paper, via closure (resp. interior) operators on a bounded lattice L , we propose a rather effective method to construct uninorms on L with a given triangular norm T (resp. triangular conorm S) on the subinterval 0 , e (resp. e , 1) of L. Our method encompasses as special cases the construction methods introduced by F. Karaçal and R. Mesiar (2015) [16] as well as by G.D. Çaylı et al. (2016) [5]. We also exploit a pair of novel closure and interior operators on complete lattices and utilize this pair of operators to construct uninorms. Several interesting examples are included to illustrate our method. [ABSTRACT FROM AUTHOR]

Recently, we presented a complete representation for nullnorms on bounded lattices, a class of ordered semigroups, in terms of two order-preserving maps, a triangular conorm, a triangular norm and a conditionally associative function, which together satisfy seven conditions. In this paper, we illustrate that each of these seven conditions is independent of the other ones by rewriting the seventh condition. We also discuss in which cases the sixth or the seventh condition becomes dependent. [ABSTRACT FROM AUTHOR]