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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.

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 Dvořak and Holcapek 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.

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.

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.

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.

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. Karacal and R. Mesiar (2015) [16] as well as by G.D. Cayli 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.

Fuzzy relations and, in particular, fuzzy order relations, have been largely studied since Zadeh first introduced the concept back in 1971. Bearing in mind that betweenness relations and their relationship with order relations have historically attracted the attention of mathematicians, it is very surprising that the study of fuzzy betweenness relations has had a very limited scientific reach so far, and barely no attention has been given to their relationship with fuzzy order relations. In this manuscript, we study fuzzy betweenness relations in depth and discuss their connection with both schools of thought in the study of fuzzy order relations: the already mentioned one introduced by Zadeh and the one initiated by Hohle and Blanchard and popularized by Bodenhofer.

First, the notion of a family of generalized L-fuzzy norms on vector spaces is proposed. Then, the L-topology induced by a family of generalized L-fuzzy norms is investigated. Finally, a necessary and sufficient condition for an L-topological vector space to be generalized L-fuzzy normable is obtained, which can be viewed as the generalized normability theorem for L-topological vector spaces compared with that by Yan and Fang in 2002.

In this paper, the concept of a family of L -fuzzy pseudo-norms on vector spaces is proposed and the characterization of L -vector topologies in terms of a family of L -fuzzy pseudo-norms is presented. As applications of the characterization, the Hausdorff separation property, convergence of molecule nets and boundedness of L -sets in L -topological vector spaces are investigated.

In this paper, the notion of generalized locally bounded L-topological vector spaces is proposed. The relationship between this kind of space and locally bounded L-topological vector space as defined by Yan and Fang [Locally bounded L-topological vector spaces, Inf. Sci. 159 (2004) 273-281] is investigated. In addition, the concept of a family of generalized L-fuzzy quasi-norms is introduced. Based on this notion, generalized locally bounded L-topological vector spaces are characterized. Finally, the Hausdorff separation property, convergence of molecule nets and boundedness of L-fuzzy sets in generalized locally bounded L-topological vector spaces are studied.

In this paper, a characterization for an I(L)-topological space to be generated by a given co-tower of L-topological spaces is obtained. Moreover, the relationship between some properties of an I(L)-topological vector space generated by a co-tower of L-topological vector spaces and the corresponding properties of the given co-tower of L-topological vector spaces is investigated. Our results show that if an I(L)-topological vector space generated by a co-tower of L-topological vector spaces has some properties, such as local convexity and local boundedness, then all L-topological vector spaces in the co-tower also have the same properties. But the converse is incorrect even in the case of I-topological vector space generated by a co-tower of classical topological vector spaces. Finally, we supply a necessary and sufficient condition for an I(L)-topological vector space generated by a co-tower of L-topological vector spaces with some properties, such as local convexity and local boundedness, to have such properties too.

In this paper, a new definition of locally convex L-topological vector spaces is given. The relationship between this new definition and the previous definition of locally convex L-topological vector spaces given by Yan and Fang in 1999 is investigated. Moreover, the concept of generalized L-fuzzy semi-norm is introduced. By using a family of generalized L-fuzzy semi-norms, a characterization of the new locally convex L-topological vector spaces is presented. Finally, as applications of this characterization, the Hausdorff separation property, convergence of molecule nets and boundedness of L-fuzzy sets in locally convex L-topological vector spaces are studied.

In this paper, we discuss the relationship between an I(L)-topological group and its level L-topological groups. We supply a method to generate an I(L)-topological group by a co-tower of group L-topologies and obtain a necessary and sufficient condition for the I(L)-topological group generated by a co-tower of group L-topologies to be L-topologically generated.

In this paper, we prove that an L-topological vector space is locally convex (locally bounded) iff its induced I(L)-topological vector space is locally convex (locally bounded), where I(L) is the L-fuzzy unit interval. Also, we give a necessary and sufficient condition for the induced I(L)-topological vector space to be I(L)-fuzzy normable.