Fuzzy Posets Research Articles

Direct and ordinal products realized by triangular norm operators with no zero divisors

In this note we continue the work of Chon, as well as Mezzomo, Bedregal, and Santiago, by studying algebraic operations on fuzzy posets and bounded fuzzy lattices. We first prove that fuzzy posets are closed under finite direct products whenever the triangular norm realizing the product construction has no zero divisors. This result is then extended to the case of bounded fuzzy lattices. Some immediate consequences are then obtained within the setting of direct products realized by triangular norms with no nilpotent elements as well as strictly monotone and cancellative triangular norms. We then introduce a triangular norm based construction of ordinal products and similarly show that fuzzy posets are closed under ordinal products whenever the triangular norm realizing the product construction has no zero divisors.

The Intrinsic Characterization of a Fuzzy Consistently Connected Domain

The concepts of a fuzzy connected set (fc set) and a fuzzy consistently connected set (fcc set) are introduced on fuzzy posets, along with a discussion of their basic properties. Inspired by some equivalent conditions of crisp connected sets, the characterizations of the fc sets are given, and we also explore fuzzy completeness and fuzzy compactness in addition to defining a new fuzzy way-below relation based on fcc complete sets. Using this relationship as a basis, the fcc domain is also provided and studied, and its equivalent characterizations are obtained. In summary, we develop a method to establish fcc completeness from a continuous poset.

C-continuous fuzzy posets

In this paper, we develop the C-continuity via Scott Q-cotopological spaces, where Q is a commutative and integral quantale.The main results are: (1) The set of all C-compact elements of a completeQ-ordered set is irreducible complete; (2) The Scott Q-cotopology of a Q-ordered set under the fuzzy inclusion order is C-continuous, evenC-prealgebraic; (3) We establish an adjunction between the category of irreducible complete Q-ordered sets and the category of C-prealgebraic complete Q-ordered sets.

Fuzzy closure operators and their applications

Fuzzy closure operators play a significant role in fuzzy order theory. This paper aims to further enrich and improve the study of fuzzy closure operators. Based on the work of Belohlavek and Yao, we shall continue to study the relative properties of fuzzy closure operators. First, we shall consider the extensions of $L$-subsets via fuzzy closure operators. Then we give an application of fuzzy closure operators, that is, by fuzzy closure operators we shall prove that the category CFPos of complete fuzzy posets and their fuzzy-join preserving maps is a reflective full subcategory of FPos, where FPos denotes the category of fuzzy posets and their fuzzy-existing-join preserving maps.

Quasi-closed elements in fuzzy posets

We generalize the notion of quasi-closed element to fuzzy posets in two stages: First, in the crisp style in which each element in a given universe either is quasi-closed or not. Second, in the graded style by defining degrees to which an element is quasi-closed. We discuss the different possible definitions and comparing them with each other. Finally, we show that the most general one has good properties to be used when we have a complete fuzzy lattice as a frame.

The ZL-completions of fuzzy posets

The ZL-completions of fuzzy posets

The Relations between Residuated Frames and Residuated Connections

We introduce the notion of (dual) residuated frames as a viewpoint of relational semantics for a fuzzy logic. We investigate the relations between (dual) residuated frames and (dual) residuated connections as a topological viewpoint of fuzzy rough sets in a complete residuated lattice. As a result, we show that the Alexandrov topology induced by fuzzy posets is a fuzzy complete lattice with residuated connections. From this result, we obtain fuzzy rough sets on the Alexandrov topology. Moreover, as a generalization of the Dedekind–MacNeille completion, we introduce R-R (resp. D R - D R ) embedding maps and R-R (resp. D R - D R ) frame embedding maps.

Prime (distributive) fuzzy posets

Prime (distributive) fuzzy posets

一致模糊极小集及其序同态

一致模糊极小集及其序同态

一致模糊偏序集及其应用

一致模糊偏序集及其应用

Categorical dualities between certain kinds of fuzzy posets

Categorical dualities between certain kinds of fuzzy posets

Fuzzy cut-stable map and its extension property1

In this paper, based on a complete residuated lattice, we introduce the concept of fuzzy cut-stable maps and discuss the relationship between fuzzy cut-stable maps and some special maps. We also obtain the extension property of fuzzy cut-stable maps. Furthermore, it is proved that the category of fuzzy complete (continuous) lattices with fuzzy complete homomorphisms is a full reflective subcategory of the category of fuzzy (precontinuous) posets with fuzzy cut-stable maps.

The L-lim-inf-convergence in fuzzy posets

In domain theory, the lim-inf-convergence in posets was introduce to characterize continuous posets. In this paper, the concept of the L-lim-inf-convergence of nets (xi)i∈I in fuzzy posets is proposed and its relationship with continuous fuzzy posets is studied. It is shown that for an arbitrary fuzzy poset, the L-lim-inf-convergence is topological if and only if the fuzzy poset is continuous.

Fuzzy order congruences on fuzzy posets

Fuzzy order congruences play an important role in studying the categoricalproperties of fuzzy posets. In this paper, the correspondence between the fuzzyorder congruences and the fuzzy order-preserving maps is discussed. We focus onthe characterization of fuzzy order congruences on the fuzzy poset in terms ofthe fuzzy preorders containing the fuzzy partial order. At last, fuzzy completecongruences on fuzzy complete lattices are discussed.

L-upper Approximation Operators and Join Preserving Maps

In this paper, we investigate the properties of join and meet preserving maps in complete residuated lattice using Zhang’s the fuzzy complete lattice which is defined by join and meet on fuzzy posets. We define L-upper (resp. L-lower) approximation operators as a generalization of fuzzy rough sets in complete residuated lattices. Moreover, we investigate the relations between L-upper (resp. L-lower) approximation operators and L-fuzzy preorders. We study various L-fuzzy preorders on L X . They are considered as an important mathematical tool for algebraic structure of fuzzy contexts.

Fuzzy posets with fuzzy order applied to fuzzy ordered groups

In the framework of lattice valued structures we investigate fuzzy sub-posets of a given poset, in which the underlying set ornand the order are fuzzy and the reflexivity is specially defined. We introduce and investigate particular fuzzy sub-posets, e.g, fuzzy up-sets and down-sets, fuzzy convex sub-posets, fuzzy intervals etc. We describe the structure of the lattice of all fuzzy orders contained in a given crisp ordering. Then we apply these in defining a fuzzy ordered subgroup of an ordered group. Main features of fuzzy ordered subgroups are introduced and investigated like fuzzy positive and negative cone and fuzzy convex subgroups.

Fuzzy -Continuous Posets

The aim of this paper is to generalize fuzzy continuous posets. The concept of fuzzy subset system on fuzzy posets is introduced; some elementary definitions such as fuzzy -continuous posets and fuzzy -algebraic posets are given. Furthermore, we try to find some natural classes of fuzzy -continuous maps under which the images of such fuzzy algebraic structures can be preserved; we also think about fuzzy -continuous closure operators in alternative ways. An extension theorem is presented for extending a fuzzy monotone map defined on the -compact elements to a fuzzy -continuous map defined on the whole set.

The Linear Discrepancy of a Fuzzy Poset

In 2001, the notion of a fuzzy poset defined on a set X via a triplet (L, G, I) of functions with domain X <TEX>${\times}$</TEX> X and range [0, 1] satisfying a special condition L+G+I = 1 is introduced by J. Negger and Hee Sik Kim, where L is the 'less than' function, G is the 'greater than' function, and I is the 'incomparable to' function. Using this approach, we are able to define a special class of fuzzy posets, and define the 'skeleton' of a fuzzy poset in view of major relation. In this sense, we define the linear discrepancy of a fuzzy poset of size n as the minimum value of all maximum of I(x, y)<TEX>${\mid}$</TEX>f(x)-f(y)<TEX>${\mid}$</TEX> for f <TEX>${\in}$</TEX> F and x, y <TEX>${\in}$</TEX> X with I(x, y) > <TEX>$\frac{1}{2}$</TEX>, where F is the set of all injective order-preserving maps from the fuzzy poset to the set of positive integers. We first show that the definition is well-defined. Then, it is shown that the optimality appears at the same injective order-preserving maps in both cases of a fuzzy poset and its skeleton if the linear discrepancy of a skeleton of a fuzzy poset is 1.

An approach to fuzzy frames via fuzzy posets

In this paper, based on a complete Heyting algebra L , we define a fuzzy version of frames, called L - frames. Then we construct an adjunction between the category of stratified L - topological spaces and that of L - locales (the opposite category of L - frames), which is a fuzzy counterpart of the Isbell-adjunction between topological spaces and locales. We also study the corresponding sobriety of stratified L - topologies and spatiality of L - frames and show the dual equivalence between them.

Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed complete posets

This paper deals with quantitative domain theory via fuzzy sets. It examines the continuity of fuzzy directed complete posets (dcpos for short) based on complete residuated lattices. First, we show that a fuzzy partial order in the sense of Fan and Zhang and an L-order in the sense of Bělohlávek are equivalent to each other. Then we redefine the concepts of fuzzy directed subsets and (continuous) fuzzy dcpos. We also define and study fuzzy Galois connections on fuzzy posets. We investigate some properties of (continuous) fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy-double-downward-arrow-operator has a right adjoint. We define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy way-below relation is the smallest approximating fuzzy auxiliary relation.