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1
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Interval-Valued Fuzzy Soft Graphs

100%
Zihni O., Çelik Y., Kara G.
Topological Algebra and its Applications
|
2017
|
tom 5
|
nr 1
19-27
EN
In this paper, we combine concepts of interval-valued fuzzy soft sets and graph theory. Then we introduce notations of interval-valued fuzzy soft graphs and complete interval-valued fuzzy soft graphs. We also present several different types operations including cartesian product, strong product and composition on interval-valued fuzzy soft graphs and investigate some properties of them.
2
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Fuzzy ideals of ordered semigroups with fuzzy orderings

76%
Huang X., Li Q.
Open Mathematics
|
2016
|
tom 14
|
nr 1
841-856
EN
The purpose of this paper is to introduce the notions of ∈, ∈ ∨qk-fuzzy ideals of a fuzzy ordered semigroup with the ordering being a fuzzy relation. Several characterizations of ∈, ∈ ∨qk-fuzzy left (resp. right) ideals and ∈, ∈ ∨qk-fuzzy interior ideals are derived. The lattice structures of all ∈, ∈ ∨qk-fuzzy (interior) ideals on such fuzzy ordered semigroup are studied and some methods are given to construct an ∈, ∈ ∨qk-fuzzy (interior) ideals from an arbitrary fuzzy subset. Finally, the characterizations of generalized semisimple fuzzy ordered semigroups in terms of ∈, ∈ ∨qk-fuzzy ideals (resp. ∈, ∈ ∨qk-fuzzy interior ideals) are developed.
3
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Basic Formal Properties of Triangular Norms and Conorms

76%
Grabowski A.
Formalized Mathematics
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2017
|
tom 25
|
nr 2
93-100
EN
In the article we present in the Mizar system [1], [8] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [13]. The name triangular emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality [2]. After defining corresponding Mizar mode using four attributes, we introduced the following t-norms: minimum t-norm minnorm (Def. 6), product t-norm prodnorm (Def. 8), Łukasiewicz t-norm Lukasiewicz_norm (Def. 10), drastic t-norm drastic_norm (Def. 11), nilpotent minimum nilmin_norm (Def. 12), Hamacher product Hamacher_norm (Def. 13), and corresponding t-conorms: maximum t-conorm maxnorm (Def. 7), probabilistic sum probsum_conorm (Def. 9), bounded sum BoundedSum_conorm (Def. 19), drastic t-conorm drastic_conorm (Def. 14), nilpotent maximum nilmax_conorm (Def. 18), Hamacher t-conorm Hamacher_conorm (Def. 17). Their basic properties and duality are shown; we also proved the predicate of the ordering of norms [10], [9]. It was proven formally that drastic-norm is the pointwise smallest t-norm and minnorm is the pointwise largest t-norm (maxnorm is the pointwise smallest t-conorm and drastic-conorm is the pointwise largest t-conorm). This work is a continuation of the development of fuzzy sets in Mizar [6] started in [11] and [3]; it could be used to give a variety of more general operations on fuzzy sets. Our formalization is much closer to the set theory used within the Mizar Mathematical Library than the development of rough sets [4], the approach which was chosen allows however for merging both theories [5], [7].
4
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A new view of relationship between atomic posets and complete (algebraic) lattices

76%
Yu B., Li Q., Wu H.
Open Mathematics
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2017
|
tom 15
|
nr 1
238-251
EN
In the context of the atomic poset, we propose several new methods of constructing the complete lattice and the algebraic lattice, and the mutual decision of relationship between atomic posets and complete lattices (algebraic lattices) is studied.
5
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Scott convergence and fuzzy Scott topology onL-posets

76%
Liu H., Chen L.
Open Mathematics
|
2017
|
tom 15
|
nr 1
815-827
EN
We firstly generalize the fuzzy way-below relation on an L-poset, and consider its continuity by means of this relation. After that, we introduce a kind of stratified L-generalized convergence structure on an L-poset. In terms of that, L-fuzzy Scott topology and fuzzy Scott topology are considered, and the properties of fuzzy Scott topology are discussed in detail. At last, we investigate the Scott convergence of stratified L-filters on an L-poset, and show that an L-poset is continuous if and only if the Scott convergence on it coincides with the convergence with respect to the corresponding topological space.
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