This survey article illustrates many important current trends and perspectives for the field and their applications, of interest to researchers in modern algebra, mathematical logic and discrete mathematics. It covers...This survey article illustrates many important current trends and perspectives for the field and their applications, of interest to researchers in modern algebra, mathematical logic and discrete mathematics. It covers a number of directions, including completeness theorem and compactness theorem for hyperidentities;the characterizations of the Boolean algebra of n-ary Boolean functions and the bounded distributive lattice of n-ary monotone Boolean functions;the functional representations of finitely-generated free algebras of various varieties of lattices via generalized Boolean functions, etc.展开更多
A finite dynamical system(FDS)over a lattice L is a pair(S(L),f),where S(L)is a left-L module and f is a mapping from S into itself.The phase space of(S(L),f)is a digraph whose vertex set is S(L)and there is an arc fr...A finite dynamical system(FDS)over a lattice L is a pair(S(L),f),where S(L)is a left-L module and f is a mapping from S into itself.The phase space of(S(L),f)is a digraph whose vertex set is S(L)and there is an arc from x to y if y=f(x).Let L be a finite distributive lattice,A an n×n matrix over L,and f(x)=Ax.The structure of the phase space of the FDS(Ln,f)is discussed.The number of limit cycles in the phase space of(Ln,f)is described in Möbius function.The phase spaces of some invertible,nilpotent,and idempotent FDS(Ln,f)are characterized explicitly.展开更多
Soft set theory has a rich potential application in several fields. A soft group is a parameterized family of subgroups and a fuzzy soft group is a parameterized family of fuzzy subgroups. The concept of fuzzy soft gr...Soft set theory has a rich potential application in several fields. A soft group is a parameterized family of subgroups and a fuzzy soft group is a parameterized family of fuzzy subgroups. The concept of fuzzy soft group is the generalization of soft group. Abdulkadir Aygunoglu and Halis Aygun introduced the notion of fuzzy soft groups in 2009[1]. In this paper, the concept of lattice ordered fuzzy soft groups and its duality has been introduced. Then distributive and modular lattice ordered fuzzy soft groups are analysed. The objective of this paper is to study the lattice theory over the collection of fuzzy soft group in a parametric manner. Some pertinent properties have been analysed and hence established duality principle.展开更多
Our main objective is to study properties of a fuzzy ideals(fuzzy dual ideals).A study of special types of fuzzy ideals(fuzzy dual ideals) is also furnished.Some properties of a fuzzy ideals(fuzzy dual ideals) are fur...Our main objective is to study properties of a fuzzy ideals(fuzzy dual ideals).A study of special types of fuzzy ideals(fuzzy dual ideals) is also furnished.Some properties of a fuzzy ideals(fuzzy dual ideals) are furnished.Properties of a fuzzy lattice homomorphism are discussed.Fuzzy ideal lattice of a fuzzy lattice is defined and discussed.Some results in fuzzy distributive lattice are proved.展开更多
In this paper,we initiate a study of S-fuzzy ideal(filter) of a lattice where S stands for a meet semilattice.A S-fuzzy prime ideal(filter) of a lattice is defined and it is proved that a S-fuzzy ideal(filter) of a la...In this paper,we initiate a study of S-fuzzy ideal(filter) of a lattice where S stands for a meet semilattice.A S-fuzzy prime ideal(filter) of a lattice is defined and it is proved that a S-fuzzy ideal(filter) of a lattice is S-fuzzy prime ideal(filter) if and only if any non-empty α-cut of it is a prime ideal(filter).Stone's theorem for a distributive lattice is extended by considering S-fuzzy ideals(filters).展开更多
This paper generalizes the Pawlak rough set method to a completely distributive lattice. The concept of a rough set has many applications in data mining. The approximation operators on a completely distributive lattic...This paper generalizes the Pawlak rough set method to a completely distributive lattice. The concept of a rough set has many applications in data mining. The approximation operators on a completely distributive lattice are studied, the rough class on a completely distributive lattice is defined and the expressional theorems of the rough class are proven. These expressional theorems are used to prove that the collection of all rough classes is an atomic completely distributive lattice.展开更多
In this paper, we generalize Clifford semirings to left Clifford semirings bymeans of the so-called band semirings. We also discuss a special case of this kind of semirings,that is, strong distributive lattices of lef...In this paper, we generalize Clifford semirings to left Clifford semirings bymeans of the so-called band semirings. We also discuss a special case of this kind of semirings,that is, strong distributive lattices of left rings.展开更多
An excellent introduction to the topic of poset matroids is due to Barnabei, Nicoletti and Pezzoli. In this paper, we investigate the rank axioms for poset matroids; thereby we can characterize poset matroids in a “g...An excellent introduction to the topic of poset matroids is due to Barnabei, Nicoletti and Pezzoli. In this paper, we investigate the rank axioms for poset matroids; thereby we can characterize poset matroids in a “global” version and a “pseudo-global” version. Some corresponding properties of combinatorial schemes are also obtained.展开更多
Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where...Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn（L） = PLn （L） for any n （n 〉 2）, in which GLn（L） is the group of all n × n invertible matrices over L and PLn（L） is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.展开更多
If K ∩ AlgL is weak. dense in AlgL, where K is the set of all compactoperators in B(H), is completely distributive? In this note, we prove that there is a reflexivesubspace lattice L on some Hilbert space, which sati...If K ∩ AlgL is weak. dense in AlgL, where K is the set of all compactoperators in B(H), is completely distributive? In this note, we prove that there is a reflexivesubspace lattice L on some Hilbert space, which satisfies the following conditions: (a) F(AlgL) isdense in AlgL in the ultrastrong operator topology, where F(AlgL) is the set of all finite rankoperators in AlgL; (b) L isnt a completely distributive lattice. The subspace lattices that satisfythe above conditions form a large class of lattices. As a special case of the result, it easy to seethat the answer to Problem 7 is negative.展开更多
文摘This survey article illustrates many important current trends and perspectives for the field and their applications, of interest to researchers in modern algebra, mathematical logic and discrete mathematics. It covers a number of directions, including completeness theorem and compactness theorem for hyperidentities;the characterizations of the Boolean algebra of n-ary Boolean functions and the bounded distributive lattice of n-ary monotone Boolean functions;the functional representations of finitely-generated free algebras of various varieties of lattices via generalized Boolean functions, etc.
基金National Natural Science Foundation of China(Nos.11671258 and 11371086)。
文摘A finite dynamical system(FDS)over a lattice L is a pair(S(L),f),where S(L)is a left-L module and f is a mapping from S into itself.The phase space of(S(L),f)is a digraph whose vertex set is S(L)and there is an arc from x to y if y=f(x).Let L be a finite distributive lattice,A an n×n matrix over L,and f(x)=Ax.The structure of the phase space of the FDS(Ln,f)is discussed.The number of limit cycles in the phase space of(Ln,f)is described in Möbius function.The phase spaces of some invertible,nilpotent,and idempotent FDS(Ln,f)are characterized explicitly.
文摘Soft set theory has a rich potential application in several fields. A soft group is a parameterized family of subgroups and a fuzzy soft group is a parameterized family of fuzzy subgroups. The concept of fuzzy soft group is the generalization of soft group. Abdulkadir Aygunoglu and Halis Aygun introduced the notion of fuzzy soft groups in 2009[1]. In this paper, the concept of lattice ordered fuzzy soft groups and its duality has been introduced. Then distributive and modular lattice ordered fuzzy soft groups are analysed. The objective of this paper is to study the lattice theory over the collection of fuzzy soft group in a parametric manner. Some pertinent properties have been analysed and hence established duality principle.
基金UGC,New Delhi for financial support through scheme F.No 33-109/2007(SR)
文摘Our main objective is to study properties of a fuzzy ideals(fuzzy dual ideals).A study of special types of fuzzy ideals(fuzzy dual ideals) is also furnished.Some properties of a fuzzy ideals(fuzzy dual ideals) are furnished.Properties of a fuzzy lattice homomorphism are discussed.Fuzzy ideal lattice of a fuzzy lattice is defined and discussed.Some results in fuzzy distributive lattice are proved.
基金UGC,New Delhi for financial support through scheme F.No33-109/2007(SR)
文摘In this paper,we initiate a study of S-fuzzy ideal(filter) of a lattice where S stands for a meet semilattice.A S-fuzzy prime ideal(filter) of a lattice is defined and it is proved that a S-fuzzy ideal(filter) of a lattice is S-fuzzy prime ideal(filter) if and only if any non-empty α-cut of it is a prime ideal(filter).Stone's theorem for a distributive lattice is extended by considering S-fuzzy ideals(filters).
基金Supported by the National Natural Science Foundation of China(No.60074015)
文摘This paper generalizes the Pawlak rough set method to a completely distributive lattice. The concept of a rough set has many applications in data mining. The approximation operators on a completely distributive lattice are studied, the rough class on a completely distributive lattice is defined and the expressional theorems of the rough class are proven. These expressional theorems are used to prove that the collection of all rough classes is an atomic completely distributive lattice.
文摘In this paper, we generalize Clifford semirings to left Clifford semirings bymeans of the so-called band semirings. We also discuss a special case of this kind of semirings,that is, strong distributive lattices of left rings.
基金Supported by the National Natural Science Foundation of China (Granted No.103710438)Education Ministry of China (Granted No.02139)
文摘An excellent introduction to the topic of poset matroids is due to Barnabei, Nicoletti and Pezzoli. In this paper, we investigate the rank axioms for poset matroids; thereby we can characterize poset matroids in a “global” version and a “pseudo-global” version. Some corresponding properties of combinatorial schemes are also obtained.
基金National Natural Science Foundation of China （60174013） Research Foundation for Doctoral Program of Higher Education （20020027013）+1 种基金 Science and Technology Key Project Foundation of Ministry of Education （03184） Major State Basic Research Development Program of China （2002CB312200）
文摘Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn（L） = PLn （L） for any n （n 〉 2）, in which GLn（L） is the group of all n × n invertible matrices over L and PLn（L） is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.
文摘If K ∩ AlgL is weak. dense in AlgL, where K is the set of all compactoperators in B(H), is completely distributive? In this note, we prove that there is a reflexivesubspace lattice L on some Hilbert space, which satisfies the following conditions: (a) F(AlgL) isdense in AlgL in the ultrastrong operator topology, where F(AlgL) is the set of all finite rankoperators in AlgL; (b) L isnt a completely distributive lattice. The subspace lattices that satisfythe above conditions form a large class of lattices. As a special case of the result, it easy to seethat the answer to Problem 7 is negative.
