Let （L, 〈, V, A） be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by V and A respectively...Let （L, 〈, V, A） be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by V and A respectively, is studied. We obtain： （i） the necessary and sufficient conditions for S（A,b）≠Ф; （ii） the necessary conditions for IS（A,b）｜ = 1. We also obtain the vector x ∈ Ln and prove that it is the largest element of S（A, b） if S（A, b）≠Ф.展开更多
In this article, we discuss several properties of the basic contact process on hexagonal lattice H, showing that it behaves quite similar to the process on d-dimensional lattice Zd in many aspects. Firstly, we constru...In this article, we discuss several properties of the basic contact process on hexagonal lattice H, showing that it behaves quite similar to the process on d-dimensional lattice Zd in many aspects. Firstly, we construct a coupling between the contact process on hexagonal lattice and the oriented percolation, and prove an equivalent finite space-time condition for the survival of the process. Secondly, we show the complete convergence theorem and the polynomial growth hold for the contact process on hexagonal lattice. Finally, we prove exponential bounds in the supercritical case and exponential decay rates in the subcritical case of the process.展开更多
Some characterizations of preregular operators between two Banach lattices are presented. Then several sufficient conditions for preregular operators being regular are given, and some related results are also obtained.
The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case ...The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation ?difined on the set ?of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation ?essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.展开更多
In this paper,we introduce a concept of principal divisor lattice and describe the structure of its elements.We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompo...In this paper,we introduce a concept of principal divisor lattice and describe the structure of its elements.We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompositions in complete principal divisor distributive lattices,and prove that the complete lower continuous,principal divisor lattices have irredundant join irreducible decompositions.In the end,we show the descriptions of lattices that have unique(resp.replaceable) irredundant join irreducible decompositions in complete lower continuous principal divisor lattices.展开更多
A completely distributive lattice is also called a molecular lattice.The least and the greatestelements in a lattice are denoted by 0 and 1,respectively.For the empty subset L of alattice L,we assume that V=0 and ∧=1...A completely distributive lattice is also called a molecular lattice.The least and the greatestelements in a lattice are denoted by 0 and 1,respectively.For the empty subset L of alattice L,we assume that V=0 and ∧=1.The set of all the molecules（i.e.non-zero∨-irreducible elements）in a lattice L is denoted by M（L）. Let L<sub>1</sub> and L<sub>2</sub> be completely distributive lattices,f:L<sub>1</sub>→L<sub>2</sub> and g:L<sub>2</sub>→L<sub>1</sub> be order-preserving maps.If for any a∈L<sub>1</sub> and any b∈L<sub>2</sub>,f（a）≤b if and only if a≤g（b）,then fis called the left adjoint of g,or g is called the right adjoint of f.The right adjoint of f展开更多
In order to study fuzzy topology and general topology in view of topological lattice, many researches have been carried out. Because of the lack of reverse order involution, there are some fundamental works, such as t...In order to study fuzzy topology and general topology in view of topological lattice, many researches have been carried out. Because of the lack of reverse order involution, there are some fundamental works, such as the theory of uniformity and metrization to be considered. In this note, we introduce the notion of uniformity on completely distributive lattice in terms of cover systems, and show that it is a good extension, we obtain some properties and prove that the equivalence of complete regularity and uniformizability on completely distributive lattices.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
The category of completely distributive lattices with Scott continuous functions is cartesian closed. Neither the category of completely distributive lattices with arbitrary union preserving mappings nor the category ...The category of completely distributive lattices with Scott continuous functions is cartesian closed. Neither the category of completely distributive lattices with arbitrary union preserving mappings nor the category of completely distributive lattices with nonempty union preserving mappings is cartesian closed.展开更多
Taking completely distributive lattices for objects and complete lattice homomorphisms for morphisms, we get a category Lat (see Refs. [1—3]). There is a primary question about Lat: Does every family of objects have ...Taking completely distributive lattices for objects and complete lattice homomorphisms for morphisms, we get a category Lat (see Refs. [1—3]). There is a primary question about Lat: Does every family of objects have product and coproduct in Lat? We prove that Lat has products and coproducts, and describe the intrinsic relation between展开更多
基金supported by the NNSF (10471035,10771056) of China
文摘Let （L, 〈, V, A） be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by V and A respectively, is studied. We obtain： （i） the necessary and sufficient conditions for S（A,b）≠Ф; （ii） the necessary conditions for IS（A,b）｜ = 1. We also obtain the vector x ∈ Ln and prove that it is the largest element of S（A, b） if S（A, b）≠Ф.
基金Supported in part by the NNSF of China (10531070,10625101)the National Basic Research Program of China (2006CB805900)
文摘In this article, we discuss several properties of the basic contact process on hexagonal lattice H, showing that it behaves quite similar to the process on d-dimensional lattice Zd in many aspects. Firstly, we construct a coupling between the contact process on hexagonal lattice and the oriented percolation, and prove an equivalent finite space-time condition for the survival of the process. Secondly, we show the complete convergence theorem and the polynomial growth hold for the contact process on hexagonal lattice. Finally, we prove exponential bounds in the supercritical case and exponential decay rates in the subcritical case of the process.
文摘Some characterizations of preregular operators between two Banach lattices are presented. Then several sufficient conditions for preregular operators being regular are given, and some related results are also obtained.
文摘The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation ?difined on the set ?of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation ?essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.
基金supported by National Natural Science Foundation of China (Grant No.10671138)
文摘In this paper,we introduce a concept of principal divisor lattice and describe the structure of its elements.We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompositions in complete principal divisor distributive lattices,and prove that the complete lower continuous,principal divisor lattices have irredundant join irreducible decompositions.In the end,we show the descriptions of lattices that have unique(resp.replaceable) irredundant join irreducible decompositions in complete lower continuous principal divisor lattices.
文摘A completely distributive lattice is also called a molecular lattice.The least and the greatestelements in a lattice are denoted by 0 and 1,respectively.For the empty subset L of alattice L,we assume that V=0 and ∧=1.The set of all the molecules（i.e.non-zero∨-irreducible elements）in a lattice L is denoted by M（L）. Let L<sub>1</sub> and L<sub>2</sub> be completely distributive lattices,f:L<sub>1</sub>→L<sub>2</sub> and g:L<sub>2</sub>→L<sub>1</sub> be order-preserving maps.If for any a∈L<sub>1</sub> and any b∈L<sub>2</sub>,f（a）≤b if and only if a≤g（b）,then fis called the left adjoint of g,or g is called the right adjoint of f.The right adjoint of f
文摘In order to study fuzzy topology and general topology in view of topological lattice, many researches have been carried out. Because of the lack of reverse order involution, there are some fundamental works, such as the theory of uniformity and metrization to be considered. In this note, we introduce the notion of uniformity on completely distributive lattice in terms of cover systems, and show that it is a good extension, we obtain some properties and prove that the equivalence of complete regularity and uniformizability on completely distributive lattices.
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The category of completely distributive lattices with Scott continuous functions is cartesian closed. Neither the category of completely distributive lattices with arbitrary union preserving mappings nor the category of completely distributive lattices with nonempty union preserving mappings is cartesian closed.
基金Project supported partly by the National Natural Science Foundation of China.
文摘Taking completely distributive lattices for objects and complete lattice homomorphisms for morphisms, we get a category Lat (see Refs. [1—3]). There is a primary question about Lat: Does every family of objects have product and coproduct in Lat? We prove that Lat has products and coproducts, and describe the intrinsic relation between
