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
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.
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
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
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
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
Using an algebraic property, the completely distributive law, we have ever given out a characterization of the semicontinuity of lattice-valued mappings. How about the inverse implication? That is to say, can we analy...Using an algebraic property, the completely distributive law, we have ever given out a characterization of the semicontinuity of lattice-valued mappings. How about the inverse implication? That is to say, can we analytically characterize the completely distributive law? Moreover, can we characterize the completely distributive law in terms of fuzzy topology? The purpose of this note is to answer affirmatively these questions for the infinitely distributive lattices. This study connecting algebra with analysis and topology seems to be rather interesting.展开更多
UP till now there has been much spectacular and creative work about the theories of uniformi-ties and metrics in topological lattice.But this is mostly generalizations and developmentsof Hutton’s and Erceg’s pointle...UP till now there has been much spectacular and creative work about the theories of uniformi-ties and metrics in topological lattice.But this is mostly generalizations and developmentsof Hutton’s and Erceg’s pointless work.They cannot directly reflect the characteristics ofpointwise topology.In this note we shall set up a theory of pointwise uniformity and a theoryof pointwise metric on fuzzy lattices.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
The tabor gives some characterizations of strongly algebraic lattices, and proves that thecategory of strongly algebraic lattices is complete and cocomplete. Finally, this paper gives thecomplete conditions under whic...The tabor gives some characterizations of strongly algebraic lattices, and proves that thecategory of strongly algebraic lattices is complete and cocomplete. Finally, this paper gives thecomplete conditions under which the minimal mapping β: L→2L on a completely distributivelattice L preserves finite infs and arbitrary infs.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
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.
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
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.
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Project supported by the National Natural Science Foundation of China
文摘Using an algebraic property, the completely distributive law, we have ever given out a characterization of the semicontinuity of lattice-valued mappings. How about the inverse implication? That is to say, can we analytically characterize the completely distributive law? Moreover, can we characterize the completely distributive law in terms of fuzzy topology? The purpose of this note is to answer affirmatively these questions for the infinitely distributive lattices. This study connecting algebra with analysis and topology seems to be rather interesting.
文摘UP till now there has been much spectacular and creative work about the theories of uniformi-ties and metrics in topological lattice.But this is mostly generalizations and developmentsof Hutton’s and Erceg’s pointless work.They cannot directly reflect the characteristics ofpointwise topology.In this note we shall set up a theory of pointwise uniformity and a theoryof pointwise metric on fuzzy lattices.
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The tabor gives some characterizations of strongly algebraic lattices, and proves that thecategory of strongly algebraic lattices is complete and cocomplete. Finally, this paper gives thecomplete conditions under which the minimal mapping β: L→2L on a completely distributivelattice L preserves finite infs and arbitrary infs.
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.
