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
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
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
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 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.
