THE LARGEST SOLUTION OF LINEAR EQUATION OVER THE COMPLETE HEYTING ALGEBRA

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.

On Concept Lattices for Numberings

The theory of numberings studies uniform computations for families of mathematical objects. In this area, computability-theoretic properties of at most countable families of sets S are typically classified via the corresponding Rogers upper semilattices. In most cases, a Rogers semilattice cannot be a lattice. Working within the framework of Formal Concept Analysis, we develop two new approaches to the classification of families S. Similarly to the classical theory of numberings, each of the approaches assigns to a family S its own concept lattice. The first approach captures the cardinality of a family S: if S contains more than 2 elements, then the corresponding concept lattice FC1(S) is a modular lattice of height 3, such that the number of its atoms to the cardinality of S. Our second approach gives a much richer environment. We prove that for any countable poset P, there exists a family S such that the induced concept lattice FC2(S) is isomorphic to the Dedekind-MacNeille completion of P. We also establish connections with the class of enumerative lattices introduced by Hoyrup and Rojas in their studies of algorithmic randomness. We show that every lattice FC2(S) is anti-isomorphic to an enumerative lattice. In addition, every enumerative lattice is anti-isomorphic to a sublattice of the lattice FC2(S) for some family S.

The Connectedness of Quantales

The concepts of connectedness and locally connectedness is introduced for rightside idempotent quantales. Some properties of connected quantale are studied, and then the equivalent characterization of connected quantale is also given.

Rough Class on a Completely Distributive Lattice

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.

A Class Of Counterexamples Concerning an Open Problem

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.