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It is known that for a nonempty topological space and a nonsingleton complete lattice endowed with the Scott topology, the partially ordered set of all continuous functions from into is a continuous lattice if and only if both and the open set lattice are continuous lattices. This result extends to certain classes of -distributive lattices, where is a subset system replacing the system of all directed subsets (for which the -distributive complete lattices are just the continuous...
We present a unified treatment of pointfree metrization theorems based on an analysis of special properties of bases. It essentially covers all the facts concerning metrization from Engelking [1] which make pointfree sense. With one exception, where the generalization is shown to be false, all the theorems extend to the general pointfree context.
We present a direct constructive proof of full normality for a class of spaces (locales) that includes, among others, all metrizable ones.
Let be an infinite cardinal. Let be the class of all lattices which are conditionally -complete and infinitely distributive. We denote by the class of all lattices such that is infinitely distributive, -complete and has the least element. In this paper we deal with direct factors of lattices belonging to . As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class .
The main goal of this paper is to construct fuzzy connectives on algebraic completely distributive lattice(ACDL) by means of extending fuzzy connectives on the set of completely join-prime elements or on the set of completely meet-prime elements, and discuss some properties of the new fuzzy connectives. Firstly, we present the methods to construct t-norms, t-conorms, fuzzy negations valued on ACDL and discuss whether De Morgan triple will be kept. Then we put forward two ways to extend fuzzy implications...
In this paper we deal with the of an -algebra , where and are nonzero cardinals. It is proved that if is singular and -distributive, then it is . We show that if is complete then it can be represented as a direct product of -algebras which are homogeneous with respect to higher degrees of distributivity.
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