Measurement of Countable Compactness and Lindelöf Property in RL -Fuzzy Topological Spaces

Based on the concepts of pseudocomplement of L -subsets and the implication operator where L is a completely distributive lattice with order-reversing involution, the definition of countable RL -fuzzy compactness degree and the Lindelöf property degree of an L -subset in RL -fuzzy topology are introduced and characterized. Since L -fuzzy topology in the sense of Kubiak and Šostak is a special case of RL -fuzzy topology, the degrees of RL -fuzzy compactness and the Lindelöf property are generalizations of the corresponding degrees in L -fuzzy topology.

On the sum of L-convex spaces

Considering L be a completely distributive lattice, the notion of the sum of L-convex spaces is introduced and its elementary properties is studied. Firstly, the connections between the sum of L-convex spaces and its factor spaces are established. Secondly, the additivity of separability (S-1, sub-S0, S0, S1, S2, S3 and S4) are investigated. Finally, the additivity of five types special L-convex spaces are examined.

Fuzzy generalized convex spaces

On completely distributive lattice, the notion of fuzzy generalized convex space is introduced. It can be characterized by many means including fuzzy generalized hull space, fuzzy generalized restricted hull space, fuzzy generalized convexly enclosed relation space and fuzzy generalized derived hull space.

A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies

In this paper, by means of the implication operator → on a completely distributive lattice M, we define the approximate degrees of M-fuzzifying convex structures, M-fuzzifying closure systems and M-fuzzifying Alexandrov topologies to interpret the approximate degrees to which a mapping is an M-fuzzifying convex structure, an M-fuzzifying closure system and an M-fuzzifying Alexandrov topology from a logical aspect. Moreover, we represent some properties of M-fuzzifying convex structures as well as its relations with M-fuzzifying closure systems and M-fuzzifying Alexandrov topologies by inequalities.

Distributive and completely distributive lattice extensions of ordered sets

It is known that a poset can be embedded into a distributive lattice if, and only if, it satisfies the prime filter separation property. We describe here a class of “prime filter completions” for posets with the prime filter separation property that are completely distributive lattices generated by the poset and preserve existing finite meets and joins. The free completely distributive lattice generated by a poset can be obtained through such a prime filter completion. We also show that every completely distributive completion of a poset with the prime filter separation property is representable as a canonical extension of the poset with respect to some set of filters and ideals. The connections between the prime filter completions and canonical extensions are described and yield the following corollary: the canonical extension of any distributive lattice is the free completely distributive lattice generated by the lattice. A construction that is a variant of the prime filter completion is given that can be used to obtain the free distributive lattice generated by a poset. In addition, it is shown that every distributive lattice extension of the poset can be represented by such a construction. Finally, we show that a poset with the prime filter separation property and the free distributive lattice generated by it generates the same free completely distributive lattice.

Torsion Radicals and Torsion Classes of Cyclically Ordered Groups

The notion of torsion radical of cyclically ordered groups is defined analogously as in the case of lattice ordered groups. We denote by T the collection of all torsion radicals of cyclically ordered groups. For τ1, τ2 ∈ T, we put τ1 ∈ τ2 if τ1(G) □ τ2(G) for each cyclically ordered group G. We show that T is a proper class; nevertheless, we apply for T the usual terminology of the theory of partially ordered sets. We prove that T is a complete completely distributive lattice. The analogous result fails to be valid for torsion radicals of lattice ordered groups. Further, we deal with products of torsion classes of cyclically ordered groups.