T. M. G. Ahsanullah

Fuzzy topology on fuzzy sets and tolerance topology

In this paper fuzzy topology has been defined on a fuzzy set. The notion of quasi-coincidence is extended in order to fit this new situation and to generalize the concept of q-neighbourhood introduced originally by Ming and Ming (1980). The generalized q-neighbourhood system has been utilized to introduce tolerance topology on a fuzzy set with a fuzzy tolerance relation.

On fuzzy neighborhood groups

Abstract Our aim is to introduce the notion of fuzzy neighborhood groups. In Section 2 we establish some basic results and prove some characterization theorems. In Section 3 we prove that every fuzzy neighborhood group is fuzzy uniformizable (in the sense of R. Lowen, J. Math. Anal. Appl. 82 (1981), 370–385).

Invariant probabilistic metrizability of fuzzy neighbourhood groups

We introduce the notions of N-local compactness and N-openess in fuzzy neighbourhood spaces and in fuzzy neighbourhood groups, together with some characterizations for them. In particular, we prove that a fuzzy neighbourhood space is N-locally compact if and only if its α-level spaces, for all 0<α<1, are locally compact. A similar criterion is established for N-open functions. We also introduce the notion of fuzzy absolute value functions on groups. We show that there exists a 1-1 correspondence between invariant fuzzy (probabilistic) pseudo-metrics on groups and fuzzy absolute value functions. We establish theorems on fuzzy (probabilistic) pseudo-metrizability of fuzzy neighbourhood groups. Finally, fuzzy (probabilistic) metrizability of quotient fuzzy neighbourhood groups is also taken into account.

On fuzzy neighborhood rings

Abstract We introduce the concept of fuzzy neighborhood rings. We give necessary and sufficient conditions for a ring structure and a fuzzy neighborhood system to be compatible. We also give necessary and sufficient conditions for a prefilter to be a neighborhood prefilter of 0 in a fuzzy neighborhood rings. The notion of bounded fuzzy set is introduced in fuzzy neighborhood rings and some results in this context are provided.

On fuzzy neighborhood modules and algebra

Abstract The notion of fuzzy neighborhood module over a ring and fuzzy neighborhood algebra over a field is introduced. Some characterization theorems with fundamental results are deduced.

On the category of fixed basis frame valued topological groups

This article gives results on fixed basis frame valued neighborhood topological groups and stratified neighborhood topological groups, includes some characterization theorems, and presents the uniformizability of stratified neighborhood topological groups within the unified approach of Gutierrez Garcia, DePrada Vicente, and Sostak. It is shown that for a frame L, the category L-NGrp is topological over Grp with respect to the forgetful functor.

Frame valued stratified generalized convergence groups

Following the notion of stratified L-fuzzy convergence space of Gunther Jäger [Quaest. Math. 24 (2001), 501–517], we introduce the notion of a stratified L-generalized convergence group, and look into some other objects, namely, stratified L-Kent convergence groups, and stratified L-principal convergence groups. We show that the category of stratified L-generalized convergence groups, S L-GCGrp is topological over the category of groups,Grp with respect to the forgetful functor, and we prove that the category S L-NeighGrp, of stratified L-neighborhood groups is isomorphic to a subcategory of S L-GCGrp. We give necessary and sufficient conditions for a group structure and a stratified L-generalized convergence structure to be a stratified L-generalized convergence group. Finally, we observe that every stratified L-generalized convergence group possessing a stratified L-principal convergence structure gives rise to a stratified L-neighborhood topological group.

Sequential convergence in fuzzy neighborhood spaces

Abstract We introduce the notions of sequential convergence and the first axiom of countability in fuzzy neighborhood spaces. We show that these notions fulfil good extension criterion attributed to Lowen. Convergent sequences and Cauchy sequences are studied in the light of fuzzy (probabilistic) pseudometric space;.

Lattice-valued convergence ring and its uniform convergence structure

Abstract Considering L a frame, we introduce the notion of stratified L-neighborhood topological ring, produce some characterization theorems including its Luniformizability. With the help of the notions of stratified convergence structures attributed to Gunther Jäger [10], we introduce and study various subcategories of stratified L-convergence rings. In doing so, we bring into light, among others, the notions of stratified L-uniform group and stratified L-uniform convergence group in an attempt to show that every stratified L-convergence ring carries in a natural way the stratified L-uniform convergence structure of Jäger and Burton [15], and the category of stratified L-uniform groups and uniformly continuous group homomorphisms, SLUnifGrp is isomorphic to the category of principal stratified L-uniform convergence groups and uniformly continuous group homomorphisms, SL-PUConvGrp. Introducing the notion of stratified L-Cauchy ring, we show that the category of stratified L-Cauchy rings and Cauchy-continuous ring homomorphisms, SL-ChyRng is topological over the category of rings, Rng with respect to the forgetful functor, and that every stratified L-Cauchy ring is a stratified L-convergence ring. We observe that if L is a Boolean algebra, then each stratified L-uniform convergence ring serves as a natural example of a stratified L-Cauchy ring. We give necessary and suficient conditions for a stratified L-convergence structure and a ring structure to be a stratified L-convergence ring.

Characterization of fuzzy neighborhood commutative division rings II

In [4] we produced a characterization of fuzzy neighborhood commutative division rings; here we present another characterization of it in a sense that we minimize the conditions so that a fuzzy neighborhood system is compatible with the commutative division ring structure. As an additional result, we show that Chadwick [5] relatively compact fuzzy set is bounded in a fuzzy neighborhood commutative division ring.