Naseem Ajmal

Level subgroups and union of fuzzy subgroups

Abstract In this paper, we investigate the conditions under which a given fuzzy subgroup of a given group can or can not be realized as a union of two proper fuzzy subgroups. Moreover, we formulate the notion of fuzzy subgroup generated by a given fuzzy subset by level subgroups. We also give characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups. Also, we discuss the relationship between level subgroups of a given fuzzy subgroup and the level subgroups of its homomorphic image.

On fuzzy rings

Abstract An internal description of the fuzzy subring/fuzzy ideal generated by a finite fuzzy subset of a ring is provided. Also a fuzzy coset of a fuzzy ideal is defined and the set of all fuzzy cosets of a fuzzy ideal is given a ring structure. A fuzzy semiprime ideal is also defined and some basic properties of this fuzzy ideal are studied.

Fuzzy ideals and fuzzy prime ideals of a ring

Abstract Besides obtaining certain ring theoretic analogues, some peculiarities of the fuzzy setting are also discussed. In particular, a study of (i) the conditions under which a given fuzzy ideal can or cannot be expressed as a union of two proper fuzzy ideals, (ii) the algebraic nature of images and preimages of fuzzy prime ideals under homomorphisms, and (iii) the ring of all the fuzzy cosets of a fuzzy prime (fuzzy maximal) ideal, is carried out.

The lattices of fuzzy subgroups and fuzzy normal subgroups

Abstract Here we discuss various types of sublattices of the lattice of fuzzy subgroups of a given group. We prove that a special class of fuzzy normal subgroups constitutes a modular sublattice of the lattice of fuzzy subgroups.

The lattice of fuzzy normal subgroups is modular

Abstract We have introduced and discussed in a recent paper [2] various sublattices such as Lf, Lt, and Lfnt, of the lattice L of fuzzy subgroups of a given group. In particular, we proved that the sublattice Lfn, of fuzzy normal subgroups with finite range and tip “t” is a modular sublattice of L. Here, we remove all the restrictions on the choice of fuzzy normal subgroups, and prove that the class of all fuzzy normal subgroups Ln forms the modular sublattice of L. As a tool, we prove an important result that the fuzzy subgroups generated by a pair of fuzzy normal subgroups of a group is fuzzy normal.

A complete study of the lattices of fuzzy congruences and fuzzy normal subgroups

In this paper, we introduce the concepts of t-fuzzy congruences and t-fuzzy equivalences. Using these ideas, we investigate completely, on one hand, the lattice structures of the set of fuzzy equivalence relations on a group and the set of fuzzy congruences and, on the other hand, the lattice structures of the set of fuzzy subgroups and fuzzy normal subgroups. Our study reveals some finer and interesting facts about these lattices. It is proved, among other results, that the set Ct of all t-fuzzy congruences of a group G forms lattice, and also the set Lnt of all those fuzzy normal subgroups, which assume the same value t at e the identity of G, forms a lattice. As an important result, we prove that the lattices Ct and Lnt are isomorphic. It is also shown that the lattices Ct and Lnt are modular. Moreover, we construct various important sublattices of the lattice Ct and exhibit their relationship by lattice diagrams. In the process, we improve and unify many results of earlier authors on fuzzy congruences.

The lattices of fuzzy ideals of a ring

We investigate the lattice structure of various sublattices of the lattice of fuzzy subrings of a given ring. We prove that a special class of fuzzy ideals forms a modular sublattice of the lattice of fuzzy ideals of a ring.

Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups

Abstract We define a notion of ‘containment’ of an ordinary kernel of a group homomorphism in a fuzzy subgroup. Using this idea, we provide the long-awaited solution of the problem of showing a one-to-one correspondence between the family of fuzzy subgroups of a group, containing the kernel of a given homomorphism, and the family of fuzzy subgroups of the homomorphic image of the given group. It is shown that an ordinary kernel gives rise to the notion of fuzzy quotient group in a natural way. Consequently, the fundamental theorem of homomorphisms is established for fuzzy subgroups. Moreover, we provide new proofs for the facts, that the homomorphic image of a fuzzy subgroup is always a fuzzy subgroup, and fuzzy normality is invariant under surjective homomorphism.

Fuzzy nets and their application

Fuzzy net-theory is enriched by the introduction of the fuzzy net of fuzzy sets. The limsup, liminf and limit of a fuzzy net of fuzzy sets are defined and their various properties are discussed. Alternative characterizations based on the notion of a fuzzy net are provided for several fuzzy topological concepts including open and closed fuzzy sets, fuzzy continuity, maps with closed fuzzy graph, open fuzzy mapping, etc. Thus the net theoretic approach is shown to be a promising tool in fuzzy topology.

Fuzzy groups with sup property

We investigate the inner structure of fuzzy subgroups with sup property. We provide a characterization of this type of fuzzy subgroups in terms of their level subsets. It is shown that the property of being a fuzzy subgroup with sup property is invariant under homomorphism and the homomorphic preimage of a fuzzy subgroup with sup property is also a fuzzy subgroup with sup property. Moreover, we investigate the lattice structure of the class Lnt of fuzzy normal subgroups of a group G, each of which assumes the same value “t” at the identity of G. It is proved that the subclass Lnst of fuzzy subgroups with sup property of Lnt constitutes a sublattice of Lnt. The modularity of Lnst is derived as a consequence of the modularity of Lnt.