Iffat Jahan

The lattice of L-ideals of a ring is modular

In this paper, we extend the notion of a tip-extended pair of fuzzy subgroups to L-ideals of a ring. We prove that the sum of two tip-extended L-ideals of an arbitrary pair of L-ideals of a ring is the least L-ideal containing the union of the given L-ideals. Using this construction of join of L-ideals, we prove that the lattice of all L-ideals of a given ring is modular.

A study of normal fuzzy subgroups and characteristic fuzzy subgroups of a fuzzy group

In this paper, we investigate the properties of normal fuzzy subgroups of a fuzzy group. We also introduce the notion of a characteristic fuzzy subgroup of a fuzzy group, of which we provide level subset and strong level subset characterizations. Then we prove that a characteristic fuzzy subgroup of a fuzzy group is a normal fuzzy subgroup. Besides we prove that the commutator subgroup generated by the commutator of two normal fuzzy subgroups of a fuzzy group is contained in their intersection. Finally, we construct many new lattices and sublattices of fuzzy subgroups and normal fuzzy subgroups of a given fuzzy group. We also construct lattices of characteristic fuzzy subgroups possessing sup-property.

An L-Point Characterization of Normality and Normalizer of an L-Subgroup of an L-Group

Abstract In this paper, we study the notion of normal -subgroup of an -group and provide its characterization by an -point. We also provide a construction of the normalizer of an -subgroup of a given -group by using -points. Moreover, we also discuss the product, homomorphic images and homomorphic preimages of normalizers.

A New Metatheorem and Subdirect Product Theorem for L-Subgroups

ABSTRACT This paper is a continuation of the work of Tom Head ‘Metatheorem for deriving fuzzy theorems from crisp versions’. The concept of natural extension is introduced which is then applied in the development of a new metatheorem in L-setting, where L is a complete chain. The application of this theorem is demonstrated through the notions of generated L-subgroups and commutator L-subgroups. Moreover, a new subdirect product theorem is developed wherein it is demonstrated that for a group G, the L-subgroup lattice can be represented as a subdirect product of copies of its associated lattice of crisp subgroups. The significance of this theorem is exhibited by applying it to a characterization of generalized cyclic groups in terms of the distributivity of the lattice of L-subgroups of a group.