K. V. Thomas

Fuzzy lattices

A genera1 development of the theory of fuzzy sublattices is provided in the paper. Several characterizations of a fuzzy sublattice, fuzzy idea1 (dual ideal), and fuzzy prime idea1 (dual ideal) are obtained. Structural theorems for fuzzy sublattices, fuzzy ideals (dual ideals) generated by a fuzzy set are proved. Moreover, the notion of fuzzy convex sublattice is introduced and discussed. Various fuzzy analogs of the results of classical lattice theory are established.

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.

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.

The join of fuzzy algebraic substructures of a group and their lattices

This is a continuation of work in previous papers [N. Ajmal and K.V. Thomas, Fuzzy Sets and Systems 58 (1993) 217; Inform. Sci. 76 (1994) 1]. Here, we provide a common technique of constructing the join of fuzzy substructures. Consequently, it leads to the formation of various types of lattices and sublattices of fuzzy substructures of a group including those of normal and quasinormal fuzzy subgroups. A lattice theoretic relationship of these sublattices is established. Moreover, we present simple and direct proofs for the fact that the lattice of fuzzy normal subgroups is modular.

Quotient of ideals of an intuitionistic fuzzy lattice

The concept of intuitionistic fuzzy ideal of an intuitionistic fuzzy lattice is introduced, and its certain characterizations are provided. We defined the quotient (or residual) of ideals of an intuitionistic fuzzy sublattice and studied their properties.

Quasinormality and fuzzy subgroups

Abstract We extend the notion of quasinormality to the fuzzy setting. We prove that a fuzzy subgroup of a group is fuzzy quasinormal if and only if all its level subsets are quasinormal in the usual group theoretic sense. We investigate the natures of homomorphic image and preimage of a fuzzy quasinormal subgroup. Moreover, we introduce the notions of core of a fuzzy subgroup, core-free fuzzy subgroup, fuzzy sylow subgroup and fuzzy maximal subgroup of a group. Consequently, we establish that in a finite group, all the fuzzy sylow subgroups of a core-free fuzzy quasinormal subgroup are fuzzy quasinormal. Furthermore, we also establish that a fuzzy maximal quasinormal subgroup with the sup property is fuzzy normal.

Intuitionistic fuzzy sublattices and ideals

We study the concept of intuitionistic fuzzy sublattices and intuitionistic fuzzy ideals of a lattice. Some characterization and properties of these intuitionistic fuzzy sublattices and ideals are established. Also we introduce the sum and product of two intuitionistic fuzzy ideals and prove that the sum and product of two Intuitionistic fuzzy ideals of a distributive lattice is again an intuitionistic fuzzy ideal. Moreover, we study the properties of intuitionistic fuzzy ideals under lattice homomorphism.

Lattice vector spaces and linear transformations

This paper introduces the concept of lattice vector space and establishes many important results. Also, this paper deals with linear transformations on lattice vector spaces and discusses their ele...

A Study on Characteristic Roots of Lattice Matrices

This paper deals with the characteristic roots of different types of lattice matrices and proves that a matrix and its transpose have the same characteristic roots. Also this paper introduces the concept of similar lattice matrices and proves that similar lattice matrices have the same characteristic roots.