THE DISTRIBUTIVE LATTICE OF QUOTIENT LATTICE-VALUED INTUITIONISTIC FUZZY SUB lGROUPS OF TYPE-3

The study of lattice-ordered groups (-groups), which elegantly unify group and lattice structures, is fundamentally constrained in contexts characterized by multi-dimensional uncertainty and vagueness. Existing generalizations, such as Lattice-Valued Intuitionistic Fuzzy Sets (LVIFSs) of Type-1 or Type-2, fail to consistently preserve the -group’s critical dual algebraic and order-theoretic properties. To address this theoretical deficiency, this paper formally introduces and axiomatically analyzes the robust concept of Lattice-Valued Intuitionistic Fuzzy Sub--groups of Type-3 (LVIFS-group-3) and their specialized counterpart, the Convex LVIFS -group-3 (C-LVIFS-group-3). A critical level set equivalence theorem is established, demonstrating that a structure is an LVIFS-group-3 if and only if all of its non-empty level sets manifest as crisp -subgroups. We rigorously prove that this structure is preserved under -homomorphisms and is closed under intersection and chain union. Meticulous analysis of the C-LVIFS-group-3 successfully models the essential order-preserving convexity property. Crucially, we investigate the quotient structure defined by the C-LVIFS-group-3, demonstrating the conditions under which it forms a distributive lattice, thereby establishing a fuzzy analog to the normal convex -subgroup, which functions as the algebraic kernel in classical -group theory. This framework furnishes the most comprehensive instrument for the algebraic study of -groups in contexts permeated by complex, lattice-valued uncertainty.