Vague groups and generalized vague subgroups on the basis of ([0, 1], ⩽, ∧)

In this paper, various elementary properties of vague groups and some properties of vague binary operations related with their associativity aspects are obtained. Furthermore, the concept of vague isomorphism is defined and some basic properties of this concept are studied. The concept of external direct product of vague groups is established. Later the definition of generalized vague subgroup, which is a generalization of the vague subgroup defined by Demirci, is introduced, and the validity of some classical results in this setting is investigated on the basis of the particular integral commutative, complete quasi-monoidal lattice ([0, 1], ⩽, ∧).