The algebraic and lattice structures of type-2 intuitionistic fuzzy sets

Type-2 intuitionistic fuzzy sets are proposed as functions from non empty set U to T-T where T = {(mu, nu) : mu + nu <= 1, mu >= 0, nu >= 0} and T-T is the set of all mappings from T to T. The members of T-T are called intuitionistic fuzzy values (IFV). In this paper, we develop a mathematical framework for IFVs by defining a set of generalized operations on T-T and proved it to be an algebra. The other important properties like convexity, normality of IFVs and many important subalgebras are also explored and studied. Furthermore, two partial orders based on generalized operations are defined, which enable us to study the lattices in T-T.