Lattice-valued finite state machines and lattice-valued transformation semigroups

This paper provides a generalization of known results about fuzzy finite state machines, fuzzy transformation semigroups and their relationship by broading the truth values domain from the interval [0,1] to a complete lattice endowed with a t-norm and a t-conorm. So, we deal with the concepts of L-fuzzy finite state machines and L-fuzzy transformation semigroups and we prove that the cited generalization is possible if and only if the t-norm and the t-conorm satisfy a distributive property. If we consider the complete lattice of the closed intervals inside the original lattice L, we give methods to obtain an interval lattice-valued finite state machine and an interval lattice-valued transformation semigroup from two L-fuzzy finite state machines or two L-fuzzy transformation semigroups, respectively. Conversely, we show two different ways to build a faithful L-fuzzy transformation semigroup from an interval lattice-valued state machine. In fact, both methods give the same result.