Lattices and Finite Machines

A finite machine is a sextuple M = (E, I, O, fE, fO, eo) formed by three non-empty finite sets E (set of states, eO is the initial state), I (set of inputs), O ( set of greetings), the transition function and the exit function . The finite machine concept is widely used in computational theories and artificial intelligence. In this work, we will prove the theorem that if M is a finite machine and S (M) is the set of all submachines of M, it is verified that it is a bounded and distributive lattice.