BCK-Algebras and Hyper BCK-Algebras Induced by a Deterministic Finite Automaton

In this note first we define a BCK-algebra on the states of a deterministic finite automaton. Then we show that it is a BCK-algebra with condition (S) and also it is a positive implicative BCK-algebra. Then we find some quotient BCK-algebras of it. After that we introduce a hyper BCK-algebra on the set of all equivalence classes of an equivalence relation on the states of a deterministic finite automaton and we prove that this hyper BCK-algebra is simple, strong normal and implicative. Finally we define a semi continuous deterministic finite automaton. Then we introduce a hyper BCK-algebra S on the states of this automaton and we show that S is a weak normal hyper BCK-algebra.