On Noncounting Regular Classes

Let A∗ be the free monoid of base A and n a fixed positive integer. For any word wϵA∗ we consider the set [w]n of all the words which are equivalent to w modulus the congruence θn generated by the relation xn∼xn+1, where x is any word of A∗. The main result of the paper is that if n>4 then for any word wϵA∗ the congruence class [w]n is a regular language. We also prove that the word problem for the quotient monoid Mn=A∗θn is recursively solvable.