Studies On Some Classes Of Free Regular Bands

The algebraic structures of some free regular bands were mainly researched in this paper. The structures of free(left, right) normal band, free left(right) regular band and free left(right) quasi-normal band were specifically described.Firstly, the free left(right) normal band was defined and these two classes of semigroups’ structure were described. By the binary relation defined on the free semigroup X?, a quotient set was obtained by a congruence relation generated by the binary relation. Then the quotient set was proven to form a semigroup and to be the free left(right) normal band on the set X under the well defined multiplication. The elements of the free left(right) normal band constructed by this way were described.By some marks defined on the free semigroup X?, the set FNB(X) was proven to form a semigroup and to be the free normal band on set X. The Green’s equivalences were described on the free normal band. Further, considering the situation of unilateral, the similar structure of free left(right) normal band was researched. Then, the?-relation and the?-relation on free left(right) normal band was described.Secondly, the free left(right) regular band was defined and its structure was described algebraic structure of free left(right) regular band was researched. The description of elements in the free left(right) regular band were studied, and the number of elements of the free left(right) regular band was given when the set was finite.Finally, the free left(right) quasi-normal band was defined and its algebraic structure was given. By the congruence relation generated by the well defined binary relation, a quotient semigroup was obtained. The quotient semigroup was proven to be the free left(right) quasi-normal band on set X. Then the elements of the free left(right) quasi-normal band were described. And the?-relation and?-relation on the free left(right) quasi-normal band was researched. Then the number of elements of the free left(right) quasi-normal band was given when the set was finite.