Congruences And Automorphisms Of The Direct Product Of Two Order-preserving Full Transformation Semigroups

Let X_n={1,2,...,9)}.The order-preserving full transformation semigroups on X_n is O_n={?(?)T_n|(?)x,y(?)X_n,x?y(?)xa?ya},where T_n is the full transformation semigroup on X_n.Since the 1960s,Aizenstat,Lavers and Solomon have described the congruences of O_nin[1,23].Next,Fernandes,Jesus,and Maltcev described the endomorphisms of O_nin[8],but no one has explored the congruences and endomorphism of the direct product O_m×O_n.In this paper,we described the principal congruences of O_m×O_nfirst,and on this basis,we further described all congruences of O_m×O_n.Next,we studied the endomorphisms of O_m×O_n,but we just obtained the automorphisms of O_m×O_n,and we proposed a further problem of describing the endomorphisms of O_m×O_n.There are mainly six parts in this paper:In Part 1,we briefly introduced the development of the theory of semigroup algebra and the research status of the semigroup O_n.In Part 2,we introduced some basic definitions and some important conclusions about the theory of semigroups.In Part 3,we described the principal congruences of O_m×O_nand obtained its all minimal congruences.In Part 4,on the basis of Part 3,we further described any congruences of O_m×O_n.In Part 5,we described the automorphisms of O_m×O_n.In Part 6,we concluded this paper and look forward to the further research issues.