Study On Special Substructures Of Three Kinds Of Semigroups

Let n,m∈N₊,S_n and T_n be the symmetric group and the full transformation semigroup on Xn={1,2,…,n},respectively.For 1≤m≤n-1,let X_m={1,2,…,m}.LetT_n,m={α∈T_n:X_mα=X_m},g_n,m={α∈T_n,m:（Xn\X_m）α=Xn\X_m},H_n,m={α∈T_n,m:（Xn\X_m）α（?）Xn\X_m},M_n,m={α∈T_n,m:Xnα（?）X_m},M_n,m^*=M_n,m∪G_n,m,then the semigroups H_n,m（r）^*,M_n,m^*,G_n,m,H_n,m,M_n,mand T_n,mare subsemigroups of the full transformation semigroup T_n and g_n,m（?）H（n,m,）（?）T_n,m,M_n,m（?）T_n,m.In this paper,by analyzing the Green’s relation of the semigroups H_n,m,we explore the characteristics of two special substructures of the maximal regular subsemigroups and the isolated subsemigroups respectively,based on the property that the top semigroup g_n,mis a symmetric group.Combining with the properties of integer splitting in combinatorial mathematics,we study the complete classification of the maximal regular subsemigroups and the maximal subsemigroups of two kinds of subsemigroups H_n,m*（r）and M_n,m^*with similar structure of the semigroup T_n,m,and get the conclusion that the structures of two kinds of maximal subsemigroups are consistent.In addition,combining with the connectedness of point graph,we construct a partition monoid Pn with a subset of the finite set[n]∪[n’]as the element.In this paper,we study the order-preserving partition monoid OP_n of its subsemigroups,and obtain Green’s relation,idempotent generative set and rank of the order-preserving partition monoid OP_n.