On the Semigroup of Order-decreasing and Order-preserving Finite Full Transformations
Pei Huisheng · Journal of Xinyang Normal University · 2006
Let T_n be the full transformation semigroup on the finite set X={1,2,…,n},S_n be the subsemigroup of all singular transformations in T_n.Denote S~-_n={f∈S_n:x∈X,f(x)≤x},and O_n={f∈T_n:x,y∈X,x≤y implies(f(x)≤f(y)}.)Then both S~-_n and O_n are subsemigroups of T_n.Let H_n=S~-_n∩O_n.Some properties for H_n,such as,(Green's) relations,Green's starred relations,rank and idempotent rank,are observed.Among other things,it is shown that H_n is idempotent-generated and that it is generated by n-1 idempotents in J~*_(n-1).