The Correlation Properties Of Semigroup OF_n(Y)

Let X be the natural number and give them the natural order,T(X)be the fully trasformation semigroup on X.Let α∈T(X),If any x,y∈X,x≤y(?)xα≤yα,Thenα is the oder-preserving,Let OT(X)is the oder-preserving set of T(x),then OT(X)be the full oder-preserving trasformation semigroup on X.Let Y is a fixed nonempty proper subset of X with |Y|=n(n≥ 3),Let OT(X,Y)={α∈OT(X):Xα(?)Y}and OT(X,Y)={α∈OT(X,Y):Xα=Yα},then OT(X,Y)and OF(X,Y)are subsemigroup of OT(X).We study the subsemigroup of OF(X,Y)defined by OFn(Y)={α∈OF(X,Y):|im(α)|≤n-1}.It is proved that the semigroup OFn(Y)is generated by idempotent elements,We stu-day the idempotent rank and rank of OFn(Y),describe a few class locally maximal idempotent-generated subsemigroups and maximal idempotent-generated subsemigroups of OFn(Y).In this paper,the main results of the paper are given as follows:Theorem 2.8 Let n≥ 3,then idrank OFn(Y)=∑i=1n-1Pi+|Γ|.Theorem 3.6 Let n≥ 3,then rank OFn(Y)=∑i=1n-1Pi.Theorem 4.10 Let 1 ≤i ≤n-1,then the following semigroups are locally maximal idempotent-generated subsemigroups of OFn(Y):(1)Mi◇=<E(Dn-1)\△ηi>;(2)Ni◇=<E(Dn-1)\△εi>;(3)Sα=(E(Dn-1)\E(Rα)>,α∈Dn-1,ker(α|Y)(>)Ωi.Theorem 4.11 Let 1 ≤i≤n-1,then the following semigroups are maximal idempotent-generated subsemigroups of OFn(Y):(1)Ai=Qn-2∪<E(Dn-1)\△ηi>;(2)Bi=Qn-2∪<E(Dn-1)\△εi>;(3)Cα=Qn-2∪<E(Dn-1)\E(Rα)>α∈Dn-1,ker(α|Y)(?)Ωi.