The Study Of One-One Partial Order-preserving Transformation Semigroups

Let [n] = 1, 2,…n, ordered in the standard way. We donote by In is the symmetric inverse semigroup and Sn is symmetric group of [n]. We say that a transformations α in In is order-preserving if, for all x, y ∈ [n],x≤ y(?)xa≤ya. Let OIn be the set of all order-perserving injective partial transformations in In, then OIn is the inverse subsemigroup of In, we call it is the order-preserving injective partial transformation semigroup.The concept of m-skewness rank of the semigroup OIn is introduced for the first time. For an arbitrary integer m such that 1≤m≤n - 1, the authors proved that the necessary and sumcient condition for the existence of m-skewness rank of the semigroup OIn is that m and n are coprime, and obtained that the m-skewness rank of the semigroup OIn is n. This article have also proved the square idempotent rank of the semigroup OIn is 2n-2, and proved the square idempotent maximal subgroups of semigroup OIn were classified.In this paper. The main results of the paper are given as follows:lemma 2.3 Let 1≤m≤n- 1 and 1≤i≤n, Let Rim = <Gm>∩R(i), then Rim={μi+kmi:k∈N}.Theorem 2.5 Let n≥ 3 and 1≤m≤n-1, then <Gm> = OIn if and only if m and n are coprime. When m and n are coprime,SrartkmOIn=n.lemma 3.1 Let a is not a square idempotent, then a is square idempotent in OIn if and only if for any of x∈ dora(a), if xa≠x, then xa(?) dora(a).lemma a.a Let n≥3, then E2(Dn-1 = {aii-1:2≤i≤n}∪{ajj+1:1≤j≤ n- 1}.Theorem 3.6 Let n≥3, (E2(Dn-1)? = OIn, and quaidrarzkOIn= 2n-2.Theorem 4.1 Let n≥3, then the maximal idempotent-generated subsemigroup of the semigroup OIn has the following forms:(1) K(n,n-2)∪{a∈Dn-1:(x∈dom(a)x≥8(?)Oxa≥i},3≤8≤n-1,(2) K(n,n-2)∪{a∈Dn-1:(x∈dom(a)x≤j(?)xa≤j},2≤j≤n-2,(3) K(n, n-2)∪(Dn-1\Bk),k= 1,n,(4) K(n, n-2)∪(Dn-1\Ll),l= ,n,lemma 4.7 Let E2*(Dn-2) = {ai:1≤i≤m} , then K(n,n-2) = <E2*<Dn-2)>.