The Quasi-idempotents Of Partial Order-preserving Transformation Semigroups

Let[n]={1,2,..., n}ordered in the standard way. We denote by Singn the semi-group （under composition） of all singular transformations of [n]. We say that a transforma-tion a in Singn is order-preserving if, for all x, y ∈[n], x≤y implies xα≤yα. We denote by On the subsemigroup of Singn of all order-preserving singular transformations. Let POn be the set of all partial order-preserving transformations in Singn, then POn is a subsemi-group of Singn and POn= On∪{a:dom（a） （?）[n], （（?）x, y∈ dom（a））x≤y=y> xa≤ya}, we call it the partial order-preserving transformation semigroups.An element a of any given semigroup is called idempotent if α2≠α and quasi-idempotent if a2≠a and a4=α2. In this paper, we mainly study the quasi-idempotents of the semigroup POn. The following results of are given:In Chapter 2, we study the quasi-idempotent of the semigroup PO of rank n-1, the following results are given:Theorem 2.4 Let n≥3, Let where a1< a2<…<an-1, A1<A2<…<An-1, then a is the quasi-idempotent if and only if there exists unique k E{1,2,…,n-1} such that ak（?）Ak.In Chapter 3, we study the quasi-idempotent rank of the semigroup POn, the follow-ing results are given:Theorem 3.1 Let n> 3, Let E2（Jn-1） be the set of all quasi-idempotents of Jn-1, thenTheorem 3.3 Let n≥3, Let E2（Jn-1） be the set of all quasi-idempotents of Jn-1, then POn= （E2（Jn-1））.Theorem 3.5 Let n≥3, then quaidrank POn=rankPOn=2n-1.In Chapter 4, we study the ideal In-2 of the semigroup POn, the following results are given:Theorem 4.7 Let n> 5, then In-2= <E2（Jn₂）>In Chapter 5, we study the quasi-idempotent-generated maximal idempotent-generated subsemigroup of the semigroup POn, the following results are given:Theorem 5.1 Let S be the maximal idempotent-generated subsemigroup of the semigroup POn, if S is quasi-idempotent-generated, then S has the following forms:...