The Order-reversing Quasi-idempotents Of The Semigroup POD_N

Let n> 4, Xn={1,2,...,n} ordered in the standard way. We denote by PJn the semigroup (under composition) of all partial transformations of Xn. We say that a transformation a in PJn is order-preserving if, for all x,y G dom(pt), x≤y implies xa≤ya. Let POn be the set of all partial order-preserving transformations in PJn, then POn is a subsemigroup of PJn, we call it the partial order-preserving transforma-tion semigroups. Conversely, we say that a transformation a in PJn is order-reversing if, for all x,y ∈dom(a), x≤ y implies xa> ya. Let PDn be the set of all partial order-reversing transformations in PJn. Let TODn= POn∪PDn, it is easy to show that PODn is a subsemigroup of PJn, we call it the order-preserving or order-reversing partial transformations semigroups. According to the nature of PODn, product of any two of the order-preserving transformations or any two of the order-reversing transformations are also the order-preserving transformation. However, product of the order-preserving transformation and the order-reversing transformation is the order-reversing transforma-tion.An element a of any given semigroup is called idempotent if a2= a and quasi-idempotent if a4= a2(a2 is idempotent). An element a of any given semigroup is called order-reversing quasi-idempotent if a is order-reversing transformation and quasi-idempotent. In this paper, we mainly study the order-reversing quasi-idempotents of the semigroup PODn. The following results of are given:In Chapter 2, we study the order-reversing quasi-idempotents of the semigroup TODn of rank n-, the following results are given:Theorem 2.3 Let n> 4, let a be the order-reversing transformation of Jn-1, then(1) When a∈[n,n-1], let A be the non-singleton kernel block of a, if A(?) im(a), then the necessary and sufficient conditions of a is order-reversing quasi-idempotent is dom(a)∩im(a)= Twin(a).(2) When a∈[n,n-1], let A be the non-singleton kernel block of a, if A (?) im(a), then the necessary and sufficient conditions of a is order-reversing quasi-idempotent is dom(a)∩im{a)= Twin(a) U A.(3) When a∈[n-1,n-1], then the necessary and sufficient conditions of a is order-reversing quasi-idempotent is dom(a)∩im{a)= Twin(a).In Chapter 3, we study the order-reversing quasi-idempotent rank of the semigroup PODn, the following results are given:In order to complete the proof of theorem, to begin with, we describe some sets of Jn 1 of PODn. LetRemark: when a > b or c > d, B(c,d)(a,b) = (?); when e > forg >h,D(g,h)(e,f)=(?). LetLemma 3.3 Let n≥4, then(1)Mi·Mi(?).Mi∪In-2.(2)Nj·Nj(?)NjIn-2Lemma3.10 Letn_>4, h= (nn-1…1 1 2…n),if PODn=<A> andAisthe set of order-reversing quasi-idempotent, then h∈A.Corollary 3.11 Let n ≥4, then Rqidrank(PODn) ≥n + p.Theorem 3.1 Let n > 4, p = [n/2], then Rqidrank(PODn) = n + p.