Sub Semigroup Study Of Partial Order-preserving Transformations Semigroup

Let Xn = {1,2,... ,n} ordered in the standard way. We denote by Singn the semi-group (under composition) of all singular transformations of Xn. We say that a transfor-mation a in Singn is order-preserving if, for all x,y G Xn, x≤ y implies xα≤ yα. We denote by 0n the subsemigroup of Singn of all order-preserving singular transformations, L(n,r) = {a ∈ 0n :| im(α) |≤r}(2≤r≤n-2) is the idea of 0n. Let POn be the set of all partial order-preserving transformations in Singn, then POn is a subsemigroup of Singn and POn = 0n ∪ {a : dom(α) (？) Xn, ((？)x, y ∈ dom(α)x≤ y(?) xα ≤yα}, we call it the partial order-preserving transformation semigroup. POr = {a ∈P0n, im(α) ≤ r,l≤r≤ n — 2} is the idea of the partial order-preserving transformation semigroup POn. SPOn =POn \ On is the Strict partial order preserving transformation semigroup of Xn, Nf(n,r) = {a ∈ SPOn :| im(α)≤r} is the idea of SPOn. Order preserving transfor-mation is an important part in the study of transformation semigroup, 0n and POn are two important Semigroups of order preserving transformation, In this paper, we mainly study four sub groups of POnIn this paper, The following results are given:In Chapter 2, we study the rank of similar partial order-preserving transformations semigroup, the following result are given:Theorem 2.8 Let Xn = {1,2,... ,n}, and gives the order of magnitude of the natural numbers, let LPOn =POn\[n,n — 1], LPOn is called similar partial order-preserving transformations semigroup on Xn, in this paper, the rank of the semigroup LPOn is proved to be (n2-n+2)/2.In Chapter 3, we study 3(LPOr), it is the idea of LPOn, the following results are given:Theorem 3.2 Let Xn = {1,2,... ,n}, and gives the order of magnitude of the natural numbers, let LPOn = POn\[n,n — 1] is the similar partial order-preserving transformations semigroup on Xn, J(LPOr) =POr\[n,r] is the idea of LPOn,lrJ(LPOr) is the top class J of 3(LPOr), an item is Theorem 3.8In Chapter 4, we study the rank of similar complete partial order-preserving trans-formations semigroup H(SPOn,r), the following result are given:Theorem 4.12Let n > 5, Xn = {1, 2,... ,n}, and gives the order of magnitude of the natural numbers, let H(SPOn,r) =SPOn∪L(n,r)(5≤n, 2≤r≤ n-3), M(SPOn,r) is complete similar partial order-preserving transformations semigroup on Xn, the rank of the semigroup H(SPOn,r) is proved to beIn Chapter 5, we study the Similar strict partial order preserving transformation semigroup L(SPOn) , the following results are given:Theorem 5.2rank(<[n-1, n-2]))=n(n-2).Theorem 5.7Let n≥5, Xn = {1,2,..., n}, and gives the order of magnitude of the natural numbers, N(n,r) = {α ∈ SPOn :| im(α) |≤ r} is the idea of the strict partial order preserving transformation semigroup SPOn, let ￡(SPOn) =N(n, 2)/[n-2,n—2]= N(n, 3)∪[n—1, n—2] is the Similar strict partial order preserving transformation semigroup of Xn, an item isnotes: [r, s] = {αa |α∈POn, | dom(α) |= r,|im(α)|= s}.