The Study Of Several Subsemigroups Of Strictly Partial One-one Order-preserving Transformations Semigroup

Let [n] = {1,2,...,n} ordered in the standard way. Let Tn(Pn) be the semi-group of full(partial) transformations of [n]. Let In(Sn) be the symmetric inverse semi-group (symmetric semigroup) on the [n]. We say that a transformation ??Pn is the order-preserving, if for all x,y E dom(?),x?y(?)x??y?. Let On be the set of all order-preserving transformations in Tn(excluding the identity map on the [n]), then On is the subsemigroup of Tn, we call it is the order-preserving transformations semigroup. Let POn be the set of all order-preserving transformations in Pn(excluding the identity map on the [n]), then POn is the subsemigroup of Pn, we call it is the order-preserving partial transformations semigroup. Let OIn be the set of all order-preserving transformations in In\Sn, then OIn is the inverse subsemigroup of In\Sn, we call it is the strictly partial one-one order-preserving transformations semigroup.Let k be a fixed point of [n]. We say that a transformation ??OIn, is the k type order-preserving, if for all x ? dam(?),x?k(?)xa? k. The collection of all k type order-preserving transformations on the [n] will be denoted by OIn(k). It is easy to verity that OIn(k) is a subsemigroup of OIn, we call it is the strictly partial one-one k type-order-preserving transformations semigroup.Let k,m be two fixed points of [n]. We say that a transformation ?? OIn is the(k,m) type order-preserving, if for all x,y E dom(?)) x?k(?)x??k,y?m(?)y??m}. The collection of all (k,m) type-oder-preserving transformations on the [n]will be denoted by OIn,(k,m). It is easy to verity that OIn(k,m) is a subsemigroup of OIn, we call it is the strictly partial one-one (k,m) type-order-preserving transformations semigroup.An element a of a semigroup S is called regular provided that there exists b ?S such that aba = a. The collection of all regular elements of S will be denoted by Reg(S). The collection of all regular elements of OIn(k)(OIn(k,m)) will be denoted by Reg(OIn(k))(Reg(OIn(k,m))). It is easy to verity that Reg(OIn(k))(Reg(OIn(k,m)))is a subsemigroup of OIn(k)(OIn(k,m), we call it is the strictly partial one-one k((k,m))type-regular order-preserving transf)ormations semigroup.We say that a transformation ??OIn is the order-decreasing(increasing), if for all x ? dom(?), implies x ? x?(x?x?). The collection of all order-decreasing(increasing)of OIn will be denoted by UOIn(MOIn).It is easy to verity that UOIn(MOIn)is a subsemigroup of OIn, we call it is the strictly partial one-one order-decreasing(increasing)transformations semigroup. Let MOIn(k) = MOIn?OIn. It is easy to verity that MOIn(k) is a subsemigroup of OIn, we call it is the strictly partial one-one k type-order-increasing transformations semigroup.In this thesis, we introduce several subsemigroup of the strictly partial one-one order-preserving transformations semigroup, in which study some properties. The main results are given in follwing:In chapter 1, we give introduction and preliminaries.In chapter 2, we study the rank and the maximal subsemigroup of the semigroup OTn(k), the main results are given in follwing:Theorem 2.1.7 Let 1 ? k ? n- 1, then rankOin(k)=n+1.Theorem 2.2.4 Let 1<k<n-1,M is proper subsemigroup of the semigroup OIn(k), then M is maximal subsemigroup of the semigroup OIn(k) if and only if exist a two-partition of certain continuous subchain on the [n] such that M = OIn(k)\E(???).In chapter 3, we study Green's relations, the rank and the maximal inverse subsemi-group of the semigroup Reg(OI(k)), the main results are given in follwing:Theorem 3.1.4 The Green's relations of the semigroup Reg(OIn(k))was de-scribed in follwing: For arbitrary ???, then(?)?R? if and only if ker(?) = ker(?).(?)?R? if and only if im(?) = im(?).(?)?R? if and only if ker(?) = ker(?)?im(?) = im(?).(?)?R? if and only if | im(?)| =| im(?)|.Theorem 3.2.5 Let 1 ? k ? n - 1? then rank Reg(OIn(k))=n.Theorem 3.3.3 Let 1<k<n-1,M is proper subsemigroup of the semigroup Reg(OIn(k)), then M is maximal inverse subsemigroup of the semigroup Reg(OIn(k))if and only if exist a two-partition of certain continuous subchain on the [n] such that M = Reg(OI(k))\(H(?,?)? H(A,??)).In chapter 4, we study the rank and quasi-idempotent rank, the maximal quasi-idempotent generated subsemigroup of the semigroup MOIn(k), the main results are given in follwing:Theorem 4.1.10 Let 1 ? k ? n - 1?then rankMOIn(k)= qidrankMOIn(k)=2n-2.Theorem 4.2.4 Let T is the maximal quasi-idempotent generated subsemigroup of the semigroup MOIn(k), then if and only if T = MOIn(k)\{?}?(?)??N(Jn-1k).Corollary 4.2.5 MOIn(k) has 2n - 2 the maximal quasi-idempotent generated subsemigroup.In chapter 5, we study the rank and the maximal subsemigroup of the semigroup OIn(k,m), the main results are given in follwing:Theorem 5.1.7 Let 1 ? k? n- 1, 2? m? n, thenTheorem 5.2.4 Let Let 1 ? k? n- 1,2?m?n adn m?k(orm?k+2), M is proper subsemigroup of the semigroup OIn(k,m) then M is maximal subsemigroup of the semigroup OIn(k,m) if and only if exist a two-partition of certain continuous subchain on the [n] such that M =OIn(k,m)\H(?, ?).In chapter 6, we study Green's relations, the rank and the maximal inverse subsemi-group of the semigroup Reg(OIn(k,m)), the main results are given in follwing:Theorem 6.1.4 The Green's relations of the semigroup Reg(OIN(k,m)) was de-scribed in follwing: For arbitrary ??, then(?)?R? if and only if ker(?) = ker(?).(?)?R? if and only if im(?) = im(?).(?)?R? if and only if ker(?) = ker(?)?im(?) = im(?).(?)?R? if and only if | im(?)| =| im(?)|.Theorem 6.2.5 Let 1 ? k ? n- 1,2?m?n, then rankReg(OIn(k, m)) = n.Theorem 6.3.3 Let 1?2?n-1, 2?m?nand m?k(or m?k+2),M is proper subsemigroup of the semigroup Reg(OInk,m)), then M is maximal inverse subsemigroup of the semigroup Reg(OIn(k,m)) if and only if exist a two-partition of certain continuous subchain on the [n] such that M = Reg(OIn(k,m))\\(H(?,?)?H(???)).