The Study Of K Type-orientation-preserving One-to-one Partial Transformations Semigroup

Let[n]={1,2,...,n} ordered in standard way.Let Pn(Tn)be the semigroup of par-tial(full)transformations on[n].Let Jn(Sn)be the symmetric inverse semigroup(symmetric semigroup)on[n].SJn=Jn\Sn is strictly partial one-to-one transformations semi-group on[n].We say that a transformation ??Jn,is the order-preserving,if for all x,y?dom(?),x?y(?)x??y?,let On the submonoid of Jn of all order-preserving transformations;we say that is orientation-preserving,if the sequence of(1?,2?·…,n?)is cyclic that can be at most one natural number i to satisfy with i?>(i+1)?,let TOPn the submonoid of Tn of all orientation-preserving transformations,Let OJn be the set of all order-preserving transformations in SJn,we call it is the strictly partial one-to-one order-preserving transformations semigroup.let POPJn be the set of all orientation-preserving transformations in SJn,we call it is the strictly partial one-to-one orientation-preserving transformations semigroup.Let k be a fixed point of[n],we say that a transformation ??S is the k-type,if for all x? dom(?),x?k(?)x??k.and a transformation a is the bilateral k-type,if for all x? dom(?),x?k(?)x??k.Let#12 then POPJn(k)is A regular subsemigroup of pOpJn,and POPJn(k)is said to be the k type-orientation-preserving one-to-one partial transformations semigroup on a finite chain[n].Let POPJ(n,r)(k)={??POPTn(k):|im(?)|?r}(0?r?n-1),it is easy to verity that POPT(n,r)“(k)is a subsemigroup of POPJn(k).An element a of a semigroup S is called regular provided that there exists ??S,such that a??=?,The collection of all regular elements of S will be denoted by Reg(S).The collection of all regular elements of;POPJ(n,r)(k)will be denoted by Reg(POPT(n,r)(k))=POPJk(n,r).Let#12 then POPJnk is A regular subsemigroup of POPJn(k),and POPJnk is said to be the all bilateral k type-orientation-preserving one-to-one partial transformations semigroup on a finite chain[n].Let#12 it is easy to verity that POPJ(n,r)k is a subsemigroup of POPJnk.In this thesis,we introduce the subsemigroups of a class of the strictly partial one-to-one orientation-preserving transformations semigroup,in which study some properties,we use egg box picture,Green's relations,combinatorial mathematics and so on,specific as follow.In chapter 1,we give introduction and preliminaries.In chapter 2,we study the regular,Green's relations and element characteristics.In chapter 3,we study the rank of the semigroup POPJ(n,r)(k)(POPJ(n,r)k).In chapter 4,we study the maximal subsemigroup of the semigroup POPJ(n,r)(k)(POPJ(n,r)k).In chapter 5,we introduce summary and prospect of my thesis.