Study Of The Kernel With Continuous Transversal Under The Semigroup Of Order-preserving Partial Transformations

Let nPO be the semigroup of all the order-preserving partial transformations on n X},...,2,1{n= for n35. Let nPOCK be the subs-emigroup consisting of kernel with continuous transversal in nPO.This thesis. The Green’s relations of nPOCK will be describe. Later, the rank of nPOCK and a class of its regular subsemigroups will be studied comprehensively discussed.In Chapter One, we firstly give the element’s standard representation, then defines the indecomposable element, The form shows below:and nr-￡￡21 among them. The number of indecomposable elements has been caulated. The result is.4321--+nnIn Chapter Two, To begin with, we describe element’s regularity in nPOCK. Let na?POCK and arim =, nr-￡￡14 among them, then a is regular if and only if in the standard representation(1) 12,...,-<<raa is a consistent sub-chain in nX. In the second place,The author describe Green’s,,DRL relations. For any nba?POCK, then ba),(?L if and only if imim =ba. ba),(?R if and only if kerker =ba and the image of ba, with synchronization. ba),(?D if and only if imim =ba and the image of ba, with synchronization.In Chapter Three, In according with the characteristics of elements of semigroup nPOCK. The author fully proves the elements of semigroup nPOCK in rJ can be generated by the union of n-1J and indecomposable elements. That it is:nrs nrsrJJ aaU and nr-￡￡23 among them. It is undifficult to draw the conclusion that VJPOCKnnè=-1 can be obtained on the premises of the natural number n35. Among the VJPOCKnnè=-1, the V is the set of indecomposable elements which are selected by the author. In the end, the rank of semigroup nPOCK will be obtained successfully,Let the natural number n35 innPOCK then we have rankn POCK53)( -=n.In Chapter Four, Adopting the similar research in chapter three, the author focuses his attention on the rank of a class regular subsemigroups of semigroup nPOCK. The ankgn POCKr12))((Re -=n can be obtai-ned on the presise of the natural number n35.