Several Problems Of Partition Order-decreasing Transformation Monoids

Let Xn={1,2,…,n},Tn be the set of all full transformations on a set Xn,it is a semigroup under the composition of transformations,which is called the full transformation semigroup on Xn. Here the composition acts of transformations from the left to the right:let S is a transformation semigroup,(x)αβ=(xα)β for all α,β∈S and all x∈Xn.Let p be an equivalence relation on Xn and≤be a total order on the partition set Xn/ρ={xρ:x∈Xn}.We consider T(ρ,≤)={α∈Tn:(xα)ρ≤πρ,(?)x∈Xn}. Obviously,T(ρ,≤)is a submonoid of Tn and which is called partition order-decreasing transformation monoid.In particular,if p is universal,we have T(ρ,≤)=Tn;if p is the identity rela-tion,then T(ρ,≤)is the order-decreasing transformation monoid[26]；T(ρ,≤)is isomorphic to PTn-1 if Xn/ρ={{1},Xn\{1}} and{1}<Xn\{1},among PTn-1 is the partial transformation semigroup of Xn\{1}.This can be it is a generaliza-tion of the knowledge Sn- Tn and PTn-1.Hence we have a natural problem:this paper mainly investigate several problems of T(ρ,≤), such as Green’s relations the Green’s* relation H*,the Green’s* relations D* and J*, regular elements, regularity and ideals.This dissertation consists of six chapters in all:Chapter 1,after introducing development background of the transformation semigroups,we present development history and research status on the partition order-decreasing transformation monoid T(ρ,≤).Chapter 2,we will give some basis concepts on semigroup algebra theory and well-known results and conclusions on the partition order-decreasing transforma-tion monoid T(p,≤).Chapter 3, we first characterize regular elements of T(ρ,≤), then discussed regularity of T(ρ,≤).finally,we consider the inverse and completely regularity of T(ρ,≤). in addition, we also studied regular elements, regularity, the inverse and completely regularity of the order-decreasing transformation monoid, at this time T(ρ,≤) is the order-decreasing transformation monoid (p is the identity relation [26]).Chapter 4, we describe Green’s relations on T(p,≤).Chapter 5, we use the relations R* and L* on T(p,≤), discussed the Green’s* relation H*, the Green’s* relations D* and J* on T(p,≤).Chapter 6, we characterize ideals of T(p,≤).