Several Questions Of The Finite Order-Preserving Extension Transformation Semigroups

Let In be the symmetric inverse semigroup on a finite set Xn={1,2,…,n},and let IOn,DPn and ODPn be semigroups of order-preserving transformation,of partial isometries and of order-preserving partial isometries of Xn,respectively.In 1992,Gomes and Howie studied the finite order-preserving transformation semigroup,and then Al-Kharousi,Kehinde and Umar studied the semigroup of partial isometries and order-preserving partial isometries.In this paper,we extend the order-preserving partial isometries in[2]to the order-preserving partial extension,namely,we study the finite order-preserving extension transformation semigroup OEX,This paper is divided into six chapters:Chapter 1:the development background of the algebraic theory of semigroups is introduced,and the study status of the sub-semigroups of IOn is also presentedChapter 2:the basic concepts and important theorems about the algebraic theory of semigroups are listedChapter 3:the Green's relations and Green's*relations on OEXn are described,and it is a non-regular and type A semigroup is also provedChapter 4:all minimal generating sets of OEXn are listed,and its rank is also calculatedChapter 5:the order,the cardinalities of L-classes,R-classes and D-classes of OEXn and the depth of the singular subsemigroup of OEXn are calculatedChapter 6:the further research related to this paper is summarized and expectedFrom left to right are all the composition of the transformations involved in this paper,i.e.(x)??=(x?)? for all ?,??In and x?Xn.