The Idempotent Ranks,Congruences And Endomorphisms Of The Principal S_n-normal Sub-semigroups Of Finite Full Transformation Semigroups

Let Tn and Sn be the full transformation semigroup and the symmetric group on a finite setXn={1,2,…,n},respectively.Let S be a sub-semigroup of Tn.If?? S,g ? Sn are arbitrary given elements,we have g-1 ?g ? S,so we define S be a Sn-normal sub-semigroup([11],[40]).In this paper,the composition acts from the left to the right,i.e.let S be a transformation semigroup,(x)? ? ?=(x?)?for all ?,? ? S and X ? Xn.Moreover,we define(g-1 ?g | g ? Sn? be the principal Sn-normal sub-semigroup provided that the minimal Sn-normal sub-semigroup which contains?,where ? ? Tn.Levi and McFadden R B.had classified the Sn-normal sub-semigroups in 1994,but the poverty of principal Sn-normal sub-semigroups hasn't be described.Thus this is a meaningful thing to investigate the idempotent ranks,congruences and endomorphisms of the principal Sn-normal sub-semigroups of the finite full transformation semigroup Tn.We divide our paper into six chapters:Chapter 1:we introduce the development background of transformation semi-groups and present the research status of the Sn-normal sub-semigroups.Chapter 2:we present some basic concepts on the algebraic theory of semi-groups and some established results and conclusions of the principal Sn-normal sub-semigroups.Chapter 3:we describe the idempotent ranks of the principal Sn-normal sub-semigroups.Chapter 4:we describe the ideals and congruences of the principal Sn-normal sub-semigroups.Chapter 5:we describe the endomorphisms of the principal Sn-normal sub-semigroups.Chapter 6:we summarise this paper and pose the further research problems related to our topics.