Several Questions Of The Order-Decreasing Transformation Semigroups

Let Xn={1,2,...,n} with a natural order 1<2<…<n.Tn be a full transformation semigroup on the set Xn,and let Singn be the singular part of the Tn.The orderdeceasing transformation semigroup on the set Xn is Sn-={α∈Singn| xα≤x,(?)x∈Xn}.A general study of Sn-was initiated by Umar in 1992,who proved that Sn-is generated by idempotents,and is generated by 1/2n(n-1)idempotents in Dn-1*in[36].We conduct further on Sn-on this basis.In this thesis,we first study the depth of the generating set of idempotents of Sn-,and then we obtain a kind of congruences p of the Sn-and study the regularity and abundance of the quotient semigroup Sn-/p,next we study the abundance of Rees quotient of Sn-.This thesis includes six chapters.In Chapter 1:We introduce the development of history of the algebraic theory of semigroups and study status of the order-deceasing transformation semigroup Sn-.In Chapter 2:We introduce the basic concepts,definitions,and important conclusions that are relevant to this paper.In Chapter 3:We prove that E(Dn-1*)is the intersection of all generating set of Sn-.The E(Dn-1*)-depth of any α∈Sn-,global E(Dn-1*)-depth of Sn-,E(Sn-)-depth of any α∈Sn-,and global E(Sn-)-depth of Sn-are obtained.In Chapter 4:We describe a kind of congruences p of the Sn-,and prove that the quotient semigroup Sn-/ρ is irregular,and neither left nor right abundant.In Chapter 5:We give necessary and sufficient conditions for the principal Rees congruences of Sn-to be abundant.Next necessary and sufficient conditions for the Rees congruences of Sn-to be abundant are given.In Chapter 6:We conclude this thesis and put the further research issues.