On Non-abundance Of Certain Semigroups Of Finite Full Transformations

Order-preserving problem is a hot issue in the research of transformation semi-groups.Let Tn be the semigroup of full transformations of Xn ＝ {1,2,...,n},A(?)Xn\{1},XE = {x ∈ Xn:x is even}.A is a collection of consecutive numbers in Xn and 1(?)A,A = {i,i + 1,…,i + m-1,i + m}(2 ≤i＜i + m ≤n).We defineSingn = {α ∈Tn:rank(α)≤ n-1},Sn-{α∈ Singn:xα≤x,(?)x ∈ Xn},Sn-(A)= {α ∈ Sn-:xa<x,(?)x A}.Then Sn-is the order-preserving transformation subsemigroup of Singn and Sn-(A)is a A-strict-decreasing subsemigroup of Singn.In recent years,some scholars study Sn-and get some good results.For example,British scholar Umar gave the Green’s relations and Green’s star relations on Sn-in[1]and characterised generating sets and rank of Sn-.However,they have little research on Sn-(A).Wangwei Li has done some research on the research of Sn-(A)in his Master Degree Thesis[2].He took A= XE for n is an even number and characterised it’s Green’s relations,their generalisations,regular part and generating sets.In this paper,we give a more general form of indecomposable elements in Sn-(A)for an arbitrary set A(?)Xn\{1}.For A,we characterise the Green’s relations on Sn-(A)and their generalisations,and show that Sn-(A)is a non-abundant semigroup.We also prove that Sn-(A)is not a weak U abundant semigroup.Meanwhile,we also get some results about regular part,generating sets and rank of Sn-(A).