Some Property Research Of Several Classes Transformation Semigroups

Let [n] = {1,2…, n} be a finite set, with the natural order, and let Sn,In, Tn be the symmetric group, the symmetric inverse semigroup and the full transformation semigroup,respectively. For any k ? [n], let CISknk = {? ?In\Sn: ????i, j ? dom??? ? [k]?i?,j? ?k, |i?-j?| = |i-j|} and the transformation semigroup of LISnk is called the local distance-preserving transformation semigroup on In. Let LOISnk = {? ? LISnk : ???x,y?dom???,x ? y ??? x??y?},and the transformation semigroup of LOISnk is called the order-preserving local distance-preserving transformation semigroup on [n].Let Singn be a singular transformation semigroup on [n], and ? ? Singn. ? is said to be monotonic increasing ?monotonic decreasing? if ???x,y ? [n],x?y??? x ?? y? ?x ?y ??? x? ? y??. The assemblage of the monotonic increasing and monotonic decreasing transformations, on [n] is, Mn. It is a regular subsemigroup on Singn. Let ? ? Mn.? is said to be compressing if x,y ? [n],|x?-y?|?|x - y|. The assemblage of all compressing elements of Mn, is MCn. The MCn is the monotonic compression singular transformation semigroup on [n].Firstly, this paper will depict the element's regularity in LXSnk. For ? ? LISnk and a is regular if and only if x ? dom??? ? ?[n] \ [k]?, then x? > k.Secondly, this paper depicts the Green's relation of LISnk,????,??LISnk,??,??? L if and only if im ??? = im???, and dom??? ? [k]?dom??? ?[k];??,??? R if and only if dom??? = dom???, im??? ? [k]?im??? ? [k] and ???i ?dom??? ? ?k + 1,k + 2, … ,n} then i?,i? ? [k], or i?,i? ? {k + 1,k + 2,…, n};??,?? ?D if and only if |im???| = Iim???|, and dom????[k]? dom????[k], im????[k]? im??? ? [k].Thirdly, through defining the equivalence relation of elements, this paper will analyze the elements characters of LOISnk, and the rank of LOISnk can be obtained. Let n ? 3,and 1 ? k ? n-1, then the rank?LOISnk? = n+1 can be obtained. Further, the structures and classification of the maximal subsemigroups of the semigroup LOISnk are obtained.The maximal subsemigroup of the semigroup LOISnk must be one of the following forms:?1?A_?= LOISnk \ {?}, ? ? H*?P1,P1??P1 = [k]?;?2?B_H= LOISnk \ H*??,?? and ??,??is a second division on [n] \ [k];?3? C = LOISnk \ H*?P1,P2??P1 = [k],P2 = [n] \ [k]?.Lastly, this paper considers the structures and classification of the maximal subsemi-groups of the semigroup MCn. The maximal subsemigroup of the semigroup MCn must be one of the following forms:?1?A₁ = MCn \ R^??r,r + 1?, 2 ? r ? n - 2.;?2?A₂ = MCn \ U, and U?{H?1,2??n,H?n-1,n??1?;?3? A₃ = MCn\V, and V? {{e2, f2,?1,?1}, (e2,f2,?2,?2}};...