Some Studies On Several Transformation Semigroups With Order-preserving Property

Assuming that the natural numbern is greater than or equal to 3,OPDn,DOPDn,RCDOn and Gn are respectively singular transformation semigroups of the ordering-preserving and spacing-preserving parts,singular descending transformation semigroup-s of the ordering-preserving and spacing-preserving parts,canonical singular transfor-mation semigroups of the ordering-preserving,spacing-preserving parts and the semi-group of all order-increasing and order-preserving partial one to one singular transfor-mations on a finite-chain[n].And let OPD(n,r)={α∈OPDn:im(α)| ≤r},DOPD(n,r)={α∈DOPDn:|im(a)|≤r},WD(n,r)={α∈RCDOn:| im(α)|≤r}and G(n,r)={α∈Gn:|im(α)| ≤r} respectively be the two-sided star ideal of the semigroup OPDn,DOPDn,RCDOn and Gn for an arbitrary integer r such that 0 ≤r≤n-1.By analyzing the elements of rank r,(0,1)-square idempotent elements and star Green’s relations,the minimal generating set,(0,1)-square idempotent generating set,rank and(0,1)-square idempotent rank of the semigroup OPDn,DOPDn,RDOn and Gn are obtained,respectively.Furthermore,the relative rank of the semigroup OPD(n,r),DOPD(n,r)and G(n,r)with respect to itself each star ideal OPD(n,l),DOPD(n,l)and G(n,l)are determined for 0 ≤l ≤r.At the same time,the complete classification of maximal subsemigroups of OPDn,,DOPDn and RCDOn is obtained.In this paper,the rank and maximal subsemigroups of some classes of transformation semigroups with order preserving properties are introduced,and some properties of them are studied:In chapter 1,introduction and basic concepts.In chapter 2,to study the semigroup to OPD(n,r)rank with the great(inverse)the son of semigroup structure,the main results areTheorem2.2.2 Set n≥ 3,0 ≤r ≤n-1,thenrank(OPD(n,r))=(rn).Theorem2.3.1 Let r ∈[1,n-1]and M be A non-null true subset of semigroup OPDn,then M is A maximal subsemigroup of semigroup OPDn if and only if(1)M=OPD(n,r)\{e},where |im(e)|=r and |[im(e)]|=1.(2)M={OPD(n,r)\Dα} ∪ Hr(A,A)∪ Hr(B,B)∪ Hr(A,B),one of A and B is[im(α)]2-division,[im(α)]={A1,A2,……,Am} and |[im(α)]|>2.Theorem2.3.2 Let r ∈[1,n-1]and M be A non-null true subset of semigroup OPDn,then M is A maximal inverse subsemigroup of semigroup OPDn if and only if(1)M=OPD(n,r)\{e},where |im(e)|=r and |[im(e)]|=1.(2)M={OPD(n,r)\Dα} ∪Hr(A,A)∪Hr(B,B),one of A and B is[im(α)]2-division,[im(α)]={A1,A2,…,Am} and |[im(α)]|≥2.In chapter 3,the rank sum maximal subgroup structure of DOPD(n,r)is studied.The main results are as follows:Theorem3.2.2 Set n≥ 3,0 ≤r ≤n-1,thentheorem3.3.1 Let r ∈[1,n-1]and M be the non-null subset of DOPD(n,r)of the semigroup if and only if(1)1=DOPD(n,r)\{e},where |im(e)|=r and |[im(e)]|=1.(2)M={DOPD(n,r)\Dα}∪{M1\α},to which α ∈ M1(?)Dr,M1 is the minimal generating set of semigroup DOPD(n,r),[im(α)]={A1,A2,…Am} and |[im(α)]|≥ 2.In chapter 4,the rank of RCDOn and the complete classification of maximal(regular)subsemigroups of ideals are described.The main results are:theorem4.2.8 Let n≥ 3,then rank(RCDn)=2.theorem4.3.8 Let r ∈[2,n-1],then the maximal regular subsemigroups of semi-group WD(n,r)have the following three types:(1)M(A,B)=WD(n,r-1)∪Hr(A,A)∪Hr(B,B),where(A,B)is some 2-partition of[1,n-r+1];(2)M(O,D)=WD(n,r-1)∪ Or(A,A)∪Or(B,B)∪Dr(A,B)∪Dr(B,A),where(A,B)is some 2-partition of[1,n-r+1];(3)M(O,r)=WD(n,r-1)∪DrO.In chapter 5,the rank and(0,1)-square idempotent rank of G(n,r)are described.The main results are:theorem5.2.2 Let n≥ 3,then rank(G(n,r))=qidrank10(G(n,r))=nr+n-r2/n(rn).theorem5.2.3 Let n≥3,then r(G(n,r)),G(n,l))=(?)In chapter 6,summary and prospect.