Maximal Substructures Of Several Transformations Semigroups

Let Xn= {1<2<…<n},a chain of n elements,and Tn be the finite full transformation semigroup under composition transformations from Xn to Xn.A trans-formation α in Tn is called order-preserving transformation if x ≤ y implies that xα≤yα for all x,y ∈ Xn.A transformation α in Tn is called an order-reversing transforma-tion if x ≤y implies that xα ≥ yα for all x,y ∈Xn.A transformation α in Tn is called a compression transformation if |xα-yα| ≤|x-y| for all x,y ∈ Xn.If On is a regular subsemigroup of Tn consisting of all order-preserving transformations on Xn,we say that On is a order-preserving transformations semigroup.If ODn is a regular subsemigroup of Tn consisting of all order-preserving transformations or order-reversing transformations on Xn,we say that ODn is a monotone transformation semigroup.If MCn is a non-abundant subsemigroup of Tn consisting of all compress transformations in ODn\{γn,idn},we say that MCn is a monotone and compressing transformation semigroup.If OCKn be a non-abundant subsemigroup of Tn consisting of transformations in On\{idn},which a kernel with a continuous transversal,we say that OCICn is transformation semigroup of kernel with a continuous transversal.Variants of Tn with respect to an element a ∈Tn,denoted Tna,is the semigroup with underlying set Tn and operation*a defined by x*a y = xay for all x,y ∈Tn.In this paper,we will consider the following:(1)Finite full transformation semigroup Tn.(2)Variants of the finite full transformation semigroups Tna and P(that is the regular part of TXa).(3)Ideals of the semigroup MCn,i.e.:MC(n,r)= {α ∈ MCn:|im(α))|≤r},1≤r ≤ n-1.(4)Ideals of the semigroup OCKn,i.e.:OCK(n,r)= {α∈OCKn:|im(α)| ≤r},1 ≤ r ≤n-1.The main results are given in following:In chapter 1,we introduced some related concepts of semigroup.In chapter 2,we studied the structure of sub-left-group(or sub-right-group)of Tn,the main results are given in following:Theorem 2.2.7 Let S be a subsemigroup of Tn,then(1)S is a sub-left-group of Tn if and only if im(α)= im(β)for all α,β∈S.(2)S is a sub-right-group of Tn if and only if ker(α)=ker(β)for all α,β∈S.Theorem 2.3.1(1)The maximal sub-left-groups of Tn must be the following forms:(2)The maximal sub-right-groups of Tn must be the following forms:Theorem 2.3.3 Some combinatorial results:(1)Let 1 ≤ r ≤ n,and A be a nonempty subset of Xn with |A| = r,then(2)The number of maximal sub-left-group of Tn be 2n-1.(3)Up to isomorphism,the number of maximal sub-left-group of Tn be n.In chapter 3,we studied the maximal subsemigroups of Tna.Also,we considered the maximal regular subsemigroups of the regular part of Tna.The main results are given in following:Theorem 3.2.2 Let S be a subsemigroups of Tna,then S be a maximal subsemigroup of Tna if and only if there exists β ∈R such that S=Tna\{β},where R = {f ∈Tn:|im(f)|>r}.Theorem 3.3.5 Let 1<r<n,then each maximal regular subsemigroup of Dra must be one of the following forms:(1)(?)= ∪(i,j)∈I×J Mij，where Mij be maximal subgroup of Hεija for each(i,j)∈I×J and the corresponding subgroups Uij of those Mij in Sr all coincide;(2)Γ= Dra\Lβa,where β∈Dra;(3)△ =Dra\Rβa,where β∈Dra.Theorem 3.3.9(?),Γ,△ as in theorem 3.3.5.(Ⅰ)If r = 1,then a subsemigroup S of P is maximal if and only if there exists β∈P such that S=P\{β}.(Ⅱ)If r = 2,then each maximal regular subsemigroup of P must be one of the following forms:1)A2 =Ir-1a∪(?);(2)B2 = Ir-1a ∪ Γ;(3)C2 = Ir-1a∪△;(4)D2 = D2a.(Ⅲ)If 2<r<n,then each maximal regular subsemigroup of P must be one of the following forms:(1)Ar = Ir-1a∪(?);(2)Br=Ir-1a ∪ Γ;(3)Cr = Ir-1a ∪ △;(4)Dr = Ir-2a ∪ Dra.In chapter 4,we studied the maximal subsemigroups of OCK(n,r)and MC(n,r),where 1 ≤ r ≤ n-1.The main results are given in following:Theorem 4.2.15(Ⅰ)If r = 1,then a subsermigroup S of MC(n,1)is maximal if and only if there existsβ ∈ MC(n,1)such that S = MC(n,1)\{β}.(Ⅱ)If r = 2,then the maximal subsemigroup of MC(n,2)must be the same as the maximal subsemigroup of OD(n,2)= {α ∈ODn:|im(α)|≤2}.(Ⅲ)If 3 ≤ r ≤ n-1,then each maximal subsemigroup of MC(n,r)must be one of the following forms:(1)MAr= MC(n,r)\Rα◇MCn,where α∈Irreg(Jr◇MCn);(2)MBr=MC(n,r)\H(∑,Λ),where(∑,Λ)be a 2-partition of[1,n-r + 1];(3)MCr = MC(n,r)\[Reg(Jr◇MCn)]↑;(4)MDr = MC(n,r)\[(H(P1,P1)∪H(P2,P2))↓∪(H(P1,P2)∪H(P2,P1))↑],where(P1,P2)be a 2-partition of[1,n-r + 1].Theorem 4.3.11(Ⅰ)If r = 1,then a subsemigroup S of OCK(n,1)is maximal if and only if there exists β ∈ OCK(n,1)such that S = OCK(n,1)\{β}.(Ⅱ)If r = 2,then the maximal subsemigroup of OCK(n,2)must be the same as the maximal subsemigroup of the O(n,2)={α∈On:|im(α)| ≤2}.(Ⅲ)If 3 ≤ r ≤ n-1,then each maximal subsemigroup of OCK(n,r)must be one of the following forms:(1)OAr = OCK(n,r)\Lα◇OCKn,where α∈Jr◇OCKn;(2)OBr =OCK(n,r)\H(P1,P2),where(P1,P2)be a 2-partition of[1,n-r+1].