A Study Of Hamilton Semigroup

In this dissertation, we characterize the Hamilton semigroup;be-sides ,we give a characterization of endomorphism semigroup and semi-direct product of Hamilton semigroup;finally,we give a definition andcharacterization of automorphism group and strong right zero band ofHamilton semigroup. The main results are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we give the proof of subsemigroup and homo-morphic image which are still Hamilton semigroup;besides we discussthe action of Hamilton semigroup on a set .The main results are givenin follow.Theorem 2.1 The subsemigroup of left (right) Hamilton semi-group is still left (right) Hamilton semigroup.Theorem 2.3 The homomorphic image set of left (right) Hamil-ton semigroup is still left (right) Hamilton semigroup.Theorem 2.6 On the left (right) Hamilton semigroup S,aρb (?)ak = bl,where a,b ∈ S,k,l ∈ Z+,then ρ is a maximum idempotent-separating congruence on S.proposition 2.9 On the left Hamilton semigroup S,the R ofGreen relation is idempotent pure left congruence;On the left Hamil-ton semigroup S,the L of Green relation is idempotent pure right con-gruence.That is Re = ES,Le = ES.In Chapter 3,we give the conclusion that the homomorphic imageof Hamilton semigroup is Hamilton semigroup;besides,we give a defi-nition descent order of Hamilton semigroup and discuss the semi-directproduct of the homomorphic semigroup of Hamilton semigroup and thedescent order of Hamilton semigroup.The main results are given in follow .Theorem 3.1 The the homomorphic set of Hamilton semigroupEndS on Hamilton semigroup S on the product(fg)(x) = f(x)g(x),f,g ∈ EndS,x ∈ S,is still a left (right) Hamilton semigroup.Theorem 3.6 S is a left Hamilton semigroup,S1 = {ai,i ∈ I},S2 = {bβ,β ∈ A},S3 = {e(p),p ∈ P},f is an endomorphism on the S,then(1)f |S1: S1 ?→ S, (2)f |S2: S2 ?→ S \ S1, (3)f |S3: S3 ?→ ESand be propitious to(4)f(aiaj) = f |S1 (ai)f |S1 (aj), ai,aj ∈ S,(5)f(bβai) = f |S2 (bβ)f |S1 (ai), ai,bβ ∈ S,(6)f(aibβ) = f |S1 (ai)f |S2 (bβ), ai,bβ ∈ S,(7)f(xe(p)) = f |Sk (x)f |S3 (e(p)), x,e(p) ∈ S,k ∈ {1,2,3}.is a map.On the other hand ,if f |S1,f |S2,f |S3 is a map as above,then f is ahomomorphism map.Theorem 3.9 The semi-direct product EndS ×α S of endomor-phic semigoup EndS of the left Hamilton semigroup S and the descentorder of left Hamilton semigroup S is a left Hamilton semigroup, wherex ∈ S ,f ∈EndS,xf = f(x).Theorem 3.13 The direct product S × S of the descent orderof left Hamilton semigroup S of the left (right) Hamilton semigroup Sand itself is a left (right) Hamilton semigroup .In chapter 4,we mainly discuss automorphism of Hamilton semi-group is a group,what'more we give the decomposition of direct productof Hamilton semigroup;besides,we characterize the relation of the au-tomorphism of automorphism of Hamilton semigroup to twe Hamiltonsemigroup. The main results are given in follow.Theorem 4.1 Let S =i∈Iaiα∈Abαp∈Pep be a left Hamiltonsemigroup.AssumeS1 = {i∈Iai }, S2 = {α∈Abα }, S3 = {p∈Pep },AS1 ,AS2, AS3 are automorphism group of S1,S2,S3 individually,thenautomorphism group of AS automorphic AS1 ×AS2 ×AS3 .That is ASAS1 × AS2 × AS3.Theorem 4.7 If the automorphism group AS,AS of left (right)Hamilton semigroup S, S are automorphic,then S/ρ S /ρ.In chapter 5,we mainly discuss left Hamilton semigroup formata strong right zero band on the product and this strong zero band isa left Hamilton semigroup,beside the idempotents of format a strongright band.Theorem 5.1 Assume S is a a strong right band of Sα,α ∈B,where B is a right band , Sα are all left Hamilton semigroup,then Sis left Hamilton semigroup.