Study On Some Problems Of Ordered Semigroups

In this dissertation, we study C-ideals in ordred semigroups, left ideals and Green's relations L in ordred semigroups, the chain of simple ordred semigroups, semilattice composition of regular ordred semigroups and quasi-separative ordered semigroups. The dissertation is divided into six sections. The main content is as follows.In the first section, we give the introductions and preliminaries.In the second section, the concept of a C-ideal is extended to an ordered semi-group. We discuss some elementary properties of C-ideals in ordered semigroups. We give the definitions of base and the greatest C-ideal of an ordered semigroup and study the relation between them. As an application, we also discuss one class ordered semigroups in which every proper ideal is a C-ideal. The main results are given in follow:Theorem 2.5 Let S be an ordered semigroup. If S contains two different proper ideals M1, M2 and M1∪M2=S, then neither of them is a C-ideal.Corollary 2.6 Let S be an ordered semigroup. If S contains more than one maximal ideal, then none of them is a C-ideal.Theorem 2.13 Let S be an ordered semigroup. I is not an empty subset of S, then I is a maximal ideal if and only if S - I≠φand S - I is a maximal J-classes about partical order (?).Theorem 2.16 Let A be a non-empty subset of an ordered semigroup S. A is a base of S if and only if:(1) For all x in S, there exists a in A such that I(x)(?)I(a);(2) For all a1, a2 in A, Ja1 (?) Ja2 implies a1= a2.Theorem 2.19 Let S be an ordered semigroup, which is not simple. If A is a base of S, then S contains the greatest C-ideal Mgand Mg=(S3]∩(?). where (?) is the intersection of all maximal ideals of S.Theorem 2.22 Let S be an ordered semigroup. All proper ideals of S are C-ideals if and only if S satisfies just one of the following conditions: (1) S contains the maxium ideal M* and M* is a C-ideal:(2) S=(S2] and for any proper ideal M and for every principal ideal I (a)(?)M, there exists b in S - M such that I (a) (?) I(b), where I(b) is a proper principal ideal of S.In the third section, we give the concept of the L-trivial ordered semigroups and discuss the relation between L-trivial ordered semigroups and nil ordered semigroups. We also discuss the relation between left△-ordered semigroups and nil ordered semigroups. The main results are given in follow:Proposition 3.11 Let (S,·,≤) be an ordered semigroup. If S is L-trivial, then the divisibility relation (?) on S is a partical order. Conversely, if the divisibil-ity (?) is antisymmetric, then S is L-trivial.Corollary 3.12 An ordered semigroup S is L-trivial if and only if the divis-ibility relation (?) on S is a partical order.Proposition 3.13 Let S be an ordered semigroup. The following statements are equivalent.(1) S is L-trivial and all the principal left ideals of S form a chain under inclusion.(2) The divisibility ordering (?) on S is a chain.Theorem 3.16 Every nil ordered semigroup is L-trivial.Theorem 3.20 Let S be a nil ordered semigroup which is a chain with respect to the divisibility ordering (?), then every complete left congruence on S is a left Rees congruence.Proposition 3.23 Let S be a nil ordered semigroup which is a chain with respect to the divisibility ordering (?), then S is a left△-ordered semigroup.In the fourth section, we study the chain of simple ordered semigroups. The main results are given in follow:Theorem 4.4 Let S be an ordered semigroup which is not simple. If S is a chain of simple ordered semigroups, then the set which is constituted by all the ideals of S endowed with the relation where A, B are ideals of S, is a chain.The main results of the fifth section are given in follow:Proposition 5.3 Let (S,·.≤) be a regular ordered semigroup, Y a semilattice and {Sα}α∈Y a family of regular ordered subsemigroups of S. Then S is a semilattice Y of the regular ordered semigroups {Sα}α∈Y if and only if there is a homomorphism φ:S→Y such thatφ-1(α)=Sαfor everyαin Y.Theorem 5.5 Let Y be a semilattice and{(Sα,oα:≤α)}α∈Y a family of disjoint regular ordered semigroups indexed by Y. Let {Dα}α∈Y be a family of ordered semigroups such that Dαis an extension of Sαfor everyαin Y.Suppose that for eachα,β∈Y,β(?)α,there exists a mappingφα,β:Sα→Dβwith the properties:(1)(?)α∈Y,(?)x∈Sα,φα,α(x)=x.(2)(?)α,β∈Y,φα,αβ(Sα)·φβ,αβ(Sβ)(?)Sαβ.(3)ifα,β,γ∈Y,(?)αβ,a∈α,b∈Sβ,thenWe assume that for eachα,β∈Y,α(?)β,there exists a set Rα,β(?)Sα×Sβsuch that(4)(?)α∈Y,Rα,α=≤α(5)If(a,b)∈Rα,β,(b,c)∈Rβ,γ,then(a,c)∈Rα,γ.(6)If(a,b)∈Rα,β,(c,d)∈Rγ,δ,thenWe put S=Uα∈Y Sα, and define operation * and relation≤* as follows: If a∈Sα,b∈Sβ,then And then(S,*,≤*)is a regular ordered semigroup and it is a semilattice Y of the regular ordered semigroups{(Sα,oα,≤α)}α∈Y.Conversely,every regular ordered semigroup which is a semilattice Y of regular ordered semigroups can be so con-structed.In the sixth section,we define a relationΩon the ordered semigroups and give the decomposition of the quasi-separative ordered semigroups into the semilattice of the quasi-cancellative ordered semigroups.The main results are given in follow:Theorem 6.7 Let the relationΩon a quasi-separative ordered semigroup S satisfy the condition(1*),(2*),(3*)Then S/～Ωis a semilattice.Theorem 6.10 If an ordered semigroup S is quasi-separative:then S is a semilattice of quasi-separative quasi-cancellative ordered semigroups.Proposition 6.11 Every separative quasi-cancellative ordered semigroup is cancellative.Proposition 6.14 Let S be a quasi-cancellative weakly balanced ordered semigroup, then S is weakly cancellative.