Some Studies On Ordered Semigroups

In this paper we study some important properties on structures, ideals, homomorphismsand Ress-quotients of ordered semigroups.The section one is the introduction and preliminaries. we give some basic definitions and properties concerned.In section two. we discuss properties of globally idempotent ordered semigroups, Archimedean ordered semigroups and maximal ideals, prime ideals and the relations between them. The main conclusions of this section are:Theorem2.8 Let S be a commutative ordered semigroup and contains maximalideals. If S satisfies any one of the following conditions, then it contains ordered idempotents.(1) S is a globally idempotent ordered semigroup:(2) S isn't a globally idempotent ordered semigroup, for any a∈S\(S²]. (S\a]≠S. and any maximal ideal of S is non-trivial.Theorem2.12 Let S be a commutative ordered semigroup containing maximalideals. If S satisfies any one of the following conditions, we have S isn't a globally idempotent ordered semigroup and (S²]= M^*.(1) S has no ordered idempotents and for any a∈S\(S²], (S\a]≠S;(2) S is an Archimedean ordered semigroup and for any a∈S\(S²], (S\a]≠S.The section three is a continuation of the results in section two. We discusssome properties of two special commutative ordered semigroups -(?)-ordered semigroup and commutative chained ordered semigroup. The main conclusions of this section are:Theorem3.1 Let S be a commutative ordered semigroup and T := {a∈(?)}, then the following are ture:(1) T is a prime ideal;(2) S\T is an Archimedean subsemigroup.Theorem3.4 Let S be a commutative ordered semigroup containing maximal ideals, then the following are ture:(1) If S is a (?)-ordered semigroup, then S is a globally idempotent ordered semigroup:(2) If S is a globally idempotent ordered semigroup, then S is a (?)-ordered semigroup or S has an unique maximal ideal which is prime.Theorem3.8 For a commutative chained ordered semigroup S, the following are ture:(1) If an ideal P of S is a filter. then P =(?)(xⁿP] for any x not in P;(2) For any ideal A of S.(?) is a prime ideal;(3) If S has no ordered idempotents, then for any a in S, (a^ωS] is a prime ideal or (a^ωS] =(?).In section four, we discuss properties of ordered groups, right ordered groups and right zero ordered semigroups and relations between them. The main conclusionsof this section are:Theorem4.2 Let G be an ordered group. E be a right zero ordered semigroup. then G×E is a right simple ordered semigroup.Theorem4.5 Let S be a right ordered group, E(S) := {x∈S|x≤x²}. Then the following are ture:(1) S is a regular ordered semigroup; (2) E(S)≠(?):(3) E(S) is a right zero subsemigroup of S:(4) b≤eb for every b in S. e in E(S);(5) If Se is a subsemigroup of S for every e in E(S), then Se is an ordered subgroup:(6) Let f be a fixed element of E(S). If Sf is a subgroup of S, then in the map (?) : Sf×E(S)→S:for any x in S. it exists y in S such that x≤(y, e)(?).In section five.we introduce an abstract definition of free ordered semigroups, and discuss properties free ordered semigroup. The main conclusions of this section are:Theorem5.3 Let A be not an empty ordered set. both F and F' are free ordered semigroups on A. then F (?) F'.Theorem5.5 Let S be an ordered semigroup, an empty ordered set A is a generation set of S. A⁺ is a free ordered semigroup on A, Then it exists an unique ordered semigroup homomorphism (?) : A⁺→S such that S(?) A⁺/Ker(?).In section six. we give some properties of QO-homomorphisms about ordered semigroups. The main conclusions of this section are:Theorem6.7 Let S be a regular ordered semigroup. T an ordered semigroup. (?) : S→T an ordered semigroup homomorphism. then the following are ture:(1) im(?) is a regular ordered semigroup:(2) Let (?) : S→T be a QO-homomorphism,f an idempotent of T, then it exists an ordered idempotent e in S such that f≤e(?).In section seven, we discuss Ress-quotient of some kinds of ordered semigroups. Especially, we obtain properties about homomorphisms, QO-homomorphisms forГ-ordered semigroups and its Ress-quotients. The main conclusions of this section Theorem7.2 Let S be a regular ordered semigroup. I be an ideal of S,ρ₁ be a Q-pseudoorder on S, then the Ress-quotient S/I is a regular ordered semigroup. too.Theorem7.8 Let S be aГ-ordered semigroup, I be an ideal of S. We define a map as following:thenθ_I is a QO-homomorphism.Theorem7.10 Let S be aГ-ordered semigroup, J be an ideal of S, T be aГ-subsemigroup and J∩T≠(?). then the following are ture:(1)We define mapθ:T/(J∩T)→(J∪T)/J,a(?)thenθis anГ-ordered semigroup homomorphism, (1-1)and onto;(2) Ifthen the mapθin (1) is reverse isotone and so it is anГ-ordered semigroup isomorphism.