On Ideals Of Ordered Γ -Semigroups

In this dissertation, we mainly study a few kinds of ideals of orderedΓ-semigroups,give some results about left(right) weakly prime iddals, weakly prime ideals, weaklysemiprime ideals, mimimal ideals, maximal ideals and C-left ideals of orderedΓ-semi-groups. In the end, give characterizations of orderedΓ-semigroups that contain noproper bi-ideal. There are five sections, the main results are as follows.In Section 1, we mainly give some definitions, symbols and lemmas that will beused in this paper.In Section 2, we mainly study the properties of left(right) weakly prime idealsin orderedΓ-semigroups and give their charaterizations, and also study the relationsbetween m-systms and weakly prime ideals, n-systems and weakly semiprime ideals.The main results are as follows:Theorem 2.1 Let S be an orderedΓ-semigroup and T a left ideal of S. Thenthe following statements are equivalent:(1)T is left weakly prime;(2)if (aΓSΓb] - T for some a and b∈S, then a∈T or b∈T ;(3)if L(a)ΓL(b) - T for some a and b∈S, then a∈T or b∈T ;(4)if A is any subset of S and B is a left ideal of S such that AΓB - T , thenA∈T or B∈T .Theorem 2.2 Let S be an orderedΓ-semigroup and T a right ideal of S. Thenthe following statements are equivalent:(1)T is right weakly prime;(2)if (aΓSΓb] - T for some a and b∈S, then a∈T or b∈T ;(3)if R(a)ΓR(b) - T for some a and b∈S, then a∈T or b∈T ;(4)if A is right ideal of S and B is any subset of S such that AΓB - T , then A∈T or B∈T .Corollary 2.1 Let S be an orderedΓ-semigroup and T an ideal of S. Then thefollowing statements are equivalent:(1)T is left weakly prime;(2)T is right weakly prime;(3)T is weakly prime;(4)if (aΓSΓb] (?)T for some a and b∈S, then a∈T or b∈T ;(5)if L(a)ΓL(b) (?)T for some a and b∈S, then a∈T or b∈T ;(6)if R(a)ΓR(b) (?)T for some a and b∈S, then a∈T or b∈T ;(7)if A is any subset of S and B is a left ideal of S such that AΓB (?)T , thenA∈T or B∈T ;(8)if A is right ideal of S and B is any subset of S such that AΓB (?)T , thenA∈T or B∈T .Theorem 2.3 Let a be a left semiregular element of an orderedΓ-semigroupS. If L is a left ideal not containing a, then there exists a left weakly prime ideal Pnot containing a.Corollary 2.2 Let S be an orderedΓ-semigroup, a a left semiregular element ofS and L a left ideal of S. If for any positive intergers n∈Z+ (n≥2),γ1,γ2, ,γn-1∈Γ, such that aγ1aγ2 aγn-1a∈/ L, then there exists a left weakly prime ideal P of Ssuch that aγ1aγ2 aγn-1a∈/ P .Corollary 2.3 Let S be an orderedΓ-semigroup, a a left semiregular elementof S, P l(?)the intersection of all left weakly prime ideals of S and L is any proper leftideal of S. If a∈P l(?)then a∈L.Theorem 2.4 Let S be an orderedΓ-semigroup. If P (?)is the intersection of allleft weakly prime ideals of S and it is left semiregular, then P (?)is left Archimedian.Theorem 2.5 Let S be an orderedΓ-semigroup, then every bi-ideal of S is anm-system.Theorem 2.6 Let S be an orderedΓ-semigroup and I an ideal of S. Then(1)If I is weakly prime and S (?)I = -, then S (?)I is an m-system. (2)If S -I is an m-system, then I is weakly prime.Corollary 2.4 (1)An ideal I of an orderedΓ-semigroup S is weakly prime ifand only if S -I = -or the set S -I is an m-system.(2)A proper ideal I of an orderedΓ-semigroup S is weakly prime if and only ifS -I is an m-system.Theorem 2.7 Let S be an orderedΓ-semigroup and I an ideal of S. Then(1)If I is weakly semiprime and S -I = -, then S -I is an n-system.(2)If S -I is an n-system, then I is weakly semiprime.Corollary 2.5 (1) An ideal I of an orderedΓ-semigroup S is weakly semiprimeif and only if S -I = -or the set S -I is an n-system.(2) A proper ideal I of an orderedΓ-semigroup S is weakly semiprime if andonly if S -I is an n-system.Theorem 2.8 Let S be an orderedΓ-semigroup. If N is an n-system of S anda∈N, then there exists an m-system M of S such that a∈M -N.In Section 3, we mainly give the characterizations of minimal ideals and maximalideals in orderedΓ-semigroups, and study the relation between maximal ideals andweakly prime ideals in commutative orderedΓ-semigroups with identity. The mainresults are as follows:Theorem 3.1 Let S be an orderedΓ-semigroup that has no zero element andA the set of all ideals of S. The following conditions are equivalent:(1) S contains minimal ideals;(2) the set {J|J∈A } is the only minimal ideal of S;(3) {J|J∈A } = -.Theorem 3.2 Let S be an orderedΓ-semigroup, a∈S, S -I(a). Let A bethe set of all proper ideals of S. If A = -, then the set {J|J∈A } is the uniquemaximal ideal of S.Theorem 3.3 Let S be an orderedΓ-semigroup with identity. If the set A ofall proper ideals of S is non-empty, then the set {J|J∈A } is the unique maximalideal of S. Theorem 3.4 Let S be a commutative orderedΓ-semigroup with identity. IfM is a maximal ideal of S, then M is a prime ideal of S. The converse statement doesnot hold, in general.In Section 4, we discuss some elementary properties of C-left ideals in orderedΓ-semigroups, give the su-cient and necessary conditions of the existence of the max-imum C-left ideal of an orderedΓ-semigroup and characterize two classes of orderedΓ-semigroups. The main results are as follows:Theorem 4.1 Let S be an orderedΓ-semigroup, e is the maximum element ofS. If eγe = e for anyγ∈Γ, then every proper left ideal of S is C-left ideal.Theorem 4.2 Let S be an orderedΓ-semigroup, then S may have C-left idealsand left ideals that are not C-left ideals.Theorem 4.3 Let S be a commutative orderedΓ-semigroup. If S is not leftsimple, then S contains C-left ideals.Theorem 4.4 Let S be an orderedΓ-semigroup, M1, M2 the proper ideals ofS. If S = M1∪M2, then neither M1 nor M2 is C-left ideal of S.Corollary 4.1 Let S be an orderedΓ-semigroup. If T is a C-left ideal of S,then any maximal left ideal of S contains T .Corollary 4.2 Let S be an orderedΓ-semigroup. If S contains more than onemaximal ideals, then none of the maximal left ideals is C-left ideal of S.Corollary 4.3 Let S be an orderedΓ-semigroup. If S contains a maximal leftideal L which is a C-left ideal of S, then L must be the maximum proper ideal of S.Theorem 4.5 Let S be an orderedΓ-semigroup, S contains the maximumproper left ideal L-, then there exists a∈S - (SΓS], such that S - [a) - L- or L- isa C-left ideal of S.Theorem 4.6 Let S be an orderedΓ-semigroup. Then all the C-left ideals ofS are both union-preserving and intersection-preserving.Theorem 4.7 Let S be an orderedΓ-semigroup, A = - a set of S. Then A isa left base of S if and only if A satisfy(1)there exist a∈A such that L(x) - L(a) for any x∈S; (2)if La1 La2, then a1 = a2 for any a1, a2∈A.Theorem 4.8 Let S be an orderedΓ?semigroup which is not C?left simple.If S contains left base A, then S contains the maximum C?left ideal L such thatL = (SΓS]∩L, L is the intersection of all the maximal left ideals of S.Theorem 4.9 Let S be an orderedΓ?semigroup which contains maximumC?left ideal L, and S = (SΓS], the elements of S ? (SΓS] can not compare with eachother, then S must contain left base.Theorem 4.10 Let S be an orderedΓ?semigroup, S is not left simple. Thenevery proper ideal of S is C?left ideal if and only if one of the following statements istrue.(1)S contains maximal proper left ideal and it is the C?left ideal.(2)S = (SΓS], and for any proper left ideal L of S and a∈L, there exist b∈S ?Lsuch that L(a) ? L(b).Theorem 4.11 Let S be an orderedΓ?semigroup. S is C?left simple if andonly if S is the disjoint union of its minimal left ideals.In Section 5, we give the characteration of orderedΓ?semigroups that have noproper bi-ideals, and also give an example showing that an orderedΓ?semigroup withproper bi-ideals need not be an orderedΓ?group. The main results are as follows:Theorem 5.1 Let S be an orderedΓ?semigroup. If S is regular, then the bi-ideals and the subidempotent bi-ideals of S are the same.Theorem 5.2 An orderedΓ?semigroup S is left simple and right simple if andonly if S does not contain proper bi-ideals.Theorem 5.3 If S is an orderedΓ?group, then S does not contain proper bi-ideals; conversly, an orderedΓ?semigroup that does not contain proper bi-ideals isnot necessarily an orderedΓ?group.