On The Study Of Some Problems Of Ordered Γ-Semigroups

In this dissertation, we first study orderedΓ-group, orderedΓ-right group and right zero orderedΓ-semigroups of orderedΓ-semigroups. Then we discuss the decomposition of orderedΓ-semigroups and we give characterizations of the semilat-tices of archimedean orderedΓ-semigroups and t-simple orderedΓ-semigroups. In the end, we discuss nil-extensions of left simple orderedΓ-semigroups and a natural ordered band of 1-archimedean orderedΓ-semigroups. There are six sections, the main results are given as following.In Section 1, we mainly give some definitions and symbols in this paper.In Section 2, we mainly discuss orderedΓ-group, orderedΓ-right group and right zero orderedΓ-semigroups, and we give some results. The main results are given as following:Theorem 2.1 Let G be an orderedΓ-group, E be a right zero orderedΓ-semigroup, then G×E is a right simple orderedΓ-semigroup.Corollary 2.1 Let G be an orderedΓ-group with identity and E be a left cancellative right zero orderedΓ-semigroup. If for any g∈G, there exist g-1∈G, such that for anyγ∈Γ, gγg-1= 1,then G×Εis a orderedΓ-right group.Theorem 2.2 Let M be a orderedΓ-right group, E(M)={x∈M｜((?)γ∈Γ) x≤xγx}, then the following are true:(1) M is a regular orderedΓ-semigroup;(2) E(M)≠(?);(3) For any b∈M,e∈E(M), there existα∈Γ.such that b≤eαb;(4) If MΓe is an ideal of M for any e∈E(M), then MΓe is a orderedΓ-subgroup of M; (5) If MΓf is a sub-orderedΓ-semigroup for any f∈E(M), then x∈(MΓfΓE(M)] for any x E M.In Section 3, we mainly discuss the semilattices of archimedean orderedΓ-semig-roups and t-archimedean orderedΓ-semigroups, and we give characterizations of them using different equivalent conditions. The main results are given as following:Theorem 3.1§is a semilattice congruence of M.Theorem 3.2 Let M be an orderedΓ-semigroup, a E M,let N(a) be the filter of M generated by a, ThenTheorem 3.3 Let M be an orderedΓ-semigroup. Then the following state-ments are equivalent(1) M is a semilattice of archimedean sub-orderedΓ-semigroups of M;(2) For any a,b∈M, if (a, b)∈τ, then for anyγ'∈Γ, there exist∈E Z+,β1,β2,···,βm∈Γsuch that (αγα, bβ1bβ2cb···bβmb)∈τ;(3) For any a, b∈M,γ∈Γ, if (a, b)∈η, then (αγα, b)∈η;(4)η2 Cη;(5)μ=§is the greatest semilattice congruence on M such that each of its con-gruence classes is an archimedean sub-orderedΓ-semigroup;(6) N(a)={b∈M｜(b,α)∈η} for any a∈M;(7) M is the greatest semilattice congruence on M such that each of its congruence classes is an archimedean sub-orderedΓ-semigroup.Theorem 3.4 A weakly commutative orderedΓ-semigroup M is a semilattice of archimedean orderedΓ-semigroups. Generally, the decomposition is not unique. But M is the unique complete semilattice of archimedean orderedΓ-semigroups.Theorem 3.5 Let M be an orderedΓ-semigroup, then the following statements are equivalent:(1) M is weakly commutative;(2)τ(?)ηt,η(?)η2;(3)μt is the greatest semilattice congruence on M such that each of its congruence classes is a t-archimedean sub-OrderedΓ-semigroup；(4)M is a semilattice of t-archimedean Sub-orderedΓ-semigroups of M；(5)N(α)={b∈M｜(6∈M)∈ηt)for anyα∈M；(6)(b,αγb)∈ηtfor anyα,6∈M；(7)N is the greatest semilattice congruence on M such that each of its congruence classes is a t-archimedean sub-orderedΓ-semigroup.In Section 4,we mainly discuss strongly left regular orderedΓ-semigroups and we give characterizations of the semilattices of left simple sub-orderedΓ-semigroups and t-simple sub-orderedΓ-semigroups.The main results are given as following:Theorem 4.1 Let M be an orderedΓ-semigroup,then the following statements are equivalent(1)M is strongly 1eft regular；(2)Every left ideal of M is strongly sem.iprime.Theorem 4.2 Let M be an orderedΓ-semigroup and M is left normal,then the following statements are equivalent(1)M is strongly left regular and left duo；(2)N(x)={y∈M｜x∈(MΓy])for any x∈M；(3)N=(?).(4)For any left ideal L of M,L=U{(x)N｜x∈L)；(5)(x)N is a left simple sub-orderedΓ—semigroup of M for any x∈M；(6)M is a semilattice of left simple sub-orderedΓ-semigroups；(7)Every left ideal of M is a right ideal and it's strongly semiprime.Theorem 4.3 Let M be an orderedΓ-semigroup,then the following statements are equivalent(1) M is a chain of left simple orderedΓ_semigroups；(2)Every left ideal of M is a right jdeal and it's strongLy prime；(3)M is strongly left regular and left duo and all left ideals of M with the inclusion relation make a chain:(4)M is left duo and x∈(MΓxγy]or y∈(MΓxkγy]for any x,y∈M ,Y∈Γ. Theorem 4.4 Let M be an orderedΓ-semigroup, then the following statements are equivalent(1) M is strongly t regular and duo;(2) M is duo and every left ideal and right ideal is strongly semiprime;(3) M is strongly t regular and [xΓM]= (MΓx] for any x∈M;(4) N{x)={y∈M｜x∈{yΓMΓy]} for any x∈M;(5)N=H=B={(x,y)∈M x M｜B(x)= B(y)};(6) Let B be a bi-ideal of M, then B=∪{(x)N｜x∈B };(7) (x)N is a B simple sub-orderedΓ-semigroup for any x∈M;(8) M is a semilattice of B simple sub-orderedΓ-semigroups.In Section 5, we mainly discuss nil-extensions of left simple orderedΓ-semigroups and we give characterizations of them using archimedes and strong regularity. The main results are given as following:Theorem 5.1 Let M be a quasi-commutative archimedean orderedΓ-semigroup, if SIntra(M)≠(?) , then(1) M has a kernal K(M) such that(2) M is a nil-extension of a simple orderedΓ-semigroup K(M);Theorem 5.2 Let M be a quasi-commutative orderedΓ-semigroup, then if I is a simple ideal of M, then I is also left simple.Theorem 5.3 Let M be a quasi-commutative orderedΓ-semigroup, then the following statements are equivalent(1) M is a nil-extension of a left simple orderedΓ-semigroup;(2) M is 1-archimedean and strongly leftπ-regular;(3) M is 1-archimedean and strongly intra-π-regular;(4) M is a 1-archimedean orderedΓ-semigroup in which SIntra(M)≠(?).In Section 6, we mainly discuss a natural orderedΓ- band of 1-archimedean orderedΓ-semigroups and we give characterizations of a natural orderedΓ- band of left simple orderedΓ-semigroups. The main results are given as following: Theorem 6.1 Let (M,≤M) be an orderedΓ-semigroup, then M is a natural orderedΓ- band of 1-archimedean orderedΓ-semigroup if and only if M satisfies the following condition:Corollary 6.1 An ordered semigroup S is a natural ordered band of 1-archimedean ordered semigroup if and only if S satisfies the following condition:Corollary 6.2 Let (M,≤) is quasi-commutative orderedΓ-semigroup. Then M is a natural orderedΓ- band of orderedΓ-semigroups, which are nil-extensions of left simple orderedΓ-semigroup, if and only if M is strongly leftπ-regular and satisfies the following conditionTheorem 6.2 Let M be an orderedΓ-semigroup, then the following conditions are equivalent(1) M is a natural orderedΓ-band of left simple ordered T.-semigroup;(2) M is strongly left regular and satisfies...