On Some Study Of Ordered Γ Semigroups

In this dissertation,we study some regularities of orderedΓ-semigroups and give some results for prime ideals,semiprime ideals,weakly prime ideals,m system and quasi-prime left ideals of orderedΓ-semigroups.In the end,we discuss the chain of right simple(left simple, t-simple)orderedΓ-semigroups in 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 the left(right) semiregular orderedΓ-semigroups, then we give characterizations of them using different type of ideals.The main results are given as following:Theorem 2.1 Let M be an orderedΓ-semigroup.Then M is left(right)semireg-ular if and only if every left(right)ideal of M is idempotent.Corollary 2.1 Let M be an orderedΓ-semigroup. If M is regular, then the left(right)ideal of M is idempotent.Conversely, let M be an orderedΓ-semigroup such that MΓX(?)XΓM (XΓM(?)MΓX) for all X(?)M, and the left (right)ideal of M be idempotent.Then M is regular.Corollary 2.2 Let M be an orderedΓ-semigroup.Then M is quasi-completely regular if and only if every left or right ideal of M is idempotent.Theorem 2.2 OrderedΓ-semigroup M is left(right)semiregular if and only if for every bi-ideal B and every ideal I,we have B∩I(?)(IΓB](B∩I(?)(BΓI]).Theorem 2.3 OrderedΓ-semigroup M is left(right) semiregular if and only if for any left ideals L1,L2(right ideals R1,R2),we have L1∩L2(?)(L1ΓL2](R1∩R2(?)(R1ΓR2]). In Section 3,we mainly discuss completely regularity, strongly regularity, quasi-completely regularity of orderedΓ-semigroups,then we give the characterizations of them respectively. The main results are given as following:Theorem 3.1 Let M be an orderedΓ-semigroup.Then the following statements are equivalent(1)M is completely regular(2) a∈(aΓaΓMΓaΓa]for any a∈M;(3)Every quasi-ideal of M is completely regular orderedΓ-semigroup;(4)Every bi-ideal of M is semiprime.Theorem 3.2 OrderedΓ-semigroup M is quasi-completely regular if and only if every ideal of M is quasi-completely regular.Theorem 3.3 Let M be an orderedΓ-semigroup.Then the following statements are equivalent(1)M is strongly regular;(2) For any a∈M,there exist y∈M,γ∈Γ,such that a≤aγyγa, y≤yγaγy,aγyy=yγa;(3) Every N-class of M is strongly regular;(4) Every left ideal L and right ideal R are semiprime,and (LΓR] is strongly regular sub-orderedΓ-semigroup of M;(5)M is left regular,right regular and (MΓaΓM]is strongly regular sub-orderedΓ-semigroup of M for any a∈M;(6)For any a∈M, there exist ea,fa∈MΓaΓaΓM,μ, a∈Γ,such that ea≤eaafa,fa≤eaμfa, a≤eaμa,a≤aαfa and(MΓeaΓM],(MΓfaΓM] are strongly regular sub-orderedΓ-semigroups of M;(7) For any a∈M, there exist ea, fa∈MΓaΓaΓM,μ, a∈Γ,such that a≤eaμ,a≤aαfa and (MΓeaΓM],(MΓfaΓM] are strongly regular sub-orderedΓ-semigroups of M;(8)For any a∈M, we have a∈(MΓa]∩(aΓM] and (MΓaΓM] is strongly regular sub-orderedΓ-semigroup of M In Section 4,we study prime ideals,semiprime ideals and weakly prime ideals of orderedΓ-semigroups,then we give the characterizations of intra-regular orderedΓ-semigroups using them.The main results are given as following:Theorem 4.1 Let M be an orderedΓ-semigroup,T be an ideal of M.Then T is prime if and only if ((?)a, b∈M) L(a)ΓR(b)(?)T(?)a∈T or b∈T.Theorem 4.2 Let M be an orderedΓ-semigroup,T be an ideal of M.Then T is prime if and only if for any left ideal A and right ideal B,if AΓB(?)T, then A(?)T or B C T.Theorem 4.3 Let M be orderedΓ-semigroup,T be an ideal of M. Then the following statements are equivalent(1)(aΓMΓa](?)T(?) a∈T for any a∈M;(2) L(a)ΓL(a)(?)T(?)a∈T for any a∈M;(3) R(a)ΓRR(a)(?) T(?)a∈T for any a∈M;(4) T is semiprime.Theorem 4.4 Let M be an orderedΓ-semigroup, T be an ideal of M.Then the following two statements hold:(1)T is weakly prime if and only if for any ideals A,B of M,if (AΓB]∩(BΓA](?)T, then A(?)T or B(?)T;(2)T is prime if and only if T is semiprime and weakly prime.Theorem 4.5 Let M be a commutative orderedΓ-semigroup,then the weakly prime ideal of M is also prime ideal of M.Theorem 4.6 All ideals of orderedΓ-semigroup M are weakly prime if and only if they make a chain and any statement of Lemma 4.1 holds.Theorem 4.7 All ideals of orderedΓ-semigroup M are prime if and only if they make a chain and M is intra-regular.Theorem 4.8 Let M be a regular (intra-regular,left regular,right regular) or-deredΓ-semigroup,then every maximal ideal of M must be weakly prime.In Section 5,we mainly discuss special properties of left ideals in orderedΓ-semigroups, and give some charactrizations when left ideals are weakly prime,quasi-prime or weakly quasi-prime.The main results are given as following:Theorem 5.1 Every quasi-prime left ideal of orderedΓ-semigroups M must be weakly prime and weakly quasi-prime.Theorem 5.2 Let M be orderedΓ-semigroup.If M is left duo or commutative, or all left ideals of M with the include ralation make a chain,then weakly prime, quasi-prime and weakly quasi-prime of left ideals of M are equivalent.Theorem 5.3 Let M'be m system of orderedΓ-semigroup M, L be a left ideal of M and L∩M'=φ.Then (M',L) must be weak m system of M.Theorem 5.4 Let M be an orderedΓ-semigroup,L be a left ideal of M. Then the following two statements are equivalent(1)Let a,b∈M, (aΓMΓb] C L, then a∈L or b∈L;(2) M\L is m system.Theorem 5.5 Let M be an orderedΓ-semigroup,L be a left ideal of M. Then the following two statements are equivalent(1)Let a, b∈M,((a∪L)ΓMΓ[b∪L)](?)L, then a∈L or b∈L;(2)(M\L,L)is weak m system.Theorem 5.6 Let L be a left ideal of orderedΓ-semigroup M, M'be a nonempty subset of M, such that (M\L) is weak m system.If P is maximal left ideal which include L and P∩M'=0,then P is weakly quasi-prime left ideal of M.Theorem 5.7 Let M be an orderedΓ-semigroup,L be a left ideal of M. Then the following statements are equivalent(1)L is weakly quasi-prime(2) Let L1,L2 be left ideals of M,(L1∪L)Γ(L2∪L) C L,then L1(?)L or L2 C L;(3) Let Li,L2 be left ideals of M and L C L1,L1ΓL2 C L,then L1=L or L2 C L;(4) Let Li,L2 be left ideals of M and(L1∪L)ΓL2 C L,then L1(?)L or L2 C L.Theorem 5.8 Let M be an orderedΓ-semigroup,L be a proper left ideal of M. If for a,6∈M,(aΓMΓb] C L,we have a∈L or b∈L,then L is quasi-prime.Theorem 5.9 Let M be an orderedΓ-semigroup with identity, L be a quasi- prime left ideal of M.If i(L)≠0,then i(L)is quasi-prime ideal of MTheorem 5.10 Let M be an orderedΓ-semigroup with identity, L be a left ideal of M but not ideal of M. Then the following statements are equivalent(1)L is weakly quasi-prime(2)L1 is a left ideal of M and LΓL1(?)L,then L1(?)L;(3)Let a∈M and LΓ(MΓa)(?)L,then a∈L;(4) L is the maximum left ideal which is included in I(L) of M.Theorem 5.11 Let M be an orderedΓ-semigroup,Then the following state-ments are equivalent(1)All the left ideals of M are idempotent;(2)Let L1,L2 be left ideals of M and L1∩L2≠φ,then L1∩L2 C(L1ΓL2];(3) L(a)=(L(a)ΓL(a)]for any a∈M;(4)a∈(MΓaΓMΓa] for any a∈M;(5)Every left ideal of M is quasi-semiprime;(6)Every left ideal of M is the intersection of all quasi-prime left ideals which include it of M.In Section 6,we mainly study the chain of right simple(left simple, t-simple) orderedΓ-semigroups in orderedΓ-semigroups.The main results are given as fol-lowing:Theorem 6.1 Let M be an orderedΓ-semigroup.If M is a chain of right simple orderedΓ-semigroups,then the set (?)(M)of right ideals of M endowed with the ralation≤:A≤B(?)A=[AΓB]=(BΓA] is a chain.Theorem 6.2 Let M be an orderedΓ-semigroup. If M is a chain of left simple orderedΓ-semigroups,then the set (?)(M) of left ideals of M endowed with the ralation≤:A≤B(?)A=(AΓB]=(BΓA] is a chain. Theorem 6.3 Let M be an orderedΓ-semigroup.If M is a chain of t-simple orderedΓ-semigroups,then the set (?)(M)of ideals of Mendowed with the ralation≤:A≤B(?)A=(AΓB]=(BΓA] is a chain.