Research On Several Classes Of Ordered Γ-Semigroups

In this dissertation, we study some classes of ordered P-semigroups . There are five 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 define J—trivial ordered P—semigroups , and prove that every nil ordered P—semigroup is J—trivial. We show that in nil ordered P—semigroups which are chains with respect to the divisibility ordering, every complete congruence is a Rees congruence, and that this type of ordered P—semigroups are A—ordered P—semigroups. Moreover, the homomorphic images of A—ordered P—semigroups are A—ordered P—semigroups as well. Finally, we prove that the ideals of a A—ordered P—semigroup M form a chain under inclusion if and only if M is a chain with respect to the divisibility ordering. The main results are given as following:Proposition 2.1 Let M be an ordered P—semigroup. If M is J—trivial, then the divisibility relation (?)on M is an order on M. Conversely, if the divisibility relation =4 on M is antisymmetric, then M is J—trivial.Corollary 2.1 An ordered P—semigroup M is J—trivial if and only if the divisibility relation =4, on M is an order on M.Proposition 2.2 Let M be an ordered P—semigroup. The following statements are equivalent:(1) M is J—trivial and the principal ideals of M form a chain under inclusion;(2) The divisibility relation (?) on M is a chain. Theorem 2.1 Every nil ordered P—semigroup is J—trivial.Corollary 2.2 An ordered P—semigroup M cannot be 0-simple and nil at the same time.Theorem 2.2 Let M be an nil ordered P—semigroup which is a chain with respect to the divisibility ordering. Then every complete congruence on M is a Rees congruence.Proposition 2.3 Let M be an ordered P—semigroup. The principal ideals of M form a chain with respect to the inclusion relation if and only if the ideals of M do so.Proposition 2.4 Let M be a nil ordered P—semigroup. The ideals of M form a chain under inclusion if and only if M is a chain with respect to the divisibility ordering.Proposition 2.5 Let M be a nil ordered T—semigroup which is a chain with respect to the divisibility ordering. Then M is a A—ordered T—semigroup.Theorem 2.3 Let (M, <) be a A—ordered P—semigroup, (T,<) an ordered T—semigroup, and / : M < T a homomorphism and onto mapping. Then T is a A—ordered P—semigroup as well.In Section 3 , we mainly study Noetherian and Artinian ordered P—semigroups. The main results are given as following:Theorem 3.1 Let M be an ordered P—semigroup. The following statements are equivalent:(1) M is Noetherian ordered T—semigroup;(2) M satisfies the ascending chain condition for ideals;(3) M satisfies the maximum condition for ideals.Theorem 3.2 An ordered P—semigroup M is Artinian if and only if it satisfies the minimum condition for ideals.Proposition 3.1 If M,T are two ordered P—semigroup ,(?) homomorphism and onto mapping and / an ideal of T , then f-1(I) is an ideal of M.Proposition 3.2 Let M be a Noetherian (Artinian) ordered P—semigroup, T an ordered P—semigroup and / : M—? T & homomorphism and onto mapping. Then T is Noetherian (Artinian).Proposition 3.3 Let (M, 2). Then M has the P/-property for each I￡N (l>2).Corollary 5.2 Let M be an ordered P—semigroup. Then the following assertions are equivalent :(1) M has the P—property ;(2) M has the Pm—property for each m￡N (m > 2) ;(3) M has the Pm—property for certain m￡N, (m > 2) .