On Ideal Extensions Of Ordered Semigroups

The problem of ideal extensions for ordered semigroups has been first given and studied by N.Kehayopulu and M.Tsinglis in 2003. They have given a theory of ideal extensions for ordered semigroups. In this dissertation, we continue studying the ideal extensions of ordered semigroups. We give the theory of the equivalence of ideal extensions of an ordered semigroup and study ideal extensions of a special ordered semigroup-weakly reductive ordered semigroup . We also use the ideal extensions of an ordered semigroup to study the semilattice compositions. There are five charpters in this paper. The main results of every charpter are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we give the definition of the equivalence of ideal extensions of an ordered semigroup and the theory of the equivalence of two ideal extensions of an ordered semigroup.Theorem 2.2 Let (S, , ≤s) be an ordered semigroup. The ideal extensions (V, *, ≤v and of S are equivalent if and only if there exists an ismorphism of Q onto Q' such thatIn Chapter 3, We give the definition of the weakly reductive ordered semigroup , the construction of its ideal extensions and the conditions of the equivalence of its ideal extensions.Theorem 3.4 Let (S, ·, ≤s) be a weakly reductive ordered semigroup, (Q, ·, ≤q) an ordered semigroup with be a partial homomorphism, in notation with the property that (S) if ab = 0. Let V = an operation and an order on V, respectively, defined by:and≤V : = ≤s ∪ r∪{(x,y)| x, y ∈ Q* x ≤Q y}, wherer = Then (V, *, ≤v) is an ordered semigroup and it is an extension of 5 by Q. Conversely, every extension of S by Q can be so constructed.Theorem 3.6 Two ideal extentions (V,*,≤V) =< S,Q;θ> and (V',o,≤V') =< S, Q';θ' > of a weakly reductive ordered semigroup (S, ·, ≤s) are equivalent if and only if there exists an ismorphism of Q onto Q' such that θ = θ'(?).In chapter four, we mainly discuss the ideal extensions of ordered semigroups determined by a partial homomorphism.Theorem 4.1 Let (S,·,≤s) be an ordered semigroup, (Q,·,≤q) be an ordered semigroup with zero, S ∩ Q* =, and η : Q* → S be a partial homomorphism. Let an operation and an order on V, respectively, defined by:, whereandThen (V, *, ≤γ,then for all a ∈ Sa, b∈ Sβ,(4) ψα,β is isotone. define a multiplication "*" and a relation " ≤S " by respectively: Then (S, *, ≤s) is a semilattice Y of ordered semigroups Sa, in notation S = (Y;Sa,ψα,β, Da) Conversely, every ordered semigroup S which is a semilattice Y of ordered semigroups Sa can be so constructed. In addition, Da can be chosen to satisfy:(6)Da is a dense extension of Sa.