On Extensions of Partial Orders of Commutative Partially Ordered Semigroups
Xie Xiang-yun · Journal of Mathematical Research and Exposition · 2005
Let (S,·,≤) be a commutative partially ordered semigroup. Sufficient and necessary conditions for extending a partial order of S to another partial order satisfying some conditions are given. Specially, if (S,·,≤) is a commutative cancellative partially ordered semigroup, the sufficient and necessary conditions such that every element of S belongs to positive cone are obtained. The cases when a partial order of S can be extended to another partial order such that some finite subsets of S form a chain are discussed.