On weak commutativity of po-semigroups and their semilattice decompositions

In this paper, concepts of left (right) weakly commutative po-semigroups and r (l or t)-archimedean subsemigroups of a po-semigroup are introduced. Six relations tau, sigma, eta, rho, mu, xi on a po-semigroup are defined. By using them, filters and radicals, fourteen necessary and sufficient conditions in order that a po-semigroup is a semilattice of archimedean subsemigroups are given. The facts that a left weakly commutative (right weakly commutative or weakly commutative) po-semigroup is a semilattice of r (l or t)-archimedean subsemigroups are proved and seven characterizations of these po-semigroups are obtained respectively.