Seminorms on ordered vector spaces that extend to Riesz seminorms on larger Riesz spaces (Q1433177)

The author deals with so-called pre-Riesz seminorms. He proves that: (a) each pre-Riesz seminorm on a directed partially ordered vector space \(E\) can be extended to a Riesz seminorm on every Riesz space in which \(E\) can be embedded as a majorizing linear subspace; (b) a seminorm on an order dense linear subspace of a Riesz space is pre-Riesz if and only if it can be extended to a Riesz seminorm on the Riesz space; (c) a seminorm on a partially ordered vector space that has a Riesz completion is pre-Riesz if and only if it is the restriction of a Riesz seminorm on the Riesz completion. Moreover, the author discusses a suitable notion of ``solid unit ball'' in partially ordered vector spaces. The author considers a joint generalization of solidness and convexity, called solvexity. The properties related to restriction and extension are studied, including extension to \(L\)- and \(M\)-seminorms.