Extension of positive functionals on certain ordered spaces (Q801242)

An ordered linear space L is said to satisfy extension property (E1) if for every directed subspace M of L and positive linear functional \(\Phi\) on M, \(\Phi\) can be extended to L. A Riesz space L is said to satisfy extension property (E2) if for every sub-Riesz space M of L and every real valued Riesz homomorphism \(\Phi\) on M, \(\Phi\) can be extended to L as a Riesz homomorphism. These properties were introduced by \textit{G. C. Schmidt}, Periodica Math. Hung. 6, 295-307 (1975; Zbl 0291.06010)]. In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic to a function space L' defined on a set X' such that if f and g belong to L' there exists a finite disjoint subset M of the set of functions on X' such that each of f and g is a linear combination of the points of M. An analogous theorem is derived for Riesz spaces with extension property (E2).