2-Universally complete Riesz spaces and inverse-closed Riesz spaces

In this paper we show mainly two results about uniformly closed Riesz subspaces of R-X containing the constant functions. First, for such a Riesz subspace E, we solve the problem of determining the properties that a real continuous function phi defined on a proper open interval of R should have in order that the conditions "E is closed under composition with phi" and "E is closed under inversion in X" become equivalent. The second result, reformulated in the more general frame of the Archimedean Riesz spaces with weak order unit e, establishes that E (e-uniformly complete and e-semisimple) is closed under inversion in C(Spec E) if and only if E is 2-universally e-complete.