Let X be a compact
Hausdorff space and let S be a point separating collection of R+-valued
lower semicontinuous functions on X which is closed under addition. Assume
that S is a lower semi-lattice with respect to the partial order ≦ (where
f ≦ g if g = f + h, for some h ∈ S). Further, assume S contains all the
nonnegative constant functions λ and is such that λ ≦ f implies λ ≦ f (where
λ ≦ f if λ ≦ f(x) for all x ∈ X). Then, the Šilov boundary of S is precisely
{x|(f ∧ g)(x) =min{f(x),g(x)}∀f,g ∈ S} if, in addition, for all f, g, and h ∈ S we
have f + (g ∧ h) = (f + g)Λ(f + h).
