Vol. 25, No. 1, 1968

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The Šilov boundary for a lattice-ordered semigroup

John C. Taylor

Vol. 25 (1968), No. 1, 185–191

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).

Mathematical Subject Classification
Primary: 46.25
Received: 15 September 1966
Published: 1 April 1968
John C. Taylor