Abstract. We show that the representation of the monoid of intervals of a simple refinement monoid in terms of affine semicontinuous functions, given by Perera in 2001, fails to be faithful in the case of strictly perforated monoids. We give some potential applications of this result in the context of monoids of intervals and K-Theory of multiplier rings.

We prove that every partially ordered simple group of rank one which is not Riesz embeds into a simple Riesz group of rank one if and only if it is not isomorphic to the additive group of the rationals. Using this result, we construct examples of simple Riesz groups of rank one G, containing unbounded intervals (D n ) n 1 and D, that satisfy: (a) for each n 1, tD n = G + for every (t < q n ), but q n D n = G + (where (q n ) is a sequence of relatively prime integers); (b) for every n 1, nD = G + . We sketch some potential applications of these results in the context of K-theory.  2004 Elsevier Inc. All rights reserved.