A Theorem and a Question about Epicomplete Archimedean Lattice-ordered Groups

G always denotes an archimedean `-group. We call G epicomplete if G is divisible, -complete, and laterally -complete. X always denotes a Tychono¤ space. D (X ) is the set of continuous f : X ! [ 1;+1] with f 1 (( 1;+1)) dense. It is familiar and useful that every G has representations in various D (X ) : It is known that (A) G is epicomplete with weak unit e if and only if there is a compact basically disconnected space K and an `-group isomorphism G D (K) with e 7 ! 1K; thus G admits a compatible f -ring multiplication with e as identity. This paper concerns the situation without weak unit and involves this particular kind of representation: H J D (X ) means H is an `-group in D (X ) situated as [x = 2 closed F =) 9h 2 H with 0 < h (x) < +1 and h (F) = f0g] and [p 2 X n X and h 2 H =) h (p) = 0 or 1]. A J-representation of G is an `-group isomorphism G H J D (X ). Our Theorem is the following analogue of (A) with D (K) replaced by D (K; p) = ff 2 D (K) : f (p) = 0g : (B) The following are equivalent for G: ) G is epicomplete with no weak units and has a J-representation; ) there is compact basically disconnected K with non-isolated P-point p and an `-isomorphism G D (K; p); ) G is epicomplete with no weak units and has a compatible reduced f -ring multiplication. Our Question is: (C) Are these conditions satis…ed by every epicomplete G with no weak units? Or, more generally, does every G have a J-representation? (We conjecture: "No.")