PSEUDOCOMPACT SUPPORTS IN POINTFREE TOPOLOGY

Let RL denote the ring of real-valued continuous functions on the completely regular frame L. The support of an element alpha is an element of RL is the closed quotient up arrow(coz alpha)*. We show that the set R-Psi(L) of elements of RL which have pseudocompact supports is an ideal. Call L supportively normal if supports are C-quotients. For supportively normal L, this ideal is the intersection of pure parts of the hyper-real maximal ideals. If L is supportively normal and lambda L is spatial, then lambda L is a continuous frame if and only if R-Psi(L) is contained in no real maximal ideal. Without the spatiality restriction, an example shows that one implication in this latter equivalence fails. We define locally pseudocompact frames conservatively, and show that if L is such a frame, then R-Psi(L) is contained in no fixed maximal ideal. The example alluded to above again shows that the converse fails - in contrast with the spatial case.