Yosida frames

Abstract

A Yosida frame is an algebraic frame in which every compact element is a meet of maximal elements. Yosida frames are used to abstractly characterize the frame of z -ideals of a ring of continuous functions C ( X ) , when X is a compact Hausdorff space. An algebraic frame in which the meet of any two compact elements is compact is Yosida precisely when it is “finitely subfit”; that is, if and only if for each pair of compact elements a < b , there is a z (not necessarily compact) such that a ∨ z < 1 = b ∨ z . This is used to prove that if L is an algebraic frame in which the meet of any two compact elements is compact, and L has disjointification and dim ( L ) = 1 , then it is Yosida. It is shown that this result fails with almost any relaxation of the hypotheses. The paper closes with a number of examples, and a characterization of the Bézout domains in which the frame of semiprime ideals is Yosida frame.