On lattices of z-ideals of function rings

An ideal I of a ring A is a z-ideal if whenever a, b ∈ A belong to the same maximal ideals of A and a ∈ I, then b ∈ I as well. On the other hand, an ideal J of A is a d-ideal if Ann2(a) ⊆ J for every a ∈ J. It is known that the lattices Z(L) and D(L) of the ring 𝓡L of continuous real-valued functions on a frame L, consisting of z-ideals and d-ideals of 𝓡L, respectively, are coherent frames. In this paper we characterize, in terms of the frame-theoretic properties of L (and, in some cases, the algebraic properties of the ring 𝓡L), those L for which Z(L) and D(L) satisfy the various regularity conditions on algebraic frames introduced by Martínez and Zenk [20]. Every frame homomorphism h : L → M induces a coherent map Z(h) : Z(L) → Z(M). Conditions are given of when this map is closed, or weakly closed in the sense Martínez [19]. The case of openness of this map was discussed in [11]. We also prove that, as in the case of the ring C(X), the sum of two z-ideals of 𝓡L is a z-ideal.

Nonmeasurable Cardinals and Pointfree Topology

This paper establishes that the familiar rôle of nonmeasurable cardinals in classical topology extends to pointfree topology, that is, the setting of frames. For this, it considers the frames which are the pointfree form of the extremally disconnected P-spaces, namely the extremally disconnected 0-dimensional frames in which any countable join of complemented elements is complemented, and shows that they(1) have discrete spectrum and(2) are realcompact whenever they have nonmeasurable cardinal.An important tool obtained for this purpose is the result that, for a Boolean frame L, any σ-frame homomorphism L → 2 preserves the joins of all subsets of nonmeasurable cardinal.