R. G. Woods

Lattice-Ordered Algebras That are Subdirect Products of Valuation Domains

An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if A/P is a valuation domain for every prime ideal P of A. If M is a maximal t-ideal of A, then the rank of A at M is the number of minimal prime ideals of A contained in M, rank of A is the sup of the ranks of A at each of its maximal ?-ideals. If the latter is a positive integer, then A is said to have finite rank, and if A = C(X) is the ring of all real-valued continuous functions on a Tychonoff space, the rank of X is defined to be the rank of the f-ring C(X) , and X is called an SV-space if C(X) is an SV-ring. X has finite rank k iff k is the maximal number of pairwise disjoint cozero sets with a point common to all of their closures. In general f-rings these two concepts are unrelated, but if A is uniformly complete (in particular, if A = C(X)) then if A is an SV-ring then it has finite rank. Showing that this latter holds makes use of the theory of finite-valued lattice-ordered (abelian) groups. These two kinds of rings are investigated with an emphasis on the uniformly complete case. Fairly powerful machinery seems to have to be used, and even then, we do not know if there is a compact space X of finite rank that fails to be an SV-space.

Separate versus joint continuity: A tale of four topologies

Abstract Several naturally occurring topologies on the product X×Y of the Tychonoff spaces X and Y are studied; each is stronger than the product topology τ. These include the cross topology γ consisting of sets meeting each horizontal and vertical fiber in a set open in the subspace topology induced by τ; the weak topology σ determined by the separately continuous real-valued functions with domain X×Y; and the weak topology determined by certain special separately continuous functions. Functorial relations between γ and σ are described. Sufficient conditions for separately continuous functions to be jointly continuous on a dense subspace of (X×Y,τ) are given. The topological structure of (X×Y,σ) is studied in detail.