Preservation properties for products and sums of metric structures

Abstract

This paper concerns product constructions within the continuous-logic framework of Ben Yaacov, Berenstein, Henson, and Usvyatsov. Continuous-logic analogues are presented for the direct product, direct sum, and almost everywhere direct product analyzed in the work of Feferman and Vaught. These constructions are shown to possess a number of preservation properties analogous to those enjoyed by their classical counterparts in ordinary first-order logic: for example, each product preserves elementary equivalence in an appropriate sense; and if for \(i\in \mathbb {N}\) \(\mathcal {M}_i\) is a metric structure and the sentence \(\theta \) is true in \(\prod _{i=0}^k\mathcal {M}_i\) for every \(k\in \mathbb {N}\), then \(\theta \) is true in \(\prod _{i\in \mathbb {N}}\mathcal {M}_i\).