Preservation properties for products and sums of metric structures

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 ∈ N M i is a metric structure and the sentence θ is true in ∏ i = 0 k M i for every k ∈ N , then θ is true in ∏ i ∈ N M i .