On Elementary Equivalence for Equality-free Logic

This paper is a contribution to the study of equality-free logic, that is, first-order logic without equality. We mainly devote ourselves to the study of algebraic characterizations of its relation of elementary equivalence by providing some Keisler-Shelah type ultrapower theorems and an Ehrenfeucht-Fraı̈ssé type theorem. We also give characterizations of elementary classes in equalityfree logic. As a by-product we characterize the sentences that are logically equivalent to an equality-free one.