Homomorphism closed vs. existential positive

Preservations theorems, which establish connection between syntactic and semantic properties of formulas, are a major topic of investigation in model theory. In the context of finite-model theory, most, but not all, preservation theorems are known to fail. It is not known, however, whether the Los-Tarski-Lyndon theorem, which asserts that a first-order sentence is preserved under homomorphisms if it is equivalent to an existential positive sentence, holds with respect to finite structures. Resolving this is an important open question in finite-model theory. In this paper we study the relationship between closure under homomorphism and positive syntax for several nonfirst-order existential logics that are of interest in computer science. We prove that the Los-Tarski-Lyndon theorem holds for these logics. The logics we consider are variable-confined existential infinitary logic, Datalog, and various fragments of second-order logic.