  In this article, we investigate the status of the homomorphism preservation
property amongst restricted classes of finite relational structures and
algebraic structures. We show that there are many homomorphism-closed classes
of finite lattices that are definable by a first-order sentence but not by
existential positive sentences, demonstrating the failure of the homomorphism
preservation property for lattices at the finite level. In contrast to the
negative results for algebras, we establish a finite-level relativised
homomorphism preservation theorem in the relational case. More specifically, we
give a complete finite-level characterisation of first-order definable finitely
generated anti-varieties relative to classes of relational structures definable
by sentences of some general forms. When relativisation is dropped, this gives
a fresh proof of Atserias's characterisation of first-order definable
constraint satisfaction problems over a fixed template, a well known special
case of Rossman's Finite Homomorphism Preservation Theorem.
