  A canonical result in model theory is the homomorphism preservation theorem
which states that a first-order formula is preserved under homomorphisms on all
structures if and only if it is equivalent to an existential-positive formula,
standardly proved compactness. Rossman (2008) established that the h.p.t.
remains valid when restricted to finite structures. This is a significant
result in the field of finite model theory, standing in contrast to other
preservation theorems and as an theorem which remains true in the finite but
whose proof uses entirely different methods. It also has importance to the
field of constraint satisfaction due to the equivalence of existential-positive
formulas and unions of conjunctive queries. Adjacently, Dellunde and Vidal
(2019) established that a version of the h.p.t. holds for a collection of
first-order many-valued logics, those whose (possibly infinite) structures are
defined over a fixed finite MTL-chain.
  In this paper we unite these two strands, showing how one can extend
Rossman's proof of a finite h.p.t. to a very wide collection of many-valued
predicate logics and simultaneously establishing a finite variant to Dellunde
and Vidal's result, one which not only applies to structures defined over
algebras more general than MTL-chains but also where we allow for those algebra
to vary between models. This investigation provides a starting point in a wider
development of finite model theory for many-valued logics and, just as the
classical finite h.p.t. has implications for constraint satisfaction, the
many-valued finite h.p.t. has implications for valued constraint satisfaction
problems.
