  Semiring semantics of first-order logic generalises classical Boolean
semantics by permitting truth values from a commutative semiring, which can
model information such as costs or access restrictions. This raises the
question to what extent classical model theoretic properties still apply, and
how this depends on the algebraic properties of the semiring.
  In this paper, we study this question for the classical locality theorems due
to Hanf and Gaifman. We prove that Hanf's Locality Theorem generalises to all
semirings with idempotent operations, but fails for many non-idempotent
semirings. We then consider Gaifman normal forms and show that for formulae
with free variables, Gaifman's Theorem does not generalise beyond the Boolean
semiring. Also for sentences, it fails in the natural semiring and the tropical
semiring. Our main result, however, is a constructive proof of the existence of
Gaifman normal forms for min-max and lattice semirings. The proof implies a
stronger version of Gaifman's classical theorem in Boolean semantics: every
sentence has a Gaifman normal form which does not add negations.
