  Quantitative algebras (QAs) are algebras over metric spaces defined by
quantitative equational theories as introduced by the same authors in a related
paper presented at LICS 2016. These algebras provide the mathematical
foundation for metric semantics of probabilistic, stochastic and other
quantitative systems. This paper considers the issue of axiomatizability of
QAs. We investigate the entire spectrum of types of quantitative equations that
can be used to axiomatize theories: (i) simple quantitative equations; (ii)
Horn clauses with no more than $c$ equations between variables as hypotheses,
where $c$ is a cardinal and (iii) the most general case of Horn clauses. In
each case we characterize the class of QAs and prove variety/quasivariety
theorems that extend and generalize classical results from model theory for
algebras and first-order structures.
