  Birkhoff's variety theorem, a fundamental theorem of universal algebra,
asserts that a subclass of a given algebra is definable by equations if and
only if it satisfies specific closure properties. In a generalized version of
this theorem, closure under filtered colimits is required. However, in some
special cases, such as finite-sorted equational theories and ordered algebraic
theories, the theorem holds without assuming closure under filtered colimits.
We call this phenomenon "filtered colimit elimination," and study a sufficient
condition for it. We show that if a locally finitely presentable category
$\mathscr{A}$ satisfies a noetherian-like condition, then filtered colimit
elimination holds in the generalized Birkhoff's theorem for algebras relative
to $\mathscr{A}$.
