An algebraic theory, sometimes called an equational theory, is a theory
defined by finitary operations and equations, such as the theories of groups
and of rings. It is well known that algebraic theories are equivalent to
finitary monads on Set. In this paper, we generalize this phenomenon
to locally finitely presentable categories using partial Horn logic. For each
locally finitely presentable category A, we define an "algebraic
concept" relative to A, which will be called an
A-relative algebraic theory, and show that A-relative
algebraic theories are equivalent to finitary monads on A. Finally,
we generalize Birkhoff's variety theorem for classical algebraic theories to
our relative algebraic theories.Comment: 34 page