Birkhoff's variety theorem for relative algebraic theories

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 $\mathbf{S}\mathbf{e}\mathbf{t}$. In this paper, we generalize this phenomenon to locally finitely presentable categories using partial Horn logic. For each locally finitely presentable category $\mathcal{A}$, we define an “algebraic concept” relative to $\mathcal{A}$, which will be called an $\mathcal{A}$-relative algebraic theory, and show that $\mathcal{A}$-relative algebraic theories are equivalent to finitary monads on $\mathcal{A}$. In establishing such equivalence, a generalized Birkhoff’s variety theorem plays an important role.