  Algebraic theories, sometimes called equational theories, are syntactic
notions given by finitary operations and equations, such as monoids, groups,
and rings. There is a well-known category-theoretic treatment of them that
algebraic theories are equivalent to finitary monads on $\mathbf{Set}$. In this
paper, using partial Horn theories, we syntactically generalize such an
equivalence to arbitrary locally presentable categories from $\mathbf{Set}$;
the corresponding algebraic concepts relative to locally presentable categories
are called relative algebraic theories. Finally, we give a framework for
universal algebra relative to locally presentable categories by generalizing
Birkhoff's variety theorem.
