Informally in mathematical logic, an algebraic theory is a theory that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences.

The notion is very close to the notion of algebraic structure, which, arguably, may be just a synonym.

Saying that a theory is algebraic is a stronger condition than saying it is elementary.

An algebraic theory consists of a collection of n-ary functional terms with additional rules (axioms).

For example, the theory of groups is an algebraic theory because it has three functional terms: a binary operation a × b, a nullary operation 1 (neutral element), and a unary operation x ↦ x⁻¹ with the rules of associativity, neutrality and inverses respectively. Other examples include:

This is opposed to geometric theory which involves partial functions (or binary relationships) or existential quantors − see e.g. Euclidean geometry where the existence of points or lines is postulated.

An algebraic theory T is a category whose objects are natural numbers 0, 1, 2,..., and which, for each n, has an n-tuple of morphisms:

This allows interpreting n as a cartesian product of n copies of 1.