An algebraic theory consists of a collection of n-ary functional terms with additional rules (axioms).
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.
Example: Let's define an algebraic theory T taking hom(n, m) to be m-tuples of polynomials of n free variables X1, ..., Xn with integer coefficients and with substitution as composition.
Note that for the case of operation 2 → 1, the appropriate algebra A will define a morphism