Algebraic theory

Informally in mathematical logic, an algebraic theory is one 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.

Informal interpretation

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

E.g. a group theory 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 xx−1 with the rules of associativity, neutrality and inversion respectively.

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.

Category-based model-theoretical interpretation

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:

proji: n → 1, i = 1,..., n

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. In this case proji is the same as Xi. This theory T is called the theory of commutative rings.

In an algebraic theory, any morphism nm can be described as m morphisms of signature n → 1. These latter morphisms are called n-ary operations of the theory.

If E is a category with finite Cartesian products, the full subcategory Alg(T, E) of the category of functors [T, E] consisting of those functors that preserve finite products is called the category of T-models or T-algebras.

Note that for the case of operation 2 → 1, the appropriate algebra A will define a morphism

A(2) ≈ A(1)×A(1) → A(1)