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.

## Informal interpretation

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:

- the theory of semigroups
- the theory of fields
- the theory of lattices

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:

*proj*:_{i}*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 *X*_{1}, ..., *X*_{n} with integer coefficients and with substitution as composition. In this case *proj _{i}* is the same as

*X*. This theory

_{i}*T*is called the theory of commutative rings.

In an algebraic theory, any morphism *n* → *m* 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 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)

## See also

## References

- Lawvere, F. W., 1963,
*Functorial Semantics of Algebraic Theories, Proceedings of the National Academy of Sciences 50, No. 5 (November 1963), 869-872* - Adámek, J., Rosický, J., Vitale, E. M.,
*Algebraic Theories. A Categorical Introduction To General Algebra* - Kock, A., Reyes, G., Doctrines in categorical logic, in Handbook of Mathematical Logic, ed. J. Barwise, North Holland 1977
- Algebraic theory in
*nLab*