Model Theory - Universal Algebra

Universal Algebra

Fundamental concepts in universal algebra are signatures σ and σ-algebras. Since these concepts are formally defined in the article on structures, the present article can content itself with an informal introduction which consists in examples of how these terms are used.

The standard signature of rings is σ_ring = {×,+,−,0,1}, where × and + are binary, − is unary, and 0 and 1 are nullary.

The standard signature of semirings is σ_smr = {×,+,0,1}, where the arities are as above.

The standard signature of groups (with multiplicative notation) is σ_grp = {×,−1,1}, where × is binary, −1 is unary and 1 is nullary.

The standard signature of monoids is σ_mnd = {×,1}.

A ring is a σ_ring-structure which satisfies the identities u + (v + w) = (u + v) + w, u + v = v + u, u + 0 = u, u + (−u) = 0, u × (v × w) = (u × v) × w, u × 1 = u, 1 × u = u, u × (v + w) = (u × v) + (u × w) and (v + w) × u = (v × u) + (w × u).

A group is a σ_grp-structure which satisfies the identities u × (v × w) = (u × v) × w, u × 1 = u, 1 × u = u, u × u−1 = 1 and u−1 × u = 1.

A monoid is a σ_mnd-structure which satisfies the identities u × (v × w) = (u × v) × w, u × 1 = u and 1 × u = u.

A semigroup is a {×}-structure which satisfies the identity u × (v × w) = (u × v) × w.

A magma is just a {×}-structure.

This is a very efficient way to define most classes of algebraic structures, because there is also the concept of σ-homomorphism, which correctly specializes to the usual notions of homomorphism for groups, semigroups, magmas and rings. For this to work, the signature must be chosen well.

Terms such as the σ_ring-term t(u,v,w) given by (u + (v × w)) + (−1) are used to define identities t = t', but also to construct free algebras. An equational class is a class of structures which, like the examples above and many others, is defined as the class of all σ-structures which satisfy a certain set of identities. Birkhoff's theorem states:

A class of σ-structures is an equational class if and only if it is not empty and closed under subalgebras, homomorphic images, and direct products.

An important non-trivial tool in universal algebra are ultraproducts, where I is an infinite set indexing a system of σ-structures A_i, and U is an ultrafilter on I.

While model theory is generally considered a part of mathematical logic, universal algebra, which grew out of Alfred North Whitehead's (1898) work on abstract algebra, is part of algebra. This is reflected by their respective MSC classifications. Nevertheless model theory can be seen as an extension of universal algebra.