Additiontype

Additiontype is a term used in some mathematical discussions to denote the qualitative profile of a single binary operation called addition on a set. More precisely, an additiontype on a set A consists of the data of a function +: A × A → A together with a specified family of equational identities that the operation is required to satisfy. The term is not standardized in the literature, but it is often used to classify structures by the axioms their addition obeys.

Common variants arise from the classical hierarchies of algebraic structures. If + is associative, the structure is a semigroup; if it is also commutative, a commutative semigroup. With an identity element, it becomes a monoid; with inverses for every element, a group. If the operation is commutative and all elements have inverses, one obtains an abelian group. Some discussions also consider weaker or alternative identities, such as idempotence (x + x = x) or absorptive properties, to define specialized additiontypes.

In universal algebra, additiontype can be viewed as the equational theory generated by the identities satisfied by the addition operation, which is captured by a signature with a single binary symbol. In model theory and category theory, additive structures are studied as abelian monoid objects or as algebraic theories, and the term additiontype is sometimes used to highlight the dependency of results on the exact axiom set.

Examples include the integers under addition as an abelian group, the natural numbers under addition as a commutative monoid, and the real numbers under addition forming a vector space structure when paired with scalar multiplication. The notion also informs the study of ordered additive structures, where addition is monotone.