additionlike

Additionlike is a term used in some mathematical and computational texts to describe a binary operation that behaves similarly to ordinary addition. When an operation ⊕ on a set S is called additionlike, it is typically required to be closed, associative, and to have an identity element e with a ⊕ e = a for all a in S. In many contexts, commutativity (a ⊕ b = b ⊕ a) is also assumed, in which case the structure is an abelian monoid. Some authors additionally consider cancellativity or order preservation as desirable, but these are not universal requirements.

Standard numerical addition on the integers (or real numbers) is the canonical example of an additionlike operation. Other additionlike operations include vector addition in any vector space, matrix addition, and the bitwise exclusive OR on bitstrings, all of which are closed, associative, and have a neutral element. Concatenation of strings can also be viewed as additionlike in the sense that it is associative and has an identity element (the empty string), though it is not commutative.

In broader contexts, the notion of additionlike supports the formation of additive structures such as monoids and, in some cases, semirings, enabling algebraic reasoning that mirrors arithmetic. The term is not standard in all mathematical literature, and when used, it generally signals that the operation shares essential additive features rather than implying a specific numerical interpretation.

See also: addition, abelian group, monoid, semiring.