Arithmetic operations

Arithmetic operations do not live in isolation, so they shouldn't be discussed alone, apart from the carrier set they endow and the axioms they jointly uphold.

The properties of an arithmetic operation cannot be studied for they don't exist per se - only the properties of the algebraic structure that embeds the operation can be considered.

For instance, addition alone is not definable; for an operation to be defined some specific carrier set must be considered at the same time; only then can we speak of, for instance, addition of the natural numbers. Usually the proposition "over" is used in these situations, as in, an operation over a set.

The properties of arithmetic operations, like associativity, are defined only on algebraic structures. An algebraic structure (or an algebra) is a carrier set together with a binary operation and a set of axioms they jointly obey.

algebraic structure = carrier set + binary operation + jointly upheld axioms

More generaly, an algebraic structure is comprised of

• a carrier (underlying) set

• a set of binary operations

• a set of axioms

An arithmetic operation apart from a carrier set, is not a well defined notion. Granted, an operation that behaves the same (obeys the same axioms), regardless of the carrier set, may be discussed in isolation if it seems that it generalize consistently accross the carrier sets. Addition is probably one such operation, although god knows all the carrier sets for which addition is defined and whether the identical set of axioms is truly respected (therefore "seems").