Here is an outline for a presentation for magnitudes. This outline doesn't have many of the details that would normally be included. Associativity and commutativity together imply that the order that addition is performed is irrelevant. An algebra satisfying only associativity is called a semigroup , while a semigroup that also satisfies commutativity is called a commutative semigroup or an Abelian semigroup . When other axioms are added for zero and negation, then the algebra is called a group , and when commutative, an Abelian group.
Groups are some of the most important algebraic structures in modern mathematics.
We can now define order in terms of addition. A number of properties of order can be easily proved. If x y and y z , then x z . With this axiom, subtraction can be defined, at least up to equality. The expected properties of subtraction, listed below, can be easily proved.