Addition in linear vector spaces

Question

In the definition of linear vector spaces, one of the axioms is that the addition must be commutative and associative. The addition of scalars and matrices are both commutative and associate. Can there be situations where addition fails to be commutative or associative? Is there any simple example?

Answer 1

By definition addition is both commutative and associative in vector spaces. A vector space is an abelian group with some extra structure that defines the multiplication by scalars. Thus you can define a mathematical structure that resembles a vector space but fails commutativity: take a non-abelian group and define multiplication by a scalar. This would result in something resembling a non commutative vector space, but I do not remember even seeing such a structure. non commutative Lie groups are pretty close to it, and I am pretty sure that what you say has already been defined in topology. If you do not get an expert's response here (mine is not, of course), you should probably try asking in https://math.stackexchange.com/, they'll know for sure.

Answer 2

In that situation the space is no longer a vector space, and if the operation is not associative it ceases to be a group. Your question as currently posed is relatively ambiguous, but it can be answered by browsing through Wikipedia's Algebraic structures page. Some examples of algebraic structures, with a single operation, are:

• Magma or groupoid: S and a single binary operation over S.
• Semigroup: an associative magma.
• Monoid: a semigroup with identity.
• Group: a monoid with a unary operation (inverse), giving rise to inverse elements.
• Abelian group: a group whose binary operation is commutative.
• Semilattice: a semigroup whose operation is idempotent and commutative. The binary operation can be called either meet or join.
• Quasigroup: a magma obeying the latin square property. A quasigroup may also be represented using three binary operations.

If the operation is not associative, you're stuck at the magma/groupoid level, and such structures are not used much in physics. Not all is lost, though: you can add a divisibility property to get a quasigroup, and an identity element to get a loop.

However, few of the algebraic structures with less structure than groups have significant use in physics. All of the ring-like structures (which have two operations, 'addition' and 'multiplication', and which include vector spaces) assume the addition to be both commutative and associative; without them, it is very hard to get anywhere.

Nevertheless, there is the whole universe of non-Abelian group theory to be had with (associative) additions which do not commute. These are very rich structures, but they don't tend to get on well with an additional 'multiplication' on the same set.

Answer 3

"Addition" or "multiplication" are just names of operators, like John, Alice... Thinking axiomatically, what define an object are the properties of the object (associative, commutative...), not its name. You should read some abstract algebra textbooks, my preference is S.T.Hu's.

However, by convention, often if some operator is named "addition" (denoted "+"), it is often commutative, associative. One simple exception to this naming convention is in C++ (which is not quite math, you may think), the "+" between two strings is not commutative: "A"+"B"="AB" while "B"+"A"="BA".