[palunon]

Aside from the fact that proofs are often difficult to verify, one cannot prove what a monoid or a vector space is. They are definitions, and to make sure the ones in the article match what is used in the mathematical community, you need citations.

And yes, sometimes there are conflicts. In France, we have two competing definition of a limit (relating to wether you include the point in its neighbourhood), one being the traditional one, taught in schools, and one being the one that's become the world standard and used from university onwards.

How do you arbiter that on wikipedia without sources ?