Do you ever think up theorums and think: "that's inetersting, I wonder if anyone's ever thought of that before?"(adsbygoogle = window.adsbygoogle || []).push({});

In this vein the other day, I thougt up these two. They are both fairly trivial and possibly it's only me that finds them worth even bothering with, but what I want to know is if any of them have ever been applied to any area of maths?

Theorum 1: The set of all functions V from a set A to a field of scalars K form a vector space over K where for any such functions f, g and h: f + g = h , where f(x) + g(x) = h(x) and for a scalar a: a.f = g, where f(a*x) = g(x).

The main reason this seems interesting to me is that the axioms governing the behaviour of +:VxV-->V and .:KxV--> V are automatically implied in their defintion, dim(V) is simply |A|, plus all isomorphism classes of vector spaces can be described by such objects.

Theorum 2: Any group (G,*) forms a subsemigroup of a semigroup (that is not a group) (G+{0},*) where for any g in G: 0*g = g*0 = 0.

The reason I find this interesting is that the muplicative semigroup in a divison algebra is such a semigroup (i.e. a group plus a '0 element'). Also when a group has some sort of toplogical structure you can add such an element and define a new topology, e.g. in the group (R,+) you can add such an elemnt in a natural way to go from an open set to one thta is neither open nor closed.

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