Setup: I know the definition of a vector space: a set $V$ over a field $F$ such that is closed under vector addition and scalar multiplication.

My question: Is there a name for spaces that are closed under addition and scalar multiplication by positive real numbers? I am in particular thinking about spaces of square integrable functions on the real line (or some compact interval of it) that are everywhere greater than or equal to zero (call it $S$), with the property that it is closed under $af_1+bf_2$, where $f_1,f_2\in S$, and $a,b\in\mathbb{R}_+$. I know this is not a vector space (for example no $f\in S$ besides $f=0$ has an additive inverse), but is there another name for this space? I want to understand if this space (and others like it) has a notion of a "basis".


One structure that you can look at are modules.

They can be thought of as a generalisation of vector spaces; instead of a field, we have a ring, and instead of a set of vectors, we have a set of elements that form an abelian group.

Since we have the additional requirement that there is no additive inverse. We call the structure a module over a semiring where a semiring is a ring without the property of an additive inverse.


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