Consider, as a motivating example, the multiset $\left(\mathbb{P}\_n,2\setminus\mathbf{0}\right)$ consisting of the underlying set of polynomials of order $n$ and lower, all with multiplicity $2$—expect for the polynomial identically equal to $0$, $\mathbf{0}$. (Heuristically, this is intended to be what occurs when you "clone" everything in $\mathbb{P}\_n$ except for the zero vector; I apologize for incorrect notation.)


If I look at the axioms that a set—combined with typical addition and scalar multiplication operations—would need to satisfy in order to qualify as a vector space, I would find that this multiset satisfies most if not all of them. For example, addition and/or scalar multiplication of the elements in this multiset will always produce another element in the multiset.


With this in mind, and given that many other such examples can be generated, **is there any meaningful extension of the notion of a vector space to multisets?**


A comment: to clarify the properties I'd like to see retained in such an extension, I'd attempt to preserve at all costs the notion of closure (that addition and scalar multiplication acting on two elements in the multiset always creates another element in the multiset) while attempting as much as possible to keep "non-unique" versions of the other axioms; for example, that there still exists at least one additive identity element $e$ in the multiset, but that identity element is perhaps not unique.



Is there any meaningful extension of the notion of a vector space for multisets?
It is hard to posit more than one additive identity by the usual proof: 


$e+f=f$ if $e$ is an additive identity and $e+f=e$ if $f$ is an additive identity hence $e=f$ if both are.


If the underlying field is $F=\{0,1\}$ then scalar multiplication is no issue and the vector space $W=F^{n+1}$ is essentially the double of $V=F^n.$ I'll start with a description which preserves some of the mystery: For each vector $x \in V$ we have two copies in $W,$ say $\color{red}x$ and $\color{blue}x.$ The rules are that for $x+y=z$ in $V$ we have


* $\color{red}x+\color{red}y=\color{blue}x+\color{blue}y=\color{red}z$
* $\color{red}x+\color{blue}y=\color{blue}x+\color{red}y=\color{blue}z$


The axioms are easy to verify. The way this diverges slightly from what you describe is that also additive identity $\mathbf{0} \in V$ yields two vectors: $\color{red}{\mathbf{0}}$ is the additive identity while $\color{blue}{\mathbf{0}}$ is the color flipper.


Of course this is basically saying that if $u \in W=F^{n+1}$ is any fixed non-zero vector then $D=\{0,u\}$ is a subspace , $W/D $ is isomorphic to $F^n$ and the map which sends $x$ to $\{x,x+u\}$ is $2$-to-$1$


A similar thing can be done over any finite field and can be extended to infinite dimensional vector spaces over finite fields. The scalar multiplication (not an issue when the only non-zero scalar is $1$) should be easy to figure out.


