Finding a model for a set of axioms

Question

Suppose $V$ is a vector space, preferably real or complex, with an additional operation $\wedge$ that sends two vectors to another vector space and obeys the following axiom:

$$a \wedge (a+b) = (a+b) \wedge b = a \wedge b$$

and for now, that's it. Note there is no distributivity requirement.

The obvious model of the above is the exterior algebra, which is anticommutative. Furthermore, if you add distributivity, anticommutative algebras seem to be the only thing you get. This is because distributivity and my axiom imply $a \wedge a = 0$, as you can see:

$$a \wedge b = a \wedge (b+a) = a \wedge b + a \wedge a \\ a \wedge a = 0$$

So my question is: is this all there is? Or are there more models, in particular without distributivity?

My questions are:

1. What non-distributive models are there of the above?
2. Does there exist any computer software, some CAS or otherwise, that can search for models for axioms like this?

Answer 1

One answer to the exotic model that I was looking for is given by the modular forms, and in particular the Eisenstein series of a complex lattice.

The axiom that I gave was motivated from an attempt to axiomatize the behavior of a lattice in a vector space. If $a$ and $b$ are vectors that generate a lattice, and $\wedge$ is supposed to represent "lattice generated by", we have $a \wedge (a+b) = a \wedge b$ and $(a+b) \wedge b = a \wedge b$.

In more familiar terms, we can also write this as $\Lambda(a,b)$ to denote the lattice generated by $a$ and $b$. This also shows why we don't want distributivity, as $\Lambda(a,2b)$ and $\Lambda(2a,b)$ are different lattices.

If we assume that $a$ and $b$ are vectors in $\Bbb R^2$, we can also treat them as complex numbers, which is equivalent to placing the algebra structure from $\Bbb C$ on the vector space. Then for any such $a$ and $b$, we can get the Eisenstein series

$$G_k(a, b) = \sum_{0 \neq \omega \in \Lambda(a, b)} \frac{1}{\omega^k}$$

which will in general by a complex number.

These are invariant for any lattice and are the simplest examples of modular forms. Furthermore, the invariants $G_4$ and $G_6$ can uniquely represent any lattice as a point in $\Bbb C^2$, and in some sense are unique in that they generate the entire ring of modular forms.

So one answer to my question is that the map

$$ G: \Bbb R^2 \times \Bbb R^2 \to \Bbb C^2\\ a \wedge b \mapsto \left(G_4(\Lambda(a,b)), G_6(\Lambda(a,b))\right) $$

is a model.

This can be extended to a pair of two vectors in an arbitrary $\Bbb R^n$, yielding a map $\Bbb R^n \times \Bbb R^n \to \Bbb C^{\frac{n(n-1)}{2}}$. To do this, just arrange the vectors $a$ and $b$ in a column matrix, and then take the original map $G$ and apply it to each of the $2 \times 2$ minors of the matrix. This is similar to the wedge product, except instead of taking the determinant of each $2 \times 2$ minor, you get the two Eisenstein invariants instead. There is likely a way to extend to higher-dimensional series as well.

Answer 2

This is not a full model for your set of axioms, just an idea of the construction, I could not proceed so much as to get all the axioms be fulfilled.

I did not succeed in defining operation $\wedge$ so that almost everything works but a model is simple, and I do not know can it be modified so as to suit your needs.

So this answer is just to perhaps give someone an idea on an approach that can be tried.

Let $(X,P(X))$ be some nonempty set $X$ and $P(X)$ the set of all subsets of $X$.

Define $+(A,B)$ to be the symmetric difference of the sets $A$ and $B$.

Let the field be a field with two elements: $0$ and $1$.

$+$ is commutative and associative.

Zero vector is the empty set.

$-A=A$

Define $0 \cdot A$ to be the empty set and $1 \cdot A$ to be $A$.

And now the problem is how to define $\wedge$ so that $a \wedge (a+b) = (a+b) \wedge b = a \wedge b$ is true and is non-distributive.

I did not succeed but someone most probably can, at least if we add additional structure on the set $X$.