Oriented matroids are abstractions of hyperplane arrangements, or equivalently vector configurations. Let me recall the definition in terms of covectors.

Let $R=\lbrace 0,+,-\rbrace$ with the monoid structure given by the table
$$\begin{array}{c|ccc}\circ& 0 &+ &-\\ \hline 0 & 0 &+&-\\ + & + &+ &+\\ - & -&-&-\end{array}$$
$R$ is the oriented matroid corresponding to the unique central arrangement in $\mathbb R$.
More generally, $R^n$ corresponds to the arrangement of coordinate hyperplanes $x_i=0$ in $\mathbb R^n$. Elements of $R^n$ are called *covectors*.

We put $0=(0,\ldots,0)$ and $-(x_1,\ldots, x_n) = (-x_1,\ldots,-x_n)$ where negation is the automorphism of $R$ fixing $0$ and switching $+,-$. We view the structure $(R^n,\circ,x\mapsto -x)$ as a unary monoid (i.e., monoid with a distinguished unary operation).

If $x,y\in R^E$, then the *separation set* is $$S(x,y)=\lbrace e\in E\mid x_e=-y_e\neq 0\rbrace.$$

An *oriented matroid* with (finite) ground set *E* is a collection $\mathcal L\subseteq R^E$ of covectors such that

- $0\in \mathcal L$
- $x\in \mathcal L\iff -x\in \mathcal L$
- $x,y\in \mathcal L\implies x\circ y\in \mathcal L$
- If $x,y\in \mathcal L$ and $e\in S(x,y)$, then there exists $z\in \mathcal L$ such that $z_e=0$ and $z_f=(x\circ y)_f=(y\circ x)_f$ for $f\notin S(x,y)$.

Axioms 1-3 simply state that $\mathcal L$ is a unary submonoid of $R^E$. Axiom 4 is a statement about the geometry of hyperplane arrangements.

The motivating example is if $\mathcal A=\lbrace H_1,\ldots, H_n\rbrace$ is a hyperplane arrangement in $\mathbb R^d$ given by forms $f_i(x)=0$ for $i=1,\ldots,n$ then the set $\mathcal L$ of covectors encoding the signs of the entries of $(f_1(x),\ldots, f_n(x))$ with $x\in \mathbb R^d$ form an oriented matroid. The covectors are in bijection with the faces of the zonotope $Z=[-f_1,f_1]+\cdots+[-f_n,f_n]$ whose normal fan is $\mathcal A$ and the face poset of $Z$ is isomorphic to $\mathcal L$ with the ordering $x\leq y$ if $y\circ x=x$.

In particular, $R$ is the oriented matroid corresponding to the hyperplane $x=0$ in $\mathbb R$. Thus it makes sense to think of a homomorphism of unary monoids $\varphi\colon \mathcal L\to R$ as a *functional* on $\mathcal L$.

Question 1.If $\mathcal L\subseteq R^E$ is an oriented matroid, is it true that every non-zero functional is of the form $x\mapsto x_e$ for some $e\in E$?

If the answer to Question 1 is no, then I am interested in the following question.

Question 2.Let $M$ be a unary monoid. We can define $\widehat M$ to be the set of non-zero homomorphisms of unary monoids $\varphi\colon M\to R$. We can then define aGelfand transformation$\gamma\colon M\to R^{\widehat M}$ by $\gamma(m)_e = e(m)$. Is it true that $M$ is isomorphic as a unary monoid to the set of covectors of an oriented matroid if and only if $\gamma(M)$ is the set of covectors of an oriented matroid on with ground set $\widehat M$?

If the answer to Question 1 is positive, then the answer to Question 2 is positive. Note that if $M$ is the set of covectors of an oriented matroid, then $\gamma$ is injective. I think it is not too difficult to describe unary monoids for which $\gamma$ is injective. (If we forget the unary operation, then I know which monoids have enough homomorphisms to *R* to separate points.)