$\gamma=(a,b,\ldots , b',a)$ is a simple cycle of length $k$, then there is an $E_2\cup\ldots \cup E_n$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ all of whose $V_n$-vertices appear in $\gamma$.

Let $T_n$ denote the $L_n$-theory expressing these axioms.

Note that the inductive nature of the definition immediately has the following consequences:

1.  The induced subgraph on $V_j\cup\ldots\cup V_{j+m}$ is a free pseudospace of dimension $m$.

2.  If $a$ is a vertex of type $V_{j+m}$ and $\gamma=(a,b,\ldots , b',a)$ is a simple cycle of length $k$ contained in $V_j\cup\ldots\cup V_{j+m}$, then there is an $E_{j+1}\cup\ldots \cup E_{j+m-2}$-path from $b$ to $b'$ of length at most $k-1$ in $R(a)$ all of whose $V_j$-vertices appear in $\gamma$.

3.  The notion of a is *self-dual*: if we put $W_i=V_{n-i}, i=0,\ldots n$, then $W_0,\ldots W_n$ with the same set of edges is again a .

Our first goal is to show that $T_n$ is consistent and complete.

Let $A$ be a finite $L_n$-graph. The following extensions are called elementary strong extensions of $A$:

1.  add a vertex of any type to $A$ which is connected to at most one vertex of $A$ of an appropriate type.

2.  If $(x,y,z)$ is 