free pseudospace of dimension $n$ is the first order theory of a Tits-building associated to a certain Coxeter diagram and we will investigate this connection in Section \[sec:building\].

In the final section we show that there are exactly two orthogonality classes of regular types.

Construction and results {#sec:construction}
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Fix a natural number $n\geq 1$. Let $L_n$ be the language for $n+1$-coloured graphs containing predicates $V_i,i=0,\ldots n$ and an edge relation $E$.

By an $L_n$-graph we mean an $n+1$-coloured graph with vertices of types $V_i, i=0,\ldots n$ and an edge relation $E\subseteq \bigcup_{i=1,\ldots n}V_{i-1}\times V_i$. We say that a path in this graph is of type $E_i$ if all its vertices are in $V_{i-1}\cup V_i$ and of type $E_i\cup \ldots \cup E_{i+j}$ if all its vertices are in $V_{i-1}\cup\ldots\cup V_{i+j}$

The free pseudospaces will be modeled along the lines of a projective space, i.e. we will think of vertices of type $V_i$ as $i$-dimensional spaces in a . Therefore we extend the notion of incidence as follows:

\[d:incidence\]

1.  \[d:incidence1\] We say that a vertex $x_i$ of type $V_i$ is *incident* to a vertex $x_j$ of type $V_j$ if there are vertice $x_l$ of type $V_l, l=i+1\ldots j$ such that 