Let $V$ be a finite dimensional vector space (over some field) with dimension $n$ . Let $PG(V)$ be its lattice of subspaces, also known as the projective geometry of $V$ . It is well-known that we can associate each element $a\in PG(V)$ a unique integer $\dim(a)$ , namely, the dimension of the $a$ as a subspace of $V$ . $\dim$ can be seen as a function from $PG(V)$ to $\mathbb{Z}$ . One property of $\dim$ is that for every $i$ between $0$ and $n$ , there is an $a\in PG(V)$ such that $\dim(a)=i$ . If we normalize $\dim$ by dividing its values by $n$ , then we get a function $d: PG(V)\to [0,1]$ . As $n$ (the dimension of $V$ ) increases, the range of $d$ begins to ``fill up'' $[0,1]$ . Of course, we know this is impossible as long as $V$ is finite dimensional.

Question: is there a ``geometry'' on which a ``dimension function'' is defined so that it is onto the closed unit interval $[0,1]$ ?

The answer is yes, and the geometry is the so-called ``continuous geometry''. However, like projective geometries, it is really just a lattice (with some special conditions). A continuous geometry $L$ is a generalization of a projective geometry so that a ``continuous'' dimension function $d$ can be defined on $L$ such that for every real number $r\in [0,1]$ there is an $a\in L$ such that $d(a)=r$ . Furthermore, $d$ takes infinite independent joins to infinite sums: $$d(\bigvee_{i=1}^{\infty} a_i)=\sum_{i=1}^{\infty} d(a_i)\mbox{ whenever }a_{j+1}\wedge (\bigvee_{i=1}^{j} a_i)=0\mbox{ for }j\ge 1.$$

Definition. A continuous geometry is a lattice $L$ that is complemented, modular, meet continuous, and join continuous.

From a continuous geometry $L$ , it can be shown that the perspective relation $\thicksim$ on elements of $L$ is a transitive relation (Von Neumann). Since $\thicksim$ is also reflexive and symmetric, it is an equivalence relation. In a projective geometry, perspective elements are exactly subspaces having the same dimension. From this equivalence relation, one can proceed to define a ``dimension'' function from $L$ into $[0,1]$ .

Continuous geometry was introduced by Von Neumann in the 1930's when he was working on the theory of operator algebras in Hilbert spaces. Write $PG(n-1)$ the projective geometry of dimension $n-1$ over $D$ (lattice of left (right) subspaces of left (right) $n$ -dimensional vector space over $D$ ). Von Neumann found that $PG(n-1)$ can be embedded into $PG(2n-1)$ in such a way that not only the lattice operations are preserved, but the values of the ``normalized dimension function'' $d$ described above are also preserved. In other words, if $\phi:PG(n-1)\to PG(2n-1)$ is the embedding, and $d_{n}$ is the dimension function on $PG(n-1)$ and $d_{2n}$ is the dimension function on $PG(2n-1)$ , then $d_{n}(a)=d_{2n}(\phi(a))$ . As a result, we get a chain of embeddings $$PG(1)\hookrightarrow PG(3)\hookrightarrow \cdots \hookrightarrow PG(2^n-1) \hookrightarrow \cdots.$$ Taking the union of these lattices, we get a lattice $PG(\infty)$ , which is complemented and modular, which has a ``normalized dimension function'' $d$ into $[0,1]$ whose values take the form $p/2^m$ ($p,m$ positive integers). This $d$ is also a valuation on $PG(\infty)$ , turning it into a metric lattice, which in turn can be completed to a lattice $CG(D)$ . This $CG(D)$ is the first example of a continuous geometry having a ``continuous'' dimension function.

Remarks.

• Any continuous geometry is a complete lattice and a topological lattice if order convergence is used to define a topology on it.
• An irreducible continuous geometry is a continuous geometry whose center is trivial (consisting of just $0$ and $1$ ). It turns out that an irreducible continuous geometry is just $CG(D)$ for some division ring $D$ .
• (Kaplansky) Any orthocomplemented complete modular lattice is a continuous geometry.

Bibliography

1

J. von Neumann, Continuous Geometry, Princeton, (1960).

2

G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).

3

G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).