continuous geometry
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 wellknown 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 socalled “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:
$$d(\underset{i=1}{\overset{\mathrm{\infty}}{\bigvee}}{a}_{i})=\sum _{i=1}^{\mathrm{\infty}}d({a}_{i})\text{whenever}{a}_{j+1}\wedge (\underset{i=1}{\overset{j}{\bigvee}}{a}_{i})=0\text{for}j\ge 1.$$ 
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 (http://planetmath.org/ComplementedLattice) relation^{} $\sim $ on elements of $L$ is a transitive relation (Von Neumann). Since $\sim $ 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 $PG(n1)$ the projective geometry of dimension $n1$ over $D$ (lattice of left (right) subspaces of left (right) $n$dimensional vector space over $D$). Von Neumann found that $PG(n1)$ can be embedded into $PG(2n1)$ 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 $\varphi :PG(n1)\to PG(2n1)$ is the embedding^{}, and ${d}_{n}$ is the dimension function on $PG(n1)$ and ${d}_{2n}$ is the dimension function on $PG(2n1)$, then ${d}_{n}(a)={d}_{2n}(\varphi (a))$. As a result, we get a chain of embeddings
$$PG(1)\hookrightarrow PG(3)\hookrightarrow \mathrm{\cdots}\hookrightarrow PG({2}^{n}1)\hookrightarrow \mathrm{\cdots}.$$ 
Taking the union of these lattices, we get a lattice $PG(\mathrm{\infty})$, 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 $PG(\mathrm{\infty})$, 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.

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Any continuous geometry is a complete lattice^{} and a topological lattice if order convergence is used to define a topology^{} on it.

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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 $D$.

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(Kaplansky) Any orthocomplemented complete^{} modular lattice is a continuous geometry.
References
 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).
Title  continuous geometry 

Canonical name  ContinuousGeometry 
Date of creation  20130322 16:42:21 
Last modified on  20130322 16:42:21 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06C20 
Classification  msc 51D30 
Synonym  von Neumann lattice 
Related topic  LatticeOfProjections 
Defines  irreducible continuous geometry 