Use this tag for questions about lattices whose subspace dimensions can be a number in the interval [0, 1].
Continuous geometry is an analogue of complex projective geometry where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be number in the interval [0, 1]. The first example of a continuous geometry other than projective space was projections of the hyperfinite type II factor.
A continuous geometry is a lattice L with the following properties:
- L is modular.
- L is complete.
- The lattice operations ∧, ∨ satisfy a certain continuity property, $\big( \bigwedge_{α \in A}$ a$_α \big)$ ∨ b = $\bigwedge_α$(a$_α$ ∨ b), where A is a directed set and if α < β then a$_α$ < a$_β,$ and the same condition with ∧ and ∨ reversed.
- Every element in L has a (not necessarily unique) complement. A complement of an element a is an element b with a ∧ b = 0, a ∨ b = 1, where 0 and 1 are the minimal and maximal elements of L.
- L is irreducible, i.e., the only elements with unique complements are 0 and 1.