Order, lattices, ordered algebraic structures Modular and complemented lattices Theorem

von Neumann coordinatization theorem

Builds on Modular lattice Complemented lattice von Neumann regular ring

Statement

Let $L$ be a complemented modular lattice with least element $0$ and greatest element $1$ . Elements $a, b \in L$ are perspective if some $x \in L$ satisfies $a \wedge x = b \wedge x = 0$ and $a \vee x = b \vee x$ . Elements $a_1, \dots, a_n \in L$ are independent if $(a_1 \vee \cdots \vee a_k) \wedge a_{k+1} = 0$ for each $k$ with $1 \le k < n$ . A homogeneous basis of $L$ is a family $a_1, \dots, a_n$ of independent elements with $a_1 \vee \cdots \vee a_n = 1$ . In addition, $a_i$ and $a_j$ must be perspective for all $i, j$ .

Suppose that $L$ has a homogeneous basis with $n \ge 4$ elements. Then there is a von Neumann regular ring $R$ such that $L$ is isomorphic to the lattice of principal right ideals of $R$ . Here the principal right ideals are ordered by inclusion.

Remarks

For a regular ring $R$ , every principal right ideal is generated by an idempotent, and the principal right ideals form a complemented modular lattice; the theorem gives the converse. The ring is built from $L$ much as a division ring is built from a projective space of dimension at least $3$ . The bound $n \ge 4$ plays the role of that dimension bound.

Example: for a division ring $D$ and $n \ge 4$ , the lattice of subspaces of $D^n$ has the homogeneous basis of coordinate lines. It is isomorphic to the lattice of principal right ideals of the matrix ring $M_n(D)$ , which is regular.

Von Neumann applied the theorem to continuous geometries, where it yields continuous regular rings.

Going deeper

For a student

The theorem identifies complemented modular lattices that have a homogeneous basis with at least $4$ elements with lattices of principal right ideals of regular rings. It explains why regular rings are the natural algebraic counterpart of complemented modular lattices. Its main application is to continuous geometries, whose coordinate rings are the continuous regular rings, including rings built from type II $_1$ factors. The independence and perspectivity conditions are lattice versions of a direct sum decomposition into isomorphic summands, like $D^n = D \oplus \cdots \oplus D$ .

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Complemented lattice concept
Modular lattice concept
von Neumann regular ring concept
von Neumann coordinatization theorem statement

Tag	`VDRCDZ048J`
URI	`m2://VDRCDZ048J`
Kind	statement, theorem
Writing	W0 stub
MSC	06C20, 16E50
Status	This copy of the site has no build data. Run `m2 status VDRCDZ048J`.
About	Modular lattice, von Neumann regular ring
Uses	Modular lattice, Complemented lattice, von Neumann regular ring