Ordered Sets in Geometry

Abstract

Partial orderings play a major role in geometry, especially in combinatorial settings. Most of these partial orderings involve lattices, and a rich profusion of lattices arise in this way. A brief review will be given of many old results, and some new ones, about lattices arising in this way.

I. Order and Incidence: Flats. As is well-known, the ‘flats’ (= points, lines, planes, etc.) in any projective or affine geometry form a geometric lattice.

In the (self-dual) projective case, this lattice is modular; in the affine case, it is a ‘Hilbert lattice’. In n ≥ 3 dimensions, these lattices can be ‘coordinatized’: for some division ring D, they are the lattices of linear subspaces of D ⁿ⁺¹ and of affine subspaces of D ⁿ, respectively. (See [Art], [B-B], [Hil], [LT1, p. 60], and [V-Y].)

Analogous lattices arise in infinite-dimensional function spaces, as the lattices of (closed) subspaces of Hilbert space left invariant by some self-adjoint linear operator. Von Neumann [JvN] coordinatized these by a brilliant extension of Wedderburn theory to regular rings. These subspace lattices also describe ‘the logic of quantum mechanics’ [JvN, pp. 105–125].

II. Order and ‘Polarity’: Orthogonality. The lattices discussed in §I are associated with homogeneous and inhomogeneous linear equations, respectively, by an obvious ‘polarity’ [LT1, §32]. Any equation

$${a_0}{x_0} + {a_1}{x_1} + \ldots + {a_n}{x_n} = 0,$$

((1))

respectively

$${a_1}{x_1} + {a_2}{x_2} + \ldots + {a_n}{x_n} = 0,$$

((2))

defines a binary relation a ρ x (meaning 〈a,x〉 = 0) between points x = (x ₀,x ₁,…,x _n ) or x = (x ₁,x ₂,…,x _n ) and coefficient vectors a = (a ₀,a ₁,…,a _n ). The sets S ⊂ D ⁿ⁺¹ that are closed under the resulting polarity are just the (affine and linear) flats in the usual geometric sense.

When D is an ordered formally real fields, the equation 〈a,x〉 = 0 defines an orthogonality relation a ⊥ x in the usual sense.

Even more interesting is the analogue of (1) for algebraic varieties of higher degree. Simplest among these (after ‘flats’) are the circles, spheres, and hyperspheres, defined by equations of the form

$${a_0}{r^2} + \sum\limits_{k = 1}^n {{a_k}{x_k} + c = 0,} $$

((3))

where ${r^2} = \sum\limits_{k = 1}^n {x_k^2}$. The analogues of ‘lines’ are here played by circles. (Since precisely one circle passes through any three points, any subset of two points is ‘closed’.)

Next simplest are conics (and quadrics, etc.), of which just one passes through any 5 coplanar points (any subset of 4 points is ‘closed’). ‘Pencils’ of circles, spheres, conics, quadrics, etc. can be handled very naturally in this context. [LT3, pp. 84, 124].

III. Order and Convexity: Segments. Much as replacing ‘inclusion’ by (self-dual) incidence is very natural geometrically, so it is natural to replace ‘order’ by betweenness. On a line, this leads to the notions of ‘interval’ and convexity. Hence, if D is an ordered division ring or field, we have a much richer, though still elementary, geometric theory. [Hil]

As one, highly combinatorial aspect of this, we have the theory of convex polyhedra, and their lattices of facets. As another, we have lattices of convex subsets. As still another, we have the theory of linear inequalities (hence linear programming). [Gru]

Homology. The combinatorial topology of Poincaré is based on the concept of a polyhedral complex, in terms of which one can define not only ‘dimension’, but also homology groups and many other topological invariants of manifolds.

Several authors have tried to define the (global) topology of complexes in terms of invariants of such complexes. ([A-H], [Tue], [Whi], [Rot]).

IV. Projective Metrics. Stemming from Laguerre, through Felix Klein and Hilbert, is the concept of a projective metric. In its simplest form, this defines d(x,y) on any line as the logarithm of the cross-ratio χ(a,x,y,b), a and b being the points where the line pierces the (strictly convex) skin of the ‘ball’ ∥x∥ ≤ a. If the ‘ball’ is an ellipsoid, this construction gives hyperbolic geometry. ([Hil], [Pog]).

From any strictly convex ‘ball’, the geodesies for this distance (‘metric’) are straight lines and form a Hilbert lattice, as they also are for the symmetric Riemann spaces of E. Cartan.

Positive Linear Operators. Finally, in any convex cone ∁ (of ‘positive’ vectors), the preceding construction defines a quasimetric, with respect to which any linear transformation such that f(∁) < ∁ is a contraction.

In the special case that ∁ is the positive orthant of all x with all x _i ≥ 0, the preceding construction gives the Perron-Frobenius theory of positive (and non-negative) matrices, and its extension to positive linear operators [LT3, Ch. XVI].