Sets of generators and chains of subspaces

TL;DR

This paper compares two definitions of rank in point-line geometries, showing that for infinite cases, only well-ordered chains accurately reflect the generating rank, especially in polar spaces.

Contribution

It demonstrates that the rank defined via well-ordered chains matches the generating rank, clarifying the correct approach for infinite geometries.

Findings

Well-ordered chains determine the rank in infinite geometries.

Arbitrary chains can overestimate the rank.

Application to polar spaces confirms the theoretical result.

Abstract

The rank of a point-line geometry G is usually defined as the generating rank of G, namely the minimal cardinality of a generating set. However, when the subspace lattice of G satisfies the Exchange Property we can also try a different definition: consider all chains of subspaces of G and take the least upper bound of their lengths as the rank of G. If G is finitely generated then these two definitions yield the same number. On the other hand, as we shall show in this paper, if infinitely many points are needed to generate G then the rank as defined in the latter way is often (perhaps always) larger than the generating rank. So, if we like to keep the first definition we should accordingly discard the second one or modify it. We can modify it as follows: consider only well ordered chains instead of arbitrary chains. As we shall prove, the least upper bound of the lengths of well ordered chains of subspaces is indeed equal to the generating rank. According to this result, the (possibly infinite) rank of a polar space can be characterized as the least upper bound of the lengths of well ordered chains of singular subspaces; referring to arbitrary chains would be an error.