Sets of generators and chains of subspaces

The rank of a point-line geometry Gamma is usually defined as the generating rank of Gamma, namely the minimal cardinality of a generating set. However, when the subspace lattice of Gamma satisfies the Exchange Property we can also try a different definition: consider all chains of subspaces of Gamma and take the least upper bound of their lengths as the rank of Gamma. If Gamma 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 Gamma 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.