Quotient sets and subset-subspace analogy (Q1961905)

Using Knuth's fundamental observation about the rank and order preserving map from the lattice of the finite linear subspace lattice \({\mathcal L}_q(n)\) to the Boolean lattice \(B_n,\) this paper exhibits an isomorphism between the Boolean lattice \(B_n\) and the quotient lattice \({\mathcal L}_q(n) / T_n(q),\) where \(T_n(q)\) is the subgroup of the general linear group \(\text{GL}_n(q)\) consisting of all upper triangular matrices. This structure theorem gives combinatorial interpretations of taking the limit \(q \rightarrow 1\) in certain \(q\)-identities.