an exceptionally simple formula for any geometric lattice. Generalizations There is also a Dowling geometry, of rank 3 only, associated with each quasigroup; see Dowling (1973b). This does not generalize in a straightforward way to higher ranks. There is a generalization due to Zaslavsky (2012) that involves n-ary quasigroups. References Peter Doubilet, Gian-Carlo Rota, and Richard P. Stanley (1972), On the foundations of combinatorial theory (VI): The idea of generating function. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, Calif., 1970/71), Vol. II: Probability Theory, pp.\ 267–318. University of California Press, Berkeley, Calif., 1972. T.A. Dowling (1973a), A q-analog of the partition lattice. Chapter 11 in: J.N. Srivastava et al., eds., A Survey of Combinatorial Theory (Proceedings of an International Symposium, Ft. Collins, Colo., 1971), pp. 101–115. North-Holland, Amsterdam, 1973. T.A. Dowling (1973b), A class of geometric lattices based on finite groups. Journal of Combinatorial Theory, Series B, Vol. 14 (1973), pp. 61–86. Kahn, Jeff, and Kung, Joseph P.S. (1982), Varieties of combinatorial geometries. Transactions of the American Mathematical Society, Vol. 271, pp. 485–499. Thomas Zaslavsky (1991), Biased graphs. II. The three matroids. Journal of Combinatorial Theory, Series B, Vol. 51, pp. 46–72. Thomas Zaslavsky (2012), Associativity in multary quasigroups: The way of biased expansions. "Aequationes Mathematicae", Vol. 83, no. 1, pp. 1–66. Matroid theory Finite groups Finite fields