Dowling geometry

==Generalizations==

==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.

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 further generalization due to Zaslavsky (2012) that involves ''n''-ary quasigroups.

A different generalization due to Zaslavsky (1991) is obtained from the full <math>G</math>-expansion of an arbitrary graph <math>\Gamma</math>. This gain graph has lattice obtained from the Dowling lattice by excluding all partial partitions such that the induced subgraph on some ''B_i'' is disconnected. The characteristic polynomial of this matroid is obtained from the chromatic polynomial <math>\chi_\Gamma(t)</math> of <math>\Gamma</math> by substituting <math>t = (y-1)/|G|</math> and normalizing to a monic polynomial.

==References==

==References==