Dowling lattices and their generalizations introduced by Hanlon are interpreted as lattices of congruences associated to certain quasi-varieties of sets with group actions. This interpretation leads, by a simple application of Mobius inversion, to polynomial identities which specialize to Hanlon's evaluation of the characteristic polynomials of generalized Dowling lattices. Analogous results ar...

A finite subgeometry of a Dowling geometry for an infinite group is exhibited, which cannot be embedded in a Dowling geometry for any finite group; this provides a negative answer to a question of Bonin. The question addressed herein is motivated by the following fundamental result of Rado, concerning the embeddability of finite geometries in projective geometries [Ra, Theorem 4]: Theorem. Ever...

In this paper we study a new class of lattices called the generalized Dowling lattices. These lattices are parametrized by a positive integer n , a finite group G , and a meet sublattice K of the lattice of subgroups of G . For an appropriate choice of K the generalized Dowling lattice Dn{G,K) agrees with the ordinary Dowling lattice Dn(G). For a different choice of K, the generalized Dowling l...

Dowling-Degos disease (DDD) is a rare autosomal dominant inherited pigmentary disorder of the flexures with a reticulate aspect and with presence of prominent comedone-like lesions and pitted scars. The diagnosis includes acanthosis nigricans as well as other reticulate pigmentary disorders classified into: dyschromatrosis symmetrica hereditaria (DSH), dyschromatosis universalis hereditaria (DU...

This paper relates the critical problem of Crapo and Rota [ " On the Foundations of Combinatorial Theory: Combinatorial Geometries, " M.I.T. Press, Cambridge, MA, 19703 to Dowling group geometries. If A is a finite group, Q,(A) is the rank-r Dowling group geometry over A, and M is a rank-r matroid embeddable as a minor of Q,(A), then it is shown that the critical exponent of M over A is well de...

We present a new combinatorial method to determine the characteristic polynomial of any subspace arrangement that is defined over an infinite field, generalizing the work of Blass and Sagan. Our methods stem from the theory of valuations and Groemer's integral theorem. As a corollary of our main theorem, we obtain a result of Zaslavsky about the number of chambers of a real hyperplane arrangeme...

The notion of exponential Dowling structures is introduced, generalizing Stanley’s original theory of exponential structures. Enumerative theory is developed to determine the Möbius function of exponential Dowling structures, including a restriction of these structures to elements whose types satisfy a semigroup condition. Stanley’s study of permutations associated with exponential structures l...

The notion of exponential Dowling structures is introduced, generalizing Stanley’s original theory of exponential structures. Enumerative theory is developed to determine the Möbius function of exponential Dowling structures, including a restriction of these structures to elements whose types satisfy a semigroup condition. Stanley’s study of permutations associated with exponential structures l...