Quasi-Varieties, Congruences, and Generalized Dowling Lattices
Andreas Blass
DOI: 10.1023/A:1022480431917
Abstract
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 Möbius inversion, to polynomial identities which specialize to Hanlon's evaluation of the characteristic polynomials of generalized Dowling lattices. Analogous results are obtained for a few other quasi-varieties.
Pages: 277–294
Keywords: dowling lattice; congruence; free algebra; characteristic polynomial; quasi-variety
Full Text: PDF
References
1. W. Burnside, Theory of Groups of Finite Order, Cambridge Univ. Press, Cambridge, England, 1911.
2. S. Burns and H. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York, Heidelberg, Berlin, 1981.
3. H. Crapo and G.C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, (preliminary edition), M.I.T. Press, Cambridge, MA, 1970.
4. T.A. Dowling, "A class of geometric lattices based on finite groups," J. Comb. Theory, Ser. B 14 (1973), 61-86.
5. P. Hanlon, "The generalized Dowling lattices," Trans. Amer. Math. Soc. 325 (1991), 1-37.
6. S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, New York, Heidelberg, Berlin, 1971.
7. R.P. Stanley, Enumerate Combinatorics, I, Wadsworth & Brooks/Cole, Monterey, CA, 1986.
2. S. Burns and H. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York, Heidelberg, Berlin, 1981.
3. H. Crapo and G.C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, (preliminary edition), M.I.T. Press, Cambridge, MA, 1970.
4. T.A. Dowling, "A class of geometric lattices based on finite groups," J. Comb. Theory, Ser. B 14 (1973), 61-86.
5. P. Hanlon, "The generalized Dowling lattices," Trans. Amer. Math. Soc. 325 (1991), 1-37.
6. S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, New York, Heidelberg, Berlin, 1971.
7. R.P. Stanley, Enumerate Combinatorics, I, Wadsworth & Brooks/Cole, Monterey, CA, 1986.