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ACKNOWLEDGMENT We thank the referee for insightful suggestions for shortening the proof of Lemma 2. ... A CHARACTERIZATION OF DOWLING LATTICES We now state our characterization of Dowling lattices. We will prove the characterization through a series of propositions. ... Among lattices in the Classification Theorem, only the Dowling lattices satisfy (1). Hence L is a Dowling lattice. 1 2. A CHARACTERIZATION OF PARTITION LATTICES THEOREM 2. ...doi:10.1016/0097-3165(91)90032-c fatcat:26sz27ws55eehe7hzqcuhgttz4
The higher-weight Dowling lattice L, is the geometric lattice consisting of those subspaces of the vector space V,(q) which have bases of weight k or less, with these subspaces ordered by inclusion. ... This paper identifies the modular elements of higher-weight Dowling lattices and applies that analysis to modular complements in Lx and to the characteristic polynomial of 4. ... Another characterization of modularly complemented geometric lattices is as follows: A geometric lattice of rank n is modularly complemented if and only if it contains a rank n Boolean sublattice of modular ...doi:10.1016/0012-365x(93)90113-8 fatcat:kfal52vmdbey5lzuzo56xkl4ce
A 56 (1991), no. 2, 195-202. The authors characterize Dowling lattices of partial G-partitions [see T. A. Dowling, same journal Ser. ... For the same order of the starting system we obtain nonisomorphic systems S™° and S™’.” 92b:05019 05B35 06C10 Bonin, Joseph E. (1-DTM); Bogart, Kenneth P. (1-DTM) A geometric characterization of Dowling ...
In this paper, we give geometric axioms and a numerical characterization of the complete principal truncations of Dowling lattices at modular flats. ... Using these characterizations, we show that these truncations are determined by their Tutte polynomials up to the order of the group. ... In  , Bogart and Bonin characterized Dowling lattices using geometric axioms. Theorem 1.1  . ...doi:10.1016/s0196-8858(03)00089-7 fatcat:w7jhpodgczbxppaqxlamiv36fa
Whitney numbers of some geometric lattices. J. Combin. Theory Ser. A 65 (1994), no. 1, 11-25. The authors extend an earlier result of T. A. Dowling [J. Com- bin. Theory Ser. ... Finally, as a consequence the authors obtain unifying proofs of some properties enjoyed by Whitney numbers of Boolean algebras, subspace lattices, partition lattices, and Dowl- ing lattices. ...
In the following we study the problem in the case of modularly complemented geometric lattices. These lattices have been completely characterized in [Kah] . ... (i) Conditions (ii) The Dowling lattice of rank n over the group G covers, in q/, the Dowling lattice of rank n-1 over the same group [Dow]. ...doi:10.1016/0097-3165(94)90034-5 fatcat:aoqcqqs2cnfs7p2g7z3tfts3wm
The M\"obius invariant of the lattice turns out to be a (nonmedian) Genocchi number. Our techniques also yield type B, and more generally Dowling arrangement, analogs of these results. ... We refine Hetyei's result by obtaining a combinatorial interpretation of the M\"obius function of this lattice in terms of variants of the Dumont permutations. ... Acknowledgements The authors thank José Samper for a valuable suggestion pertaining to bipartite graphs. They also thank the referees for their comments. ...arXiv:1811.06882v3 fatcat:tm7iogpov5cwneadey6mzrmgeq
Whitney numbers of both kinds of Dowling lattices (a class of geometric lattices over a finite group [T. A. Dowling, J. Combina- torial Theory Ser. ... of a finite Weyl group and the class of supersolvable lattices. ...
We present some results on combinatorial geometries (geometric lattices) in which closure is fine-closure. We prove that every interval of a line-closed geometry is line-closed. ... Furthermore, if every rank 3 interval of a geometry is line-closed, then the geometry is line-closed. This impfies that every supersolvable geometry is line-closed. ... It is well known that [0, G] is a geometric lattice as are the intervals of [0, G]. ...doi:10.1016/0012-365x(87)90056-2 fatcat:l4mpbv4d7fdapdw3wdgjroxnzi
A geometric lattice is a frame if its matroid, possibly after enlargement, has a basis such that every atom lies under a join of at most two basis elements. ... A geometric lattice is a graphic lift if it can be extended to contain an atom whose upper interval is graphic. ... Acknowledgement I thank Marge Pratt for expert and patient typing of many and messy changes to get the manuscript completed. ...doi:10.1006/eujc.2000.0418 fatcat:7mjlmlonazafbbnqp4sue2ugtu
We give a combinatorial description of the Dowling lattice via enriched partitions. Since the Dowling lattice is a geometric lattice, it has many EL-labelings. ... By applying this result to the Dowling lattice recursion, we obtain a recursive formula for the cd-index of the lattice of regions of the braid arrangements A n and B n . ... Since it is an intersection lattice, it follows that the Dowling lattice is a geometric lattice of rank n. Observe L n,1 is isomorphic to n+1 , the partition lattice of rank n. ...doi:10.1007/pl00009428 fatcat:wb4q7hv3j5g2jlor2rlbop3s2q
In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice. ... In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. ... Acknowledgement: Theorem 2.6 was suggested by one of the referees; it weakens the hypotheses of a theorem from the original version of the paper. ...doi:10.1006/eujc.1999.0245 fatcat:fdzv7rmru5bwpar3dnwfmo7jvy
A continuous analogue to the partition lattices is presented. This is the metric completion of the direct limit of a system of embeddings of the finite partition lattices. ... The construction is analogous to von Neumann's construction of a continuous geometry over a field F from the finite-dimensional projective geometries over F. ... As a lattice, L(4) inherits many of the typical features of a geometric lattice-e.g., it is relatively complemented. ...doi:10.1073/pnas.84.18.6327 pmid:16593874 pmcid:PMC299068 fatcat:p6hsqsvulnclphh5uzygxoa2zi
They show that such a lattice is either a Dowling lattice or the lattice of flats of a projective geometry with some of its points deleted. R. P. Dilworth (Oroville, Calif.) 87i:06026 ... A geometric lattice is said to be modularly complemented if every point has a modular complement. The authors classify connected modularly complemented geometric lattices of rank at least four. ...
Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of Boolean representable simplicial complexes. ... Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. ... Acknowledgments The first author acknowledges support from the Binational Science Foundation (BSF) of the United States and Israel, grant number 2012080. ...arXiv:1710.05314v1 fatcat:znw3v46lmra2naxld537slapjq
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