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A geometric characterization of dowling lattices

1991
*
Journal of combinatorial theory. Series A
*

ACKNOWLEDGMENT We thank the referee for insightful suggestions for shortening the proof

doi:10.1016/0097-3165(91)90032-c
fatcat:26sz27ws55eehe7hzqcuhgttz4
*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. ...##
###
Modular elements of higher-weight dowling lattices

1993
*
Discrete Mathematics
*

The higher-weight

doi:10.1016/0012-365x(93)90113-8
fatcat:kfal52vmdbey5lzuzo56xkl4ce
*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 ...##
###
Page 632 of Mathematical Reviews Vol. , Issue 92b
[page]

1992
*
Mathematical Reviews
*

*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*...

##
###
Complete principal truncations of Dowling lattices

2004
*
Advances in Applied Mathematics
*

In this paper, we give

doi:10.1016/s0196-8858(03)00089-7
fatcat:w7jhpodgczbxppaqxlamiv36fa
*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 [2] , Bogart and Bonin*characterized**Dowling**lattices*using*geometric*axioms. Theorem 1.1 [1] . ...##
###
Page 2581 of Mathematical Reviews Vol. , Issue 95e
[page]

1995
*
Mathematical Reviews
*

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*. ...##
###
Whitney numbers of some geometric lattices

1994
*
Journal of combinatorial theory. Series A
*

In the following we study the problem in the case

doi:10.1016/0097-3165(94)90034-5
fatcat:aoqcqqs2cnfs7p2g7z3tfts3wm
*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]. ...##
###
On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers
[article]

2019
*
arXiv
*
pre-print

The M\"obius invariant

arXiv:1811.06882v3
fatcat:tm7iogpov5cwneadey6mzrmgeq
*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. ...##
###
Page 59 of Mathematical Reviews Vol. , Issue 98A
[page]

1998
*
Mathematical Reviews
*

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*. ...##
###
Line-closed combinatorial geometries

1987
*
Discrete Mathematics
*

We present some results on combinatorial geometries (

doi:10.1016/0012-365x(87)90056-2
fatcat:l4mpbv4d7fdapdw3wdgjroxnzi
*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]. ...##
###
Supersolvable Frame-matroid and Graphic-lift Lattices

2001
*
European journal of combinatorics (Print)
*

*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. ...

##
###
On Flag Vectors, the Dowling Lattice, and Braid Arrangements

1999
*
Discrete & Computational Geometry
*

We give

doi:10.1007/pl00009428
fatcat:wb4q7hv3j5g2jlor2rlbop3s2q
*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. ...##
###
Characterizing Combinatorial Geometries by Numerical Invariants

1999
*
European journal of combinatorics (Print)
*

In particular, we show that any matroid with the same Tutte polynomial as

doi:10.1006/eujc.1999.0245
fatcat:fdzv7rmru5bwpar3dnwfmo7jvy
*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. ...##
###
Continuous partition lattice

1987
*
Proceedings of the National Academy of Sciences of the United States of America
*

*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. ...

##
###
Page 4760 of Mathematical Reviews Vol. , Issue 87i
[page]

1987
*
Mathematical Reviews
*

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. ...##
###
On the Dowling and Rhodes lattices and wreath products
[article]

2017
*
arXiv
*
pre-print

Although the Rhodes

arXiv:1710.05314v1
fatcat:znw3v46lmra2naxld537slapjq
*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. ...
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