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Oriented Matroids and Associated Valuations

Jim Lawrence
2004 Discrete & Computational Geometry  
Additionally, the connection between the valuations and the total polynomials associated with uniform oriented matroids is noted.  ...  It is possible to associate a valuation on the "orthant lattice" with each oriented matroid.  ...  Oriented Matroids and Associated Valuations 459 Valuations from Uniform Oriented Matroids Let O be an oriented matroid and let ν be the associated valuation, as in Section 4.  ... 
doi:10.1007/s00454-004-1114-6 fatcat:g52dx4t3cvhkdhatyaxesytgvm

Pfaffian forms and Δ-matroids

Walter Wenzel
1993 Discrete Mathematics  
This suggests a definition of orientable and valuated d-matroids or, more generally, of d-matroids with coefficients which is analogous to the corresponding concept studied in matroid theory.  ...  ., Pfaffian forms and d-matroids, Discrete Mathematics 115 (1993) 253-266.  ...  Acknowledgement The idea to connect the theory of matroids with coefficients with the theories of representable A-matroids and of (W, P)-matroids was kindly suggested to me by Andreas Dress who also had  ... 
doi:10.1016/0012-365x(93)90494-e fatcat:2oaszvvzkrexfdwt3ejvarrls4

Pfaffian forms and Δ-matroids with coefficients

Walter Wenzel
1996 Discrete Mathematics  
Particular examples are A-matroids representable by some skew-symmetric matrix with coefficients in a field, oriented A-matroids, and valuated A-matroids.  ...  In this paper I begin to build up a theory of A-matroids with coefficients parallel to the theory of matroids with coefficients.  ...  Finally, as suggested by results concerning (valuated) matroids and (affine) buildings associated with linear geometries, relations between valuated (W;P)matroids and affine buildings for several W and  ... 
doi:10.1016/0012-365x(94)00172-f fatcat:4oz2hpquljc7bgq7tcwa2op2gu

On Circuit Valuation of Matroids

Kazuo Murota, Akihisa Tamura
2001 Advances in Applied Mathematics  
The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids.  ...  Minty's characterization for matroids by the painting property is generalized for valuated matroids.  ...  For any oriented matroid, the underlying matroid is obtained by ignoring signs. Conversely, a matroid is said to be orientable if there exists an oriented matroid associated in this way.  ... 
doi:10.1006/aama.2000.0716 fatcat:3ofdhyeb2facffywbm7k7xox7y

Combinatorics and Nonparametric mathematics

Joseph P. S. Kung
1997 Annals of Combinatorics  
Pushing this further, by retaining a 2, 4, or 8-tuple of bits or trits, one obtains complex, quarternionic, and octonionic matroids or oriented matroids.  ...  Similarly, a theory of octonionic matroids would uncover the precise role played by associativity in linear algebra or, equivalently, Desargues' theorem in geometry.  ...  The paper of Terhalle [9] , studies the role played by a valuated matroid in "realizing" trees.  ... 
doi:10.1007/bf02558467 fatcat:pcwsihoceffjzlewkcmwq2aahi

Flag matroids with coefficients [article]

Manoel Jarra, Oliver Lorscheid
2022 arXiv   pre-print
We establish duality of flag matroids and construct minors.  ...  This paper is a direct generalization of Baker-Bowler theory to flag matroids, including its moduli interpretation as developed by Baker and the second author for matroids.  ...  We would like to thank Matthew Baker for several discussions and Christopher Eur for pointing us to the work on strong maps of oriented matroids.  ... 
arXiv:2204.04658v1 fatcat:hs3epp3oyfcmhnxm54kfbvigba

Perfect matroids over hyperfields [article]

Nathan Bowler, Rudi Pendavingh
2020 arXiv   pre-print
Also, we present vector axioms for matroids over stringent skew hyperfields which generalize the vector axioms for oriented matroids and valuated matroids.  ...  We investigate valuated matroids with an additional algebraic structure on their residue matroids. We encode the structure in terms of representability over stringent hyperfields.  ...  are as expected (and as for oriented and valuated matroids) the restrictions of the vectors of M to X.  ... 
arXiv:1908.03420v2 fatcat:7dbcpc73izgmxezavex3uhf6y4

Generalized permutahedra and positive flag Dressians [article]

Michael Joswig, Georg Loho, Dante Luber, Jorge Alberto Olarte
2022 arXiv   pre-print
We study valuated matroids, their tropical incidence relations, flag matroids and total positivity.  ...  Further, we get a characterization of those subdivisions arising from positive valuated flag matroids.  ...  Further, we want to thank Jonathan Boretsky, Federico Castillo, Daniel Corey, Christopher Eur, Melissa Sherman-Bennett and Ben Smith for their feedback and Lauren Williams for pointing  ... 
arXiv:2111.13676v3 fatcat:ma376jdiazb67eoc4g7uaakbfu

The geometry of geometries: matroid theory, old and new [article]

Federico Ardila
2021 arXiv   pre-print
We survey some recent successes, stemming from three geometric models of a matroid: the matroid polytope, the Bergman fan, and the conormal fan.  ...  The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization  ...  for the great fun we've had understanding matroids together.  ... 
arXiv:2111.08726v1 fatcat:7s3e4abrabglzidij7dsk7ee5m

The positive Dressian equals the positive tropical Grassmannian

David Speyer, Lauren K. Williams
2021 Transactions of the American Mathematical Society. Series B  
in 2017 by Ardila-Rincón-Williams) that all positively oriented matroids are realizable.  ...  Matroid and positroid polytopes. In what follows, we set e I := i∈I e i ∈ R n , where {e 1 , . . . , e n } is the standard basis of R n . Definition 2.3.  ...  It is easy to verify that the dual of a positively oriented matroid is again a positively oriented matroid, and moreover, [ARW17, Lemma 4.11] showed that positively oriented matroids are closed under restriction  ... 
doi:10.1090/btran/67 fatcat:h2qhbv7djjennjlwgzvszvakse

The positive Dressian equals the positive tropical Grassmannian [article]

David Speyer, Lauren K. Williams
2020 arXiv   pre-print
in 2017 by Ardila-Rincon-Williams) that all positively oriented matroids are realizable.  ...  We also show that the finest regular positroidal subdivisions of the hypersimplex consist of series-parallel matroid polytopes, and achieve equality in Speyer's f-vector theorem.  ...  It is easy to verify that the dual of a positively oriented matroid is again a positively oriented matroid, and moreover, [ARW17, Lemma 4.11] showed that positively oriented matroids are closed under restriction  ... 
arXiv:2003.10231v1 fatcat:ho2rt6c56rfsfgg2j3zeglq3nq

Page 317 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews  
Does every oriented matroid have an encoding in terms of a pseudoconfiguration of points?  ...  The main result of this paper is to exhibit an oriented matroid that admits an adjoint, but for which any adjoint of the underlying matroid does not itself admit an adjoint.  ... 

Valuated Matroid Intersection I: Optimality Criteria

Kazuo Murota
1996 SIAM Journal on Discrete Mathematics  
Specifically, the problem considered is: Given a bipartite graph G = (V + , V − ; A) with arc weight w : A → R and matroid valuations ω + and ω − on V + and V − respectively, find a matching M (⊆ A) that  ...  The independent assignment problem (or the weighted matroid intersection problem) is extended using Dress-Wenzel's matroid valuations, which are attached to the vertex set of the underlying bipartite graph  ...  He also thanks Satoru Iwata for careful reading of the manuscript and for fruitful discussion, which resulted in the extension given in Section 5.  ... 
doi:10.1137/s0895480195279994 fatcat:te5ryqupdbhkfbiveo7swvvgfy

Matroids over partial hyperstructures [article]

Matthew Baker, Nathan Bowler
2018 arXiv   pre-print
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids.  ...  We also give examples of tracts F and weak F-matroids which are not strong.  ...  There are a number of important enhancements of the notion of matroid, including oriented matroids, valuated matroids, and regular matroids.  ... 
arXiv:1709.09707v2 fatcat:r2ufkj563zhyphwzaltbsgrzka

Shortest bibranchings and valuated matroid intersection

Kenjiro Takazawa
2012 Japan journal of industrial and applied mathematics  
The valuated matroid intersection problem, introduced by Murota, is a weighted generalization of the independent matching problem, including the independent assignment problem and the weighted matroid  ...  This reduction suggests one answer to why the shortest S-T bibranching problem is tractable, and implies new combinatorial algorithms for the shortest S-T bibranching problem based on the valuated matroid  ...  This work was supported by Grant-in-Aid for Young Scientists from the Ministry of Education, Culture, Sports, Science and Technology of Japan.  ... 
doi:10.1007/s13160-012-0072-2 fatcat:duze36pukbazfoprpndloaba6a
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