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A LatticeTheoretical Characterization of Oriented Matroids
 EUROP. J. COMBINATORICS
, 1997
"... If P is the big face lattice of the covectors of an oriented matroid, it is well known that the zero map is a coverpreserving, orderreversing surjection onto the geometric lattice of the underlying (unoriented) matroid. In this paper we give a (necessary and) sufficient condition for such maps ..."
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If P is the big face lattice of the covectors of an oriented matroid, it is well known that the zero map is a coverpreserving, orderreversing surjection onto the geometric lattice of the underlying (unoriented) matroid. In this paper we give a (necessary and) sufficient condition for such maps to come from the face lattice of an oriented matroid.
Adjoints and Duals of Matroids Linearly Representable over a Skewfield
, 1994
"... Following an approach suggested by B. Lindstrom we prove that the dual of a matroid representable over a skewfield is itself representable over the same field. Along the same line we show that any matroid within this class has an adjoint. As an application we derive an adjoint for the dual of the No ..."
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Following an approach suggested by B. Lindstrom we prove that the dual of a matroid representable over a skewfield is itself representable over the same field. Along the same line we show that any matroid within this class has an adjoint. As an application we derive an adjoint for the dual of the NonPappusMatroid. Furthermore, we reprove a result by Alfter and Hochstättler concerning the existence of an adjoint for a certain eight point configuration and show that this configuration is linearly representable over a field if and only if the field is skew.
About the TicTacToe Matroid
, 1997
"... The purpose of this note is to make a problem, already mentioned in [3], more tangible. We introduce a matroid which has "the" combinatorial properties of algebraic matroids as derived in [4], the dual of which is nonalgebraic. Therefore, it seems to be a good candidate for a negative ..."
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The purpose of this note is to make a problem, already mentioned in [3], more tangible. We introduce a matroid which has "the" combinatorial properties of algebraic matroids as derived in [4], the dual of which is nonalgebraic. Therefore, it seems to be a good candidate for a negative answer to the old problem whether algebraic matroids are closed under duality (see eg. [8] 6.7.15).
A Pseudoconfiguration of Points without Adjoint
, 1995
"... We give an example of a simple oriented matroid D that admits an oriented adjoint. Already any adjoint of the underlying matroid D, however, does itself not admit an adjoint. D arises from the wellknown NonDesarguesMatroid by a coextension by a coparallel element and, hence, has rank 4. The ori ..."
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We give an example of a simple oriented matroid D that admits an oriented adjoint. Already any adjoint of the underlying matroid D, however, does itself not admit an adjoint. D arises from the wellknown NonDesarguesMatroid by a coextension by a coparallel element and, hence, has rank 4. The orientability of D and some of its adjoints follows from an apparantly new oriented matroid construction given in the paper that is a very special case of an amalgam of two copies of one oriented matroid.