On Pseudomodular Matroids and Adjoints on Pseudomodular Matroids and Adjoints
نویسنده
چکیده
1992 Mathematisches Institut Universitt at zu KK oln Weyertal 86-90 D{5000 KK oln 41 (Lindenthal) Germany Telefon (0221) 470{3937 Telex 8882291 unik d Telefax (0221) 470{5160 Abstract There are two concepts of duality in combinatorial geometry. A set theoretical one, generalizing the structure of two orthocomplementary vector spaces and a lattice theoretical concept of an adjoint, that mimics duality between points and hyperplanes. The latter { usually called polarity { seems to make sense almost only in the linear case. In fact the only non-linear combinatorial geometries known to admit an adjoint were of rank 3. Moreover, N. E. Mn ev conjectured that in higher ranks there would exist no non-linear oriented matroid that has an oriented adjoint. At least with unoriented matroids this is not true. In this paper we present a class of rank 4 matroids with adjoint including a non-linear example.
منابع مشابه
Gammoids, Pseudomodularity and Flatness Degree
We introduce the concept of flatness degree for matroids, as a generalization of submodularity. This represents weaker variations of the concept of flatness which characterize strict gammoids for finite matroids. We prove that having flatness degree 3, which is the smallest non-trivial flatness degree, implies pseudomodularity on the lattice of flats of the matroid. We show however an example o...
متن کاملCATEGORICAL RELATIONS AMONG MATROIDS, FUZZY MATROIDS AND FUZZIFYING MATROIDS
The aim of this paper is to study the categorical relations betweenmatroids, Goetschel-Voxman’s fuzzy matroids and Shi’s fuzzifying matroids.It is shown that the category of fuzzifying matroids is isomorphic to that ofclosed fuzzy matroids and the latter is concretely coreflective in the categoryof fuzzy matroids. The category of matroids can be embedded in that offuzzifying matroids as a simul...
متن کاملBASES AND CIRCUITS OF FUZZIFYING MATROIDS
In this paper, as an application of fuzzy matroids, the fuzzifying greedy algorithm is proposed and an achievableexample is given. Basis axioms and circuit axioms of fuzzifying matroids, which are the semantic extension for thebasis axioms and circuit axioms of crisp matroids respectively, are presented. It is proved that a fuzzifying matroidis equivalent to a mapping which satisfies the basis ...
متن کاملM-FUZZIFYING MATROIDS INDUCED BY M-FUZZIFYING CLOSURE OPERATORS
In this paper, the notion of closure operators of matroids is generalized to fuzzy setting which is called $M$-fuzzifying closure operators, and some properties of $M$-fuzzifying closure operators are discussed. The $M$-fuzzifying matroid induced by an $M$-fuzzifying closure operator can induce an $M$-fuzzifying closure operator. Finally, the characterizations of $M$-fuzzifying acyclic matroi...
متن کاملStructural properties of fuzzy graphs
Matroids are important combinatorial structures and connect close-lywith graphs. Matroids and graphs were all generalized to fuzzysetting respectively. This paper tries to study connections betweenfuzzy matroids and fuzzy graphs. For a given fuzzy graph, we firstinduce a sequence of matroids from a sequence of crisp graph, i.e.,cuts of the fuzzy graph. A fuzzy matroid, named graph fuzzy matro...
متن کامل