Pseudo-Matroids and Cuts of Matroids

Pseudo-Matroids and Cuts of Matroids

by Sergey A. Gizunov and V. N. Lyamin
English | 2016 | ISBN: 1634848810 | 146 Pages | PDF | 4.67 MB

The matroid theory was begun in the 1930s, when B. L. Van der Waerden in his work "Modern Algebra" examined algebraic dependance, along with linear dependance, and H. Whitney (H. Whitney, 1935) introduced the abstract notion of a matroid, in order to generalize the notion of the dual graph. Later on, S. MacLane (S. MacLane, 1936) gave an interpretation of matroids in terms of projective geometry (which served as a reason for matroids to be referred to as combinatory geometrics), while G. Birkhoff (G. Birkhoff, 1935) introduced the notion of the "M-structure" and noted that it includes projective geometries.

Matroids appear in various combinatory algebraic contexts. Such notions as independence and basis in vector spaces, algebraic dependance, cycles and cuts in graphs, surfaces in projective geometries, and point semi-modular lattices all come down to the structure of the matroid. Owing to the possibility of implementing the lattice theory, graphs, vector spaces and geometry language in the description of matroids, some unexpected similarities between the results of graph theory, coding theory, algebra, topology, electro-technics, geometry and combinatorial mathematics were found.

This study examines both strong and weak mappings of matroids. The notion of the "pseudo-matroid" is introduced as a particular type of matroid generated by elementary strong mappings. Various properties of pseudo-matroids are studied. It is remarkable that pseudo-matroids unequivocally define the mappings by which they are generated. This property has allowed the authors of the study to obtain an array of fresh results.

Pseudo-matroids, introduced by the authors of this work, and their properties present an instrument for further studies of matroid categories and mappings.

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