{"pk":64961,"title":"Foundations of matroids Part 2: Further theory, examples, and computational methods","subtitle":null,"abstract":"In this sequel to \"Foundations of matroids - Part 1,\" we establish several presentations of the foundation of a matroid in terms of small building blocks. For example, we show that the foundation of a matroid \\(M\\) is the colimit of the foundations of all embedded minors of \\(M\\) isomorphic to one of the matroids \\(U^2_4\\), \\(U^2_5\\), \\(U^3_5\\), \\(C_5\\), \\(C_5^\\ast\\), \\(U^2_4\\oplus U^1_2\\), \\(F_7\\), \\(F_7^\\ast\\), and we show that this list is minimal. We establish similar minimal lists of building blocks for the classes of 2-connected and 3-connected matroids. We also establish a presentation for the foundation of a matroid in terms of its lattice of flats. Each of these presentations provides a useful method to compute the foundation of certain matroids, as we illustrate with a number of concrete examples. Combining these techniques with other results in the literature, we are able to compute the foundations of several interesting classes of matroids, including whirls, rank-2 uniform matroids, and projective geometries. In an appendix, we catalogue various `small' pastures which occur as foundations of matroids, most of which were found with the assistance of a computer, and we discuss some of their interesting properties.\n \nMathematics Subject Classifications: 05B35, 12K99\n \nKeywords: Matroid representation, cross ratio, inner Tutte group, foundations","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Matroid representation"},{"word":"cross ratio"},{"word":"inner Tutte group"},{"word":"foundations"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/78w5q7gt","frozenauthors":[{"first_name":"Matthew","middle_name":"","last_name":"Baker","name_suffix":"","institution":"School of Mathematics, Georgia Tech, Atlanta, U.S.A.","department":""},{"first_name":"Oliver","middle_name":"","last_name":"Lorscheid","name_suffix":"","institution":"Mathematics Department, University of Groningen, The Netherlands","department":""},{"first_name":"Tianyi","middle_name":"","last_name":"Zhang","name_suffix":"","institution":"School of Mathematics, Georgia Tech, Atlanta, U.S.A.","department":""}],"date_submitted":"2025-03-14T21:11:24+05:00","date_accepted":"2025-03-14T21:11:24+05:00","date_published":"2025-03-15T12:00:00+05:00","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64961/galley/49771/download/"}]}