Abstract
The aim of this paper is to extend the notions of geometric lattices, semimodularity and matroids in the framework of finite posets and related systems of sets. We define a geometric poset as one which is atomistic and which satisfies particular conditions connecting elements to atoms. Next, by using a suitable partial closure operator and the corresponding partial closure system, we define a partial matroid. We prove that the range of a partial matroid is a geometric poset under inclusion, and conversely, that every finite geometric poset is isomorphic to the range of a particular partial matroid. Finally, by introducing a new generalization of semimodularity from lattices to posets, we prove that a poset is geometric if and only if it is atomistic and semimodular.
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Presented by W. DeMeo.
Dedicated to Ralph Freese, Bill Lampe, and J.B. Nation.
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This work of second and third authors was supported by the Serbian Ministry of Education, Science and Technological Development through Faculty of Science, University of Novi Sad, (Grant No. 451-03-68/2020-14/200125) and of third author through Mathematical Institute of the Serbian Academy of Sciences and Arts.
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Šešelja, B., Slivková, A. & Tepavčević, A. On geometric posets and partial matroids. Algebra Univers. 81, 42 (2020). https://doi.org/10.1007/s00012-020-00673-7
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DOI: https://doi.org/10.1007/s00012-020-00673-7