## On geometric posets and partial matroids.(English)Zbl 1484.06016

Summary: 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.

### MSC:

 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06A06 Partial orders, general
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### References:

 [1] Aigner, M., Combinatorial Theory (1979), Berlin: Springer-Verlag, Berlin [2] Birkhoff, G., Lattice Theory (1948), Soc: Amer. Math, Soc [3] Caspard, N.; Leclerc, B.; Monjardet, B., Finite ordered sets: concepts, results and uses (2012), Cambridge: Cambridge University Press, Cambridge · Zbl 1238.06001 [4] Caspard, N.; Monjardet, B., The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey, Discret. Appl. Math., 127, 241-269 (2003) · Zbl 1026.06008 [5] Caspard, N.; Monjardet, B., The lattices of Moore families and closure operators on a finite set: A survey, Electron. Notes in Discret. Math., 2, 25-50 (1999) [6] Erné, M., Compact generation in partially ordered sets, J. Austral. Math. Soc. (Series A), 42, 69-83 (1987) · Zbl 0614.06004 [7] Erné, M.; Rosenberg, IG; Sabidussi, G., Algebraic ordered sets and their generalizations, Algebras and Orders, 113-192 (1993), Dordrecht: Springer, Dordrecht · Zbl 0791.06007 [8] Grätzer, G., Lattice Theory: Foundation (2011), Basel: Springer, Basel · Zbl 1233.06001 [9] Maeda, F.; Maeda, S., Theory of Symmetric Lattices (1970), Berlin: Springer-Verlag, Berlin · Zbl 0219.06002 [10] Ore, O., Chains in partially ordered sets, Bull. Am. Math. Soc., 49, 558-566 (1943) · Zbl 0060.06104 [11] Oxley, JG, Matroid Theory (1992), New York: Oxford University Press, New York [12] Pfaltz, JL; Jamison, RE, Closure systems and their structure, Inf. Sci., 139, 275-286 (2001) · Zbl 0993.06004 [13] Ronse, C., Closures on partial partitions from closures on sets, Math. Slovaca, 63, 959-978 (2013) · Zbl 1324.06004 [14] Šešelja, B.; Slivková, A.; Tepavčević, A., Sharp partial closure operator, Miskolc Math. Notes, 19, 569-579 (2018) [15] Šešelja, B.; Tepavčević, A., Posets via partial closure operators, Contributions to General Algebra, 12, 371-375 (2000) · Zbl 0965.06003 [16] Slivková, A.: Partial closure operators and applications in ordered set theory. PhD thesis, University of Novi Sad (2018) [17] Stern, M.: Semimodular Lattices. Vieweg+Teubner Verlag (2013) · Zbl 0654.06006 [18] White, N., Theory of matroids (1986), Cambridge: Cambridge University Press, Cambridge
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