{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T17:19:12Z","timestamp":1778692752004,"version":"3.51.4"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2013,12,13]],"date-time":"2013-12-13T00:00:00Z","timestamp":1386892800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formalism based on two basic ingredients, namely an orthogonality relation and matroids which are a very generic algebraic structure permitting to define a notion of dimension. Having obtained what we call orthomatroids, we then show that, in high enough dimension, the basic constituants of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbert lattices, so that their presence is a direct consequence of an orthogonality-based characterization of dimension.<\/jats:p>","DOI":"10.3390\/axioms2040477","type":"journal-article","created":{"date-parts":[[2013,12,16]],"date-time":"2013-12-16T06:18:40Z","timestamp":1387174720000},"page":"477-489","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Orthogonality and Dimensionality"],"prefix":"10.3390","volume":"2","author":[{"given":"Olivier","family":"Brunet","sequence":"first","affiliation":[{"name":"CAPP, Laboratoire d'Informatique de Grenoble, B\u00e2timent IMAG C, 220, rue de la Chimie, 38400 Saint Martin d'H\u00e8res, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,12,13]]},"reference":[{"key":"ref_1","unstructured":"Mackey, G. (1957). The Mathematical Foundations of Quantum Mechanics, Benjamin."},{"key":"ref_2","unstructured":"Beltrametti, E., and Cassinelli, G. (2010). The Logic of Quantum Mechanics\u2014Encyclopedia of Mathematics and its Applications, Cambridge University Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/s10702-005-1129-0","article-title":"Information-theoretic principle entails orthomodularity of a lattice","volume":"18","author":"Grinbaum","year":"2005","journal-title":"Found. Phys. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2147","DOI":"10.1007\/s10773-005-8010-6","article-title":"Representation systems and quantum structures","volume":"44","author":"Brunet","year":"2005","journal-title":"Int. J. Theor. Phys."},{"key":"ref_5","first-page":"439","article-title":"Axiomatique quantique","volume":"37","author":"Piron","year":"1964","journal-title":"Helv. Phys. Acta"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Piron, C. (1976). Foundations of Quantum Physics, Benjamin.","DOI":"10.1007\/978-94-010-1440-3_7"},{"key":"ref_7","unstructured":"Engesser, K., Gabbay, D.M., and Lehmann, D. (2007). Handbook of Quantum Logic and Quantum Structures, Elsevier."},{"key":"ref_8","unstructured":"Zalta, E.N. The Stanford Encyclopedia of Philosophy, [Fall 2012 ed.]. Available online: http:\/\/plato.stanford.edu\/archives\/fall2012\/entries\/qt-quantlog\/."},{"key":"ref_9","unstructured":"Hardy, L. (2001). Quantum theory from five reasonable axioms."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1637","DOI":"10.1007\/BF02302261","article-title":"Relational quantum mechanics","volume":"35","author":"Rovelli","year":"1996","journal-title":"Int. J. Theor. Phys."},{"key":"ref_11","unstructured":"Kung, J.P.S. (1982). A Source Book in Matroid Theory, Birkha\u00fcser."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Oxley, J.G. (2011). Matroid Theory, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198566946.001.0001"},{"key":"ref_13","unstructured":"Welsh, D.J.A. (1976). Matroid Theory, Dover."},{"key":"ref_14","unstructured":"White, N. (1986). Theory of Matroids\u2014Encyclopedia of Mathematics and its Applications, Cambridge University Press."},{"key":"ref_15","unstructured":"Engesser, K., Gabbay, D.M., and Lehmann, D. (2007). Handbook of Quantum Logic and Quantum Structures, Elsevier."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Gabbay, D., and Guenthner, F. (2001). Handbook of Philosophical Logic, Kluwer.","DOI":"10.1007\/978-94-017-0454-0"},{"key":"ref_17","unstructured":"Engesser, K., Gabbay, D.M., and Lehmann, D. (2007). Handbook of Quantum Logic and Quantum Structures, Elsevier."},{"key":"ref_18","unstructured":"Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures, Springer."},{"key":"ref_19","unstructured":"Davey, B.A., and Priestley, H.A. (1990). Introduction to Lattices and Order, Cambridge University Press. Cambridge Mathematical Textbooks."},{"key":"ref_20","unstructured":"Ern\u00e9, M., Koslowski, J., Melton, A., and Strecker, G.E. (1991, January 26\u201329). A Primer on Galois Connections. Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin, Madison, WI, USA."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Gratzer, G. (1978). General Lattice Theory, Academic Press.","DOI":"10.1007\/978-3-0348-7633-9"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.aim.2013.01.011","article-title":"Axioms for infinite matroids","volume":"239","author":"Bruhn","year":"2013","journal-title":"Adv. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"215","DOI":"10.4064\/cm-20-2-215-220","article-title":"Matroids and duality","volume":"20","author":"Higgs","year":"1969","journal-title":"Colloq. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"861","DOI":"10.4153\/CMB-1969-112-6","article-title":"Equicardinality of bases in B-matroids","volume":"12","author":"Higgs","year":"1969","journal-title":"Can. Math. Bull."},{"key":"ref_25","unstructured":"Engesser, K., Gabbay, D.M., and Lehmann, D. (2007). Handbook of Quantum Logic and Quantum Structures, Elsevier."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1080\/00927879508825218","article-title":"Characterization of Hilbert spaces by orthomodular spaces","volume":"23","year":"1995","journal-title":"Commun. Algebr."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2887","DOI":"10.1007\/s10773-007-9400-8","article-title":"An intrinsic topology for orthomodular lattices","volume":"46","author":"Brunet","year":"2007","journal-title":"Int. J. Theor. Phys."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/2\/4\/477\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:51:23Z","timestamp":1760219483000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/2\/4\/477"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,12,13]]},"references-count":27,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,12]]}},"alternative-id":["axioms2040477"],"URL":"https:\/\/doi.org\/10.3390\/axioms2040477","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,12,13]]}}}