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The restriction to the finitary case (FAOMs) allows us to study tope graphs and covector posets, as well as to view FAOMs as oriented finitary semimatroids. We show shellability of FAOMs and single out the FAOMs that are affinely homeomorphic to <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Finally, we study group actions on AOMs, whose quotients in the case of FAOMs are a stepping stone towards a general theory of affine and toric pseudoarrangements. Our results include applications of the multiplicity Tutte polynomial of group actions of semimatroids, generalizing enumerative properties of toric arrangements to a combinatorially defined class of arrangements of submanifolds. This answers partially a question by Ehrenborg and Readdy.<\/jats:p>","DOI":"10.1007\/s00454-024-00651-z","type":"journal-article","created":{"date-parts":[[2024,5,1]],"date-time":"2024-05-01T14:50:56Z","timestamp":1714575056000},"page":"208-257","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Finitary Affine Oriented Matroids"],"prefix":"10.1007","volume":"73","author":[{"given":"Emanuele","family":"Delucchi","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8151-2184","authenticated-orcid":false,"given":"Kolja","family":"Knauer","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,5,1]]},"reference":[{"key":"651_CR1","doi-asserted-by":"publisher","first-page":"1135","DOI":"10.1007\/s10801-015-0620-3","volume":"42","author":"M Aguiar","year":"2015","unstructured":"Aguiar, M., Petersen, T.K.: The Steinberg torus of a Weyl group as a module over the Coxeter complex. 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