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Unexpectedly, these groups (except for some special cases) are either trivial, cyclic or dihedral. In this context, we define and study monoids that generalize both monoids of order-preserving mappings and monoids of orientation-preserving mappings.<\/jats:p>","DOI":"10.1142\/s0218196725500407","type":"journal-article","created":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T03:02:24Z","timestamp":1759287744000},"page":"1-15","source":"Crossref","is-referenced-by-count":0,"title":["Groups of permutations that are even on subsets of a fixed size, and related monoids"],"prefix":"10.1142","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1057-4975","authenticated-orcid":false,"given":"V\u00edtor","family":"Fernandes","sequence":"first","affiliation":[{"name":"Center for Mathematics and Applications (NOVA Math) and Department of Mathematics, Faculdade de Ci\u00eanciase Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, Caparica 2829-516, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0179-9099","authenticated-orcid":false,"given":"Alexei","family":"Vernitski","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Actuarial Science University of Essex, Colchester, UK"}]}],"member":"219","published-online":{"date-parts":[[2025,10,16]]},"reference":[{"key":"S0218196725500407BIB001","first-page":"161","volume":"3","author":"Aizenstat A. 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