{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T05:20:29Z","timestamp":1776316829047,"version":"3.50.1"},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2022,1,10]],"date-time":"2022-01-10T00:00:00Z","timestamp":1641772800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2022,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>It is well known that classical varieties of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000463_inline1.png\"\/><jats:tex-math>\n$\\Sigma$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-algebras correspond bijectively to finitary monads on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000463_inline2.png\"\/><jats:tex-math>\n$\\mathsf{Set}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We present an analogous result for varieties of ordered <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000463_inline3.png\"\/><jats:tex-math>\n$\\Sigma$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-algebras, that is, categories of algebras presented by inequations between <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000463_inline4.png\"\/><jats:tex-math>\n$\\Sigma$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-terms. We prove that they correspond bijectively to strongly finitary monads on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000463_inline5.png\"\/><jats:tex-math>\n$\\mathsf{Pos}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. That is, those finitary monads which preserve reflexive coinserters. We deduce that strongly finitary monads have a coinserter presentation, analogous to the coequalizer presentation of finitary monads due to Kelly and Power. We also show that these monads are liftings of finitary monads on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000463_inline6.png\"\/><jats:tex-math>\n$\\mathsf{Set}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Finally, varieties presented by equations are proved to correspond to extensions of finitary monads on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000463_inline7.png\"\/><jats:tex-math>\n$\\mathsf{Set}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> to strongly finitary monads on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000463_inline8.png\"\/><jats:tex-math>\n$\\mathsf{Pos}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s0960129521000463","type":"journal-article","created":{"date-parts":[[2022,1,10]],"date-time":"2022-01-10T08:58:49Z","timestamp":1641805129000},"page":"349-373","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":11,"title":["A categorical view of varieties of ordered algebras"],"prefix":"10.1017","volume":"32","author":[{"given":"J.","family":"Ad\u00e1mek","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4373-0471","authenticated-orcid":false,"given":"M.","family":"Dost\u00e1l","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"Velebil","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,1,10]]},"reference":[{"key":"S0960129521000463_ref2","volume-title":"Abstract and Concrete Categories","author":"Ad\u00e1mek","year":"1990"},{"key":"S0960129521000463_ref19","first-page":"1153","article-title":"Quasivarieties and varieties of ordered algebras: Regularity and exactness","volume":"27","author":"Kurz","year":"2017","journal-title":"Logical Methods in Computer Science"},{"key":"S0960129521000463_ref10","volume-title":"Lattices and Ordered Algebraic Structures","author":"Blyth","year":"2005"},{"key":"S0960129521000463_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2019.05.016"},{"key":"S0960129521000463_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/BF00872987"},{"key":"S0960129521000463_ref12","unstructured":"Ford, C. , Milius, S. and Schr\u00f6der, L. (2021). Monads on categories of relational structures, arXiv:2107.03880."},{"key":"S0960129521000463_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01111838"},{"key":"S0960129521000463_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(93)90092-8"},{"key":"S0960129521000463_ref23","volume-title":"Categories for the Working Mathematician","author":"Mac Lane","year":"1988"},{"key":"S0960129521000463_ref24","volume-title":"Lecture Notes Computer Science","volume":"3629","author":"Power","year":"2005"},{"key":"S0960129521000463_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(99)00019-5"},{"key":"S0960129521000463_ref15","unstructured":"Kelly, G. M. (1982). Structures defined by finite limits in the enriched context I, Cahiers de Topologie et G\u00e9om\u00e9trie Diff\u00e9rentielle Cat\u00e9goriques XXIII(1) 3\u201342."},{"key":"S0960129521000463_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0000(76)80030-X"},{"key":"S0960129521000463_ref4","first-page":"251","article-title":"What are sifted colimits?","volume":"23","author":"Ad\u00e1mek","year":"2010","journal-title":"Theory and Applications of Categories"},{"key":"S0960129521000463_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0060485"},{"key":"S0960129521000463_ref13","author":"Golan","year":"2003"},{"key":"S0960129521000463_ref1","doi-asserted-by":"crossref","unstructured":"Ad\u00e1mek, J. , Ford, C. , Milius, S. and Schr\u00f6der, L. (2021). Finitary monads on the category of posets. arXiv:2011.14796, to appear in Math. Struct. Comput. 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Metric Monads, arXiv:2012.14641."},{"key":"S0960129521000463_ref26","first-page":"339","article-title":"Free algebras, input processes and free monads","volume":"16","author":"Trnkov\u00e1","year":"1975","journal-title":"Commentationes Mathematicae Universitatis Carolinae"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129521000463","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T12:26:59Z","timestamp":1671193619000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129521000463\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,10]]},"references-count":26,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,4]]}},"alternative-id":["S0960129521000463"],"URL":"https:\/\/doi.org\/10.1017\/s0960129521000463","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,10]]},"assertion":[{"value":"\u00a9 The Author(s), 2022. 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