{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:38Z","timestamp":1747198058170,"version":"3.40.5"},"reference-count":18,"publisher":"Walter de Gruyter GmbH","issue":"0","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Given a regular language <jats:italic>L<\/jats:italic>, we effectively construct a\nunary semigroup that recognizes the topological closure of <jats:italic>L<\/jats:italic> in the\nfree unary semigroup relative to the variety of unary semigroups\ngenerated by the pseudovariety <jats:inline-formula id=\"j_forum-2019-0158_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>\ud835\uddb1<\/m:mi>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forum-2019-0158_eq_0365.png\"\/>\n                        <jats:tex-math>{\\mathsf{R}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> of all finite <jats:inline-formula id=\"j_forum-2019-0158_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi mathvariant=\"script\">\u211b<\/m:mi>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forum-2019-0158_eq_0327.png\"\/>\n                        <jats:tex-math>{\\mathcal{R}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-trivial\nsemigroups. In particular, we obtain a new effective solution of the\nseparation problem of regular languages by <jats:inline-formula id=\"j_forum-2019-0158_ineq_9997\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>\ud835\uddb1<\/m:mi>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forum-2019-0158_eq_0365.png\"\/>\n                        <jats:tex-math>{\\mathsf{R}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-languages.<\/jats:p>","DOI":"10.1515\/forum-2019-0158","type":"journal-article","created":{"date-parts":[[2022,7,22]],"date-time":"2022-07-22T13:41:56Z","timestamp":1658497316000},"source":"Crossref","is-referenced-by-count":0,"title":["Recognizing pro-\ud835\uddb1 closures of regular languages"],"prefix":"10.1515","volume":"0","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3829-2382","authenticated-orcid":false,"given":"Jorge","family":"Almeida","sequence":"first","affiliation":[{"name":"CMUP , Departamento de Matem\u00e1tica , Faculdade de Ci\u00eancias , Universidade do Porto , Rua do Campo Alegre s\/n, 4169-007 Porto , Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8733-9474","authenticated-orcid":false,"given":"Jos\u00e9 Carlos","family":"Costa","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica , Universidade do Minho , Campus de Gualtar, 4700-320 Braga , Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4101-8437","authenticated-orcid":false,"given":"Marc","family":"Zeitoun","sequence":"additional","affiliation":[{"name":"Universit\u00e9s de Bordeaux , CNRS, Bordeaux INP, LaBRI, UMR 5800, 33400 Talence Cedex , France"}]}],"member":"374","published-online":{"date-parts":[[2022,7,23]]},"reference":[{"key":"2022072213415334322_j_forum-2019-0158_ref_001","doi-asserted-by":"crossref","unstructured":"D.  Albert, R.  Baldinger and J.  Rhodes,\nUndecidability of the identity problem for finite semigroups,\nJ. Symb. Log. 57 (1992), no. 1, 179\u2013192.","DOI":"10.2307\/2275184"},{"key":"2022072213415334322_j_forum-2019-0158_ref_002","doi-asserted-by":"crossref","unstructured":"J.  Almeida,\nFinite Semigroups and Universal Algebra,\nSer. Algebra 3,\nWorld Scientific, River Edge, 1995.","DOI":"10.1142\/2481"},{"key":"2022072213415334322_j_forum-2019-0158_ref_003","unstructured":"J.  Almeida,\nSome algorithmic problems for pseudovarieties,\nPubl. Math. Debrecen 54 (1999), 531\u2013552."},{"key":"2022072213415334322_j_forum-2019-0158_ref_004","doi-asserted-by":"crossref","unstructured":"J.  Almeida,\nProfinite semigroups and applications,\nStructural Theory of Automata, Semigroups, and Universal Algebra,\nNATO Sci. Ser. II Math. Phys. Chem. 207,\nSpringer, Dordrecht (2005), 1\u201345.","DOI":"10.1007\/1-4020-3817-8_1"},{"key":"2022072213415334322_j_forum-2019-0158_ref_005","doi-asserted-by":"crossref","unstructured":"J.  Almeida and A.  Costa,\nProfinite topologies,\nHandbook of Automata Theory. Vol. I. Theoretical Foundations,\nEuropean Mathematical Society, Berlin (2021), 615\u2013652.","DOI":"10.4171\/Automata-1\/17"},{"key":"2022072213415334322_j_forum-2019-0158_ref_006","doi-asserted-by":"crossref","unstructured":"J.  Almeida, J. C.  Costa and M.  Zeitoun,\nTameness of pseudovariety joins involving R,\nMonatsh. Math. 146 (2005), no. 2, 89\u2013111.","DOI":"10.1007\/s00605-005-0324-1"},{"key":"2022072213415334322_j_forum-2019-0158_ref_007","doi-asserted-by":"crossref","unstructured":"J.  Almeida, J. C.  Costa and M.  Zeitoun,\nComplete reducibility of systems of equations with respect to \n                  \n                     \n                        \ud835\uddb1\n                     \n                     \n                     \\mathsf{R}\n                  \n               ,\nPort. Math. (N.\u2009S.) 64 (2007), no. 4, 445\u2013508.","DOI":"10.4171\/PM\/1792"},{"key":"2022072213415334322_j_forum-2019-0158_ref_008","doi-asserted-by":"crossref","unstructured":"J.  Almeida, J. C.  Costa and M.  Zeitoun,\nPointlike sets with respect to \n                  \n                     \n                        \ud835\udc11\n                     \n                     \n                     {\\mathbf{R}}\n                  \n                and \n                  \n                     \n                        \ud835\udc09\n                     \n                     \n                     {\\mathbf{J}}\n                  \n               ,\nJ. Pure Appl. Algebra 212 (2008), no. 3, 486\u2013499.","DOI":"10.1016\/j.jpaa.2007.06.007"},{"key":"2022072213415334322_j_forum-2019-0158_ref_009","doi-asserted-by":"crossref","unstructured":"J.  Almeida, J. C.  Costa and M.  Zeitoun,\nClosures of regular languages for profinite topologies,\nSemigroup Forum 89 (2014), no. 1, 20\u201340.","DOI":"10.1007\/s00233-014-9574-3"},{"key":"2022072213415334322_j_forum-2019-0158_ref_010","doi-asserted-by":"crossref","unstructured":"J.  Almeida and B.  Steinberg,\nOn the decidability of iterated semidirect products with applications to complexity,\nProc. Lond. Math. Soc. (3) 80 (2000), no. 1, 50\u201374.","DOI":"10.1112\/S0024611500012144"},{"key":"2022072213415334322_j_forum-2019-0158_ref_011","doi-asserted-by":"crossref","unstructured":"J.  Almeida and P.  Weil,\nFree profinite \n                  \n                     \n                        \u211c\n                     \n                     \n                     \\mathfrak{R}\n                  \n               -trivial monoids,\nInternat. J. Algebra Comput. 7 (1997), no. 5, 625\u2013671.","DOI":"10.1142\/S0218196797000289"},{"key":"2022072213415334322_j_forum-2019-0158_ref_012","doi-asserted-by":"crossref","unstructured":"J.  Almeida and M.  Zeitoun,\nAn automata-theoretic approach to the word problem for \u03c9-terms over \n                  \n                     \n                        \ud835\uddb1\n                     \n                     \n                     \\mathsf{R}\n                  \n               ,\nTheoret. Comput. Sci. 370 (2007), no. 1\u20133, 131\u2013169.","DOI":"10.1016\/j.tcs.2006.10.019"},{"key":"2022072213415334322_j_forum-2019-0158_ref_013","doi-asserted-by":"crossref","unstructured":"C. J.  Ash,\nInevitable graphs: A proof of the type \n                  \n                     \n                        II\n                     \n                     \n                     {\\rm II}\n                  \n                conjecture and some related decision procedures,\nInternat. J. Algebra Comput. 1 (1991), no. 1, 127\u2013146.","DOI":"10.1142\/S0218196791000079"},{"key":"2022072213415334322_j_forum-2019-0158_ref_014","doi-asserted-by":"crossref","unstructured":"K.  Auinger and B.  Steinberg,\nOn the extension problem for partial permutations,\nProc. Amer. Math. Soc. 131 (2003), no. 9, 2693\u20132703.","DOI":"10.1090\/S0002-9939-03-06860-6"},{"key":"2022072213415334322_j_forum-2019-0158_ref_015","doi-asserted-by":"crossref","unstructured":"T.  Coulbois and A.  Khelif,\nEquations in free groups are not finitely approximable,\nProc. Amer. Math. Soc. 127 (1999), no. 4, 963\u2013965.","DOI":"10.1090\/S0002-9939-99-04747-4"},{"key":"2022072213415334322_j_forum-2019-0158_ref_016","unstructured":"S.  Eilenberg,\nAutomata, Languages, and Machines. Vol. B,\nPure Appl. Math. 58,\nAcademic Press, New York, 1976."},{"key":"2022072213415334322_j_forum-2019-0158_ref_017","doi-asserted-by":"crossref","unstructured":"M.  Kufleitner and J. P.  W\u00e4chter,\nThe word problem for omega-terms over the Trotter-Weil hierarchy,\nTheory Comput. Syst. 62 (2018), no. 3, 682\u2013738.","DOI":"10.1007\/s00224-017-9763-z"},{"key":"2022072213415334322_j_forum-2019-0158_ref_018","doi-asserted-by":"crossref","unstructured":"T.  Place and M.  Zeitoun,\nSeparating regular languages with first-order logic,\nLog. Methods Comput. Sci. 12 (2016), no. 1, Paper No. 5.","DOI":"10.2168\/LMCS-12(1:5)2016"}],"container-title":["Forum Mathematicum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/forum-2019-0158\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/forum-2019-0158\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,22]],"date-time":"2022-07-22T13:42:06Z","timestamp":1658497326000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/forum-2019-0158\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,23]]},"references-count":18,"journal-issue":{"issue":"0","published-online":{"date-parts":[[2022,7,23]]}},"alternative-id":["10.1515\/forum-2019-0158"],"URL":"https:\/\/doi.org\/10.1515\/forum-2019-0158","relation":{},"ISSN":["0933-7741","1435-5337"],"issn-type":[{"type":"print","value":"0933-7741"},{"type":"electronic","value":"1435-5337"}],"subject":[],"published":{"date-parts":[[2022,7,23]]}}}