{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:45:16Z","timestamp":1740109516662,"version":"3.37.3"},"reference-count":7,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,10,1]],"date-time":"2023-10-01T00:00:00Z","timestamp":1696118400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T00:00:00Z","timestamp":1696982400000},"content-version":"vor","delay-in-days":10,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006752","name":"Universidade do Porto","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006752","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Semigroup Forum"],"published-print":{"date-parts":[[2023,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we study the regular semigroups weakly generated by a single element <jats:italic>x<\/jats:italic>, that is, with no proper regular subsemigroup containing <jats:italic>x<\/jats:italic>. We show there exists a regular semigroup <jats:inline-formula><jats:alternatives><jats:tex-math>$$F_{1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> weakly generated by <jats:italic>x<\/jats:italic> such that all other regular semigroups weakly generated by <jats:italic>x<\/jats:italic> are homomorphic images of <jats:inline-formula><jats:alternatives><jats:tex-math>$$F_{1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We define <jats:inline-formula><jats:alternatives><jats:tex-math>$$F_{1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> using a presentation where both sets of generators and relations are infinite. Nevertheless, the word problem for this presentation is decidable. We describe a canonical form for the congruence classes given by this presentation, and explain how to obtain it. We end the paper studying the structure of <jats:inline-formula><jats:alternatives><jats:tex-math>$$F_{1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In particular, we show that the \u2018free regular semigroup <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\textrm{FI}}_2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mtext>FI<\/mml:mtext>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> weakly generated by two idempotents\u2019 is isomorphic to a regular subsemigroup of <jats:inline-formula><jats:alternatives><jats:tex-math>$$F_{1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> weakly generated by <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{xx',x'x\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:msup>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00233-023-10389-4","type":"journal-article","created":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T16:01:50Z","timestamp":1697040110000},"page":"525-563","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Regular semigroups weakly generated by one element"],"prefix":"10.1007","volume":"107","author":[{"given":"Lu\u00eds","family":"Oliveira","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,10,11]]},"reference":[{"key":"10389_CR1","unstructured":"The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.10.0, (2018), http:\/\/www.gap-system.org"},{"key":"10389_CR2","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1017\/S000497270000349X","volume":"40","author":"T Hall","year":"1989","unstructured":"Hall, T.: Identities for existence varieties of regular semigroups. Bull. Austral. Math. Soc. 40, 59\u201377 (1989)","journal-title":"Bull. Austral. Math. Soc."},{"key":"10389_CR3","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/BF02573274","volume":"40","author":"J Kad\u2019ourek","year":"1990","unstructured":"Kad\u2019ourek, J., Szendrei, M.B.: A new approach in the theory of orthodox semigroups. Semigroup Forum 40, 257\u2013296 (1990)","journal-title":"Semigroup Forum"},{"key":"10389_CR4","unstructured":"Mitchell, J.\u00a0D.: Semigroups - GAP package, Version 3.0.20, (2018), http:\/\/www.gap-system.org"},{"key":"10389_CR5","doi-asserted-by":"publisher","first-page":"223","DOI":"10.2307\/1968867","volume":"43","author":"MHA Newman","year":"1942","unstructured":"Newman, M.H.A.: On theories with a combinatorial definition of equivalence. Ann. Math. 43, 223\u2013243 (1942)","journal-title":"Ann. Math."},{"key":"10389_CR6","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1142\/S0218196723500388","volume":"33","author":"L Oliveira","year":"2023","unstructured":"Oliveira, L.: Regular semigroups weakly generated by idempotents. Internat. J. Algebra Comput. 33, 851\u2013891 (2023)","journal-title":"Internat. J. Algebra Comput."},{"key":"10389_CR7","first-page":"471","volume":"164","author":"YT Yeh","year":"1992","unstructured":"Yeh, Y.T.: The existence of e-free objects in e-varieties of regular semigroups. J. Algebra 164, 471\u2013484 (1992)","journal-title":"J. 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