{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T13:59:49Z","timestamp":1773237589849,"version":"3.50.1"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,4,19]],"date-time":"2019-04-19T00:00:00Z","timestamp":1555632000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bull. Malays. Math. Sci. Soc."],"published-print":{"date-parts":[[2020,3]]},"DOI":"10.1007\/s40840-019-00762-4","type":"journal-article","created":{"date-parts":[[2019,4,19]],"date-time":"2019-04-19T20:24:50Z","timestamp":1555705490000},"page":"1623-1645","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Ranks of Monoids of Endomorphisms of a Finite Undirected Path"],"prefix":"10.1007","volume":"43","author":[{"given":"I.","family":"Dimitrova","sequence":"first","affiliation":[]},{"given":"V. H.","family":"Fernandes","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1687-4656","authenticated-orcid":false,"given":"J.","family":"Koppitz","sequence":"additional","affiliation":[]},{"given":"T. M.","family":"Quinteiro","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,4,19]]},"reference":[{"key":"762_CR1","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1080\/00927872.2014.984838","volume":"44","author":"F Al-Kharousi","year":"2016","unstructured":"Al-Kharousi, F., Kehinde, R., Umar, A.: On the semigroup of partial isometries of a finite chain. Commun. Algebra 44, 639\u2013647 (2016)","journal-title":"Commun. Algebra"},{"key":"762_CR2","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1017\/S0305004115000389","volume":"159","author":"J Ara\u00fajo","year":"2015","unstructured":"Ara\u00fajo, J., Bentz, W., Mitchell, J.D., Schneider, C.: The rank of the semigroup of transformations stabilising a partition of a finite set. Math. Proc. Camb. Philos. Soc. 159, 339\u2013353 (2015)","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"762_CR3","doi-asserted-by":"publisher","first-page":"498","DOI":"10.1007\/s00233-008-9122-0","volume":"78","author":"J Ara\u00fajo","year":"2009","unstructured":"Ara\u00fajo, J., Schneider, C.: The rank of the endomorphism monoid of a uniform partition. Semigroup Forum 78, 498\u2013510 (2009)","journal-title":"Semigroup Forum"},{"key":"762_CR4","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/j.disc.2007.12.049","volume":"309","author":"S Arworn","year":"2009","unstructured":"Arworn, S.: An algorithm for the numbers of endomorphisms on paths. Discrete Math. 309, 94\u2013103 (2009)","journal-title":"Discrete Math."},{"key":"762_CR5","doi-asserted-by":"publisher","first-page":"2525","DOI":"10.1016\/j.disc.2007.06.007","volume":"308","author":"S Arworn","year":"2008","unstructured":"Arworn, S., Knauer, U., Leeratanavalee, S.: Locally strong endomorphisms of paths. Discrete Math. 308, 2525\u20132532 (2008)","journal-title":"Discrete Math."},{"key":"762_CR6","doi-asserted-by":"publisher","first-page":"532","DOI":"10.1007\/s00233-014-9629-5","volume":"90","author":"S Cical\u00f2","year":"2015","unstructured":"Cical\u00f2, S., Fernandes, V.H., Schneider, C.: Partial transformation monoids preserving a uniform partition. Semigroup Forum 90, 532\u2013544 (2015)","journal-title":"Semigroup Forum"},{"key":"762_CR7","unstructured":"Fan, S.: On end-regular bipartite graphs, combinatorics and graph theory. In: Proceedings of SSICC\u2019s92, pp. 117\u2013130. World Scientific, Singapore (1993)"},{"key":"762_CR8","first-page":"419","volume":"40","author":"S Fan","year":"1997","unstructured":"Fan, S.: The regularity of the endomorphism monoid of a split graph. Acta Math. Sin. 40, 419\u2013422 (1997)","journal-title":"Acta Math. Sin."},{"key":"762_CR9","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/S0012-365X(02)00388-6","volume":"257","author":"S Fan","year":"2002","unstructured":"Fan, S.: Retractions of split graphs and End-orthodox split graphs. Discrete Math. 257, 161\u2013164 (2002)","journal-title":"Discrete Math."},{"key":"762_CR10","unstructured":"Fernandes, V.H.: Presentations for some monoids of partial transformations on a finite chain: a survey. In: Gomes, G.M.S., Pin, J.-\u00c9., Silva, P.V. (eds.) Semigroups, Algorithms, Automata and Languages, pp. 363\u2013378. World Scientific Publishing, River Edge, NJ (2002). https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/9789812776884_0015"},{"key":"762_CR11","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1007\/s00233-013-9548-x","volume":"89","author":"VH Fernandes","year":"2014","unstructured":"Fernandes, V.H., Honyam, P., Quinteiro, T.M., Singha, B.: On semigroups of endomorphisms of a chain with restricted range. Semigroup Forum 89, 77\u2013104 (2014)","journal-title":"Semigroup Forum"},{"key":"762_CR12","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1080\/00927872.2014.975345","volume":"44","author":"VH Fernandes","year":"2016","unstructured":"Fernandes, V.H., Honyam, P., Quinteiro, T.M., Singha, B.: On semigroups of orientation-preserving transformations with restricted range. Commun. Algebra 44, 253\u2013264 (2016)","journal-title":"Commun. Algebra"},{"key":"762_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/s40840-017-0598-1","author":"VH Fernandes","year":"2018","unstructured":"Fernandes, V.H., Koppitz, J., Musunthia, T.: The rank of the semigroup of all order-preserving transformations on a finite fence. Bull. Malays. Math. Sci. Soc. (2018). https:\/\/doi.org\/10.1007\/s40840-017-0598-1 . (in Press)","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"762_CR14","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1080\/00927872.2012.720322","volume":"42","author":"VH Fernandes","year":"2014","unstructured":"Fernandes, V.H., Quinteiro, T.M.: On the ranks of certain monoids of transformations that preserve a uniform partition. Commun. Algebra 42, 615\u2013636 (2014)","journal-title":"Commun. Algebra"},{"key":"762_CR15","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1142\/S1005386714000431","volume":"21","author":"VH Fernandes","year":"2014","unstructured":"Fernandes, V.H., Sanwong, J.: On the rank of semigroups of transformations on a finite set with restricted range. Algebra Colloq. 21, 497\u2013510 (2014)","journal-title":"Algebra Colloq."},{"key":"762_CR16","first-page":"79","volume":"127","author":"H Hou","year":"2016","unstructured":"Hou, H., Gu, R.: Split graphs whose completely regular endomorphisms form a monoid. Ars Comb. 127, 79\u201388 (2016)","journal-title":"Ars Comb."},{"key":"762_CR17","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1080\/00927872.2012.724746","volume":"42","author":"H Hou","year":"2014","unstructured":"Hou, H., Gu, R., Shang, Y.: The join of split graphs whose regular endomorphisms form a monoid. Commun. Algebra 42, 795\u2013802 (2014)","journal-title":"Commun. Algebra"},{"key":"762_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S000497271400046X","volume":"91","author":"H Hou","year":"2015","unstructured":"Hou, H., Gu, R., Shang, Y.: The join of split graphs whose quasi-strong endomorphisms form a monoid. Bull. Aust. Math. Soc. 91, 1\u201310 (2015)","journal-title":"Bull. Aust. Math. Soc."},{"key":"762_CR19","doi-asserted-by":"publisher","first-page":"1173","DOI":"10.1016\/j.ejc.2007.06.026","volume":"29","author":"H Hou","year":"2008","unstructured":"Hou, H., Luo, Y., Cheng, Z.: The endomorphism monoid of $${\\bar{P}}_{n}$$ P \u00af n . Eur. J. Comb. 29, 1173\u20131185 (2008)","journal-title":"Eur. J. Comb."},{"key":"762_CR20","first-page":"305","volume":"105","author":"H Hou","year":"2012","unstructured":"Hou, H., Luo, Y., Fan, S.: End-regular and end-orthodox joins of split graphs. Ars Comb. 105, 305\u2013318 (2012)","journal-title":"Ars Comb."},{"key":"762_CR21","first-page":"833","volume":"15","author":"H Hou","year":"2017","unstructured":"Hou, H., Song, Y., Gu, R.: The join of split graphs whose completely regular endomorphisms form a monoid. De Gruyter Open 15, 833\u2013839 (2017)","journal-title":"De Gruyter Open"},{"key":"762_CR22","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198511946.001.0001","volume-title":"Fundamentals of Semigroup Theory","author":"JM Howie","year":"1995","unstructured":"Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)"},{"key":"762_CR23","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s00233-004-0135-z","volume":"70","author":"P Huisheng","year":"2005","unstructured":"Huisheng, P.: On the rank of the semigroup $$T_\\rho (X)$$ T \u03c1 ( X ) . Semigroup Forum 70, 107\u2013117 (2005)","journal-title":"Semigroup Forum"},{"key":"762_CR24","doi-asserted-by":"publisher","DOI":"10.1515\/9783110255096","volume-title":"Algebraic Graph Theory: Morphisms, Monoids, and Matrices","author":"U Knauer","year":"2011","unstructured":"Knauer, U.: Algebraic Graph Theory: Morphisms, Monoids, and Matrices. De Gruyter, Berlin (2011)"},{"key":"762_CR25","first-page":"357","volume":"115","author":"U Knauer","year":"2014","unstructured":"Knauer, U., Wanichsombat, A.: Completely regular endomorphisms of split graphs. Ars Comb. 115, 357\u2013366 (2014)","journal-title":"Ars Comb."},{"key":"762_CR26","first-page":"253","volume":"26","author":"W Li","year":"2006","unstructured":"Li, W.: Split graphs with completely regular endomorphism monoids. J. Math. Res. Expos. 26, 253\u2013263 (2006)","journal-title":"J. Math. Res. Expos."},{"key":"762_CR27","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1006\/eujc.2000.0443","volume":"22","author":"W Li","year":"2001","unstructured":"Li, W., Chen, J.: Endomorphism\u2014regularity of split graphs. Eur. J. Comb. 22, 207\u2013216 (2001)","journal-title":"Eur. J. Comb."},{"key":"762_CR28","first-page":"357","volume":"11","author":"D Lu","year":"2013","unstructured":"Lu, D., Wu, T.: Endomorphism monoids of generalized split graphs. Ars Comb. 11, 357\u2013373 (2013)","journal-title":"Ars Comb."},{"issue":"1988","key":"762_CR29","first-page":"367","volume":"37","author":"L Marki","year":"1987","unstructured":"Marki, L.: Problem raised at the problem session of the Colloquium on Semigroups in Szeged. Semigroup Forum 37(1988), 367\u2013373 (1987)","journal-title":"Semigroup Forum"},{"key":"762_CR30","doi-asserted-by":"publisher","first-page":"5352","DOI":"10.1016\/j.disc.2008.11.022","volume":"309","author":"MA Michels","year":"2009","unstructured":"Michels, M.A., Knauer, U.: The congruence classes of paths and cycles. Discrete Math. 309, 5352\u20135359 (2009)","journal-title":"Discrete Math."},{"key":"762_CR31","first-page":"284","volume":"22\u20132","author":"N Pipattanajinda","year":"2016","unstructured":"Pipattanajinda, N., Knauer, U., Gyurov, B., Panma, S.: The endomorphism monoids of $$(n-3)$$ ( n - 3 ) -regular graphs of order $$n$$ n . Algebra Discrete Math. 22\u20132, 284\u2013300 (2016)","journal-title":"Algebra Discrete Math."},{"key":"762_CR32","doi-asserted-by":"publisher","first-page":"344","DOI":"10.1007\/BF01207836","volume":"66","author":"E Wilkeit","year":"1996","unstructured":"Wilkeit, E.: Graphs with a regular endomorphism monoid. Arch. Math. 66, 344\u2013352 (1996)","journal-title":"Arch. Math."},{"key":"762_CR33","doi-asserted-by":"publisher","first-page":"4195","DOI":"10.1080\/00927872.2010.521933","volume":"39","author":"P Zhao","year":"2011","unstructured":"Zhao, P.: On the ranks of certain semigroups of orientation preserving transformations. Commun. Algebra 39, 4195\u20134205 (2011)","journal-title":"Commun. Algebra"},{"key":"762_CR34","doi-asserted-by":"publisher","first-page":"674","DOI":"10.1080\/00927872.2013.847946","volume":"43","author":"P Zhao","year":"2015","unstructured":"Zhao, P., Fernandes, V.H.: The ranks of ideals in various transformation monoids. Commun. Algebra 43, 674\u2013692 (2015)","journal-title":"Commun. Algebra"}],"container-title":["Bulletin of the Malaysian Mathematical Sciences Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40840-019-00762-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40840-019-00762-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40840-019-00762-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,17]],"date-time":"2024-07-17T03:28:37Z","timestamp":1721186917000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40840-019-00762-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,19]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["762"],"URL":"https:\/\/doi.org\/10.1007\/s40840-019-00762-4","relation":{},"ISSN":["0126-6705","2180-4206"],"issn-type":[{"value":"0126-6705","type":"print"},{"value":"2180-4206","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,4,19]]},"assertion":[{"value":"24 October 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 April 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}