{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T18:05:31Z","timestamp":1773338731815,"version":"3.50.1"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stud Logica"],"published-print":{"date-parts":[[2022,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we extend to various classes of subreducts of hoops some results about splitting algebras. In particular we prove that every finite chain in the purely implicational fragment of basic hoops is splitting and that every finite chain in the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{\\wedge ,\\rightarrow \\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mo>\u2227<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> fragment of hoops is splitting. We also produce explicitly the splitting equations in most cases.<\/jats:p>","DOI":"10.1007\/s11225-022-09997-y","type":"journal-article","created":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T12:11:12Z","timestamp":1651752672000},"page":"1155-1187","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Splittings in Subreducts of Hoops"],"prefix":"10.1007","volume":"110","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2860-6157","authenticated-orcid":false,"given":"Paolo","family":"Aglian\u00f2","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,5]]},"reference":[{"key":"9997_CR1","first-page":"397","volume":"64","author":"P Aglian\u00f2","year":"1998","unstructured":"Aglian\u00f2, P., Ternary deduction terms in residuated structures, Acta Scientiarum Mathematicarum 64:397\u2013429, 1998.","journal-title":"Acta Scientiarum Mathematicarum"},{"key":"9997_CR2","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/s00500-016-2367-8","volume":"21","author":"P Aglian\u00f2","year":"2017","unstructured":"Aglian\u00f2, P., Varieties of BL-algebras I, revisited, Soft Computing 21:153\u2013163, 2017.","journal-title":"Soft Computing"},{"key":"9997_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.fss.2019.06.015","volume":"373","author":"P Aglian\u00f2","year":"2019","unstructured":"Aglian\u00f2, P., Splittings in GBL-algebras I: the general case, Fuzzy Sets and Systems 373:1\u201318, 2019.","journal-title":"Fuzzy Sets and Systems"},{"key":"9997_CR4","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/j.fss.2018.07.003","volume":"373","author":"P Aglian\u00f2","year":"2019","unstructured":"Aglian\u00f2, P., Splittings in GBL-algebras II: the representable case, Fuzzy Sets and Systems 373:19\u201336, 2019.","journal-title":"Fuzzy Sets and Systems"},{"key":"9997_CR5","doi-asserted-by":"publisher","first-page":"1235","DOI":"10.1007\/s11225-018-9836-2","volume":"107","author":"P Aglian\u00f2","year":"2019","unstructured":"Aglian\u00f2, P., Varieties of BL-algebras III: splitting algebras, Studia Logica 107:1235\u20131259, 2019.","journal-title":"Studia Logica"},{"issue":"1","key":"9997_CR6","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s11225-007-9078-1","volume":"87","author":"P Aglian\u00f2","year":"2007","unstructured":"Aglian\u00f2, P., I.M.A. Ferreirim, and F. Montagna, Basic hoops: an algebraic study of continuous t-norms, Studia Logica 87(1):73\u201393, 2007.","journal-title":"Studia Logica"},{"key":"9997_CR7","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/S0022-4049(02)00329-8","volume":"181","author":"P Aglian\u00f2","year":"2003","unstructured":"Aglian\u00f2, P., and F. Montagna, Varieties of BL-algebras I: general properties, The Journal of Pure and Applied Algebra 181:105\u2013129, 2003.","journal-title":"The Journal of Pure and Applied Algebra"},{"key":"9997_CR8","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1007\/s000120050156","volume":"43","author":"WJ Blok","year":"2000","unstructured":"Blok, W.J., and I.M.A. Ferreirim, On the structure of hoops, Algebra Universalis 43:233\u2013257, 2000.","journal-title":"Algebra Universalis"},{"key":"9997_CR9","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1007\/BF01190766","volume":"33","author":"WJ Blok","year":"1995","unstructured":"Blok, W.J., and J.G. Raftery, On the quasivariety of BCK-algebras and its subvarieties, Algebra Universalis 33:68\u201390, 1995.","journal-title":"Algebra Universalis"},{"key":"9997_CR10","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1006\/jabr.1996.6834","volume":"190","author":"WJ Blok","year":"1997","unstructured":"Blok, W.J., and J.G. Raftery, Varieties of commutative residuated integral pomonoids and their residuation subreducts, Journal of Algebra 190:280\u2013328, 1997.","journal-title":"Journal of Algebra"},{"key":"9997_CR11","doi-asserted-by":"publisher","first-page":"589","DOI":"10.1142\/S0218196708004627","volume":"18","author":"WJ Blok","year":"2008","unstructured":"Blok, W.J., and J.G. Raftery, Assertionally equivalent quasivarieties, International Journal of Algebra and Computation 18:589\u2013681, 2008.","journal-title":"International Journal of Algebra and Computation"},{"issue":"4","key":"9997_CR12","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1142\/S0218196703001511","volume":"13","author":"K Blount","year":"2003","unstructured":"Blount, K., and C. Tsinakis, The structure of residuated lattices, International Journal of Algebra and Computation 13(4):437\u2013461, 2003.","journal-title":"International Journal of Algebra and Computation"},{"key":"9997_CR13","doi-asserted-by":"crossref","unstructured":"Burris, S., and H.P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, Springer, Berlin, 1981.","DOI":"10.1007\/978-1-4613-8130-3"},{"key":"9997_CR14","first-page":"309","volume":"29","author":"WH Cornish","year":"1984","unstructured":"Cornish, W.H., T. Sturm, and T. Traczyk, Embedding of commutative BCK-algebras into distributive lattice BCK-algebras, Mathematica Japonica 29:309\u2013320, 1984.","journal-title":"Mathematica Japonica"},{"key":"9997_CR15","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1007\/BF02944983","volume":"1","author":"A Day","year":"1971","unstructured":"Day, A., A note on the congruence extension property, Algebra Universalis 1:234\u2013235, 1971.","journal-title":"Algebra Universalis"},{"key":"9997_CR16","unstructured":"Diego, A., Sur les alg\u00e8bres de Hilbert, Collection Logique Mathematique, Series A, no.\u00a021, Gauthiers-Villars, Paris, 1966."},{"key":"9997_CR17","first-page":"119","volume":"53","author":"IMA Ferreirim","year":"2001","unstructured":"Ferreirim, I.M.A., On a conjecture by Andrzej Wro\u0144ski for BCK-algebras and subreducts of hoops, Scientiae Mathematicae Japonicae 53:119\u2013132, 2009.","journal-title":"Scientiae Mathematicae Japonicae"},{"key":"9997_CR18","first-page":"839","volume":"29","author":"PM Idziak","year":"1984","unstructured":"Idziak, P.M., Lattice operations on BCK-algebras, Mathematica Japonica 29:839\u2013846, 1984.","journal-title":"Mathematica Japonica"},{"key":"9997_CR19","first-page":"26","volume":"41","author":"K Is\u00e9ki","year":"1962","unstructured":"Is\u00e9ki, K., An algebra related with a propositional calculus, Proceedings of the Japan Academy 41:26\u201329, 1962.","journal-title":"Proceedings of the Japan Academy"},{"key":"9997_CR20","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1017\/S0027763000018249","volume":"72","author":"Y Komori","year":"1978","unstructured":"Komori, Y., Super-\u0141ukasiewicz implicational logics, Nagoya Mathematical Journal 72: 127\u2013133, 1978.","journal-title":"Nagoya Mathematical Journal"},{"key":"9997_CR21","first-page":"87","volume":"29","author":"T Kowalski","year":"1995","unstructured":"Kowalski, T., The bottom of the lattice of BCK-varieties, Reports on Mathematical Logic 29: 87\u201393, 1995.","journal-title":"Reports on Mathematical Logic"},{"key":"9997_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0002-9947-1972-0313141-1","volume":"174","author":"R McKenzie","year":"1972","unstructured":"McKenzie, R., Equational bases and nonmodular lattice varieties, Transactions of the American Mathematical Society 174: 1\u201343, 1972.","journal-title":"Transactions of the American Mathematical Society"},{"key":"9997_CR23","unstructured":"McKenzie, R., G. McNulty, and W. Taylor, Algebras, Lattices, Varieties, Volume 1, Wadsworth & Brooks\/Cole, Belmont, CA, 1987."},{"key":"9997_CR24","doi-asserted-by":"publisher","first-page":"169","DOI":"10.2307\/2273798","volume":"50","author":"H Ono","year":"1985","unstructured":"Ono, H., and Y. Komori, Logics without the contraction rule, The Journal of Symbolic Logic 50:160\u2013201, 1985.","journal-title":"The Journal of Symbolic Logic"},{"key":"9997_CR25","first-page":"137","volume":"8","author":"M Pa\u0142asi\u0144ski","year":"1980","unstructured":"Pa\u0142asi\u0144ski, M., Some remarks on BCK-algebras, Mathematics Seminar Notes (Kobe University) 8:137\u2013144, 1980.","journal-title":"Mathematics Seminar Notes (Kobe University)"},{"key":"9997_CR26","first-page":"543","volume":"26","author":"M Pa\u0142asi\u0144ski","year":"1981","unstructured":"Pa\u0142asi\u0144ski, M., On ideals and congruence lattices of BCK-algebras, Mathematica Japonica 26:543\u2013544, 1981.","journal-title":"Mathematica Japonica"},{"key":"9997_CR27","first-page":"465","volume":"32","author":"JG Raftery","year":"1987","unstructured":"Raftery, J.\u00a0G., and T. Sturm, On ideal and congruence lattices of BCK-semilattices, Mathematica Japonica 32:465\u2013474, 1987.","journal-title":"Mathematica Japonica"},{"key":"9997_CR28","doi-asserted-by":"publisher","first-page":"179","DOI":"10.2307\/2371781","volume":"65","author":"PM Whitman","year":"1943","unstructured":"Whitman, P.M., Splittings of a lattice, American Journal of Mathematics 65:179\u2013196, 1943.","journal-title":"American Journal of Mathematics"},{"key":"9997_CR29","first-page":"211","volume":"28","author":"A Wro\u0144ski","year":"1983","unstructured":"Wro\u0144ski, A., BCK-algebras do not form a variety, Mathematica Japonica 28:211\u2013213, 1983.","journal-title":"Mathematica Japonica"}],"container-title":["Studia Logica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11225-022-09997-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11225-022-09997-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11225-022-09997-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T07:40:47Z","timestamp":1664437247000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11225-022-09997-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,5]]},"references-count":29,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["9997"],"URL":"https:\/\/doi.org\/10.1007\/s11225-022-09997-y","relation":{},"ISSN":["0039-3215","1572-8730"],"issn-type":[{"value":"0039-3215","type":"print"},{"value":"1572-8730","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,5]]},"assertion":[{"value":"20 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 August 2022","order":3,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":4,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Missing Open Access funding information has been added in the Funding Note.","order":5,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}