{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:21:48Z","timestamp":1759335708862},"reference-count":10,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2009,2]]},"abstract":"<jats:p> For a class [Formula: see text] of algebras, denote by Con<jats:sub>c<\/jats:sub>[Formula: see text] the class of all (\u2228, 0)-semilattices isomorphic to the semilattice Con<jats:sub>c<\/jats:sub>A of all compact congruences of A, for some A in [Formula: see text]. For classes [Formula: see text] and [Formula: see text] of algebras, we denote by [Formula: see text] the smallest cardinality of a (\u2228, 0)-semilattices in Con<jats:sub>c<\/jats:sub>[Formula: see text] which is not in Con<jats:sub>c<\/jats:sub>[Formula: see text] if it exists, \u221e otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties [Formula: see text] and [Formula: see text], [Formula: see text] is either finite, or \u2135<jats:sub>n<\/jats:sub> for some natural number n, or \u221e. We also find two finitely generated modular lattice varieties [Formula: see text] and [Formula: see text] such that [Formula: see text], thus answering a question by J. T\u016fma and F. Wehrung. <\/jats:p>","DOI":"10.1142\/s0218196709004932","type":"journal-article","created":{"date-parts":[[2009,2,25]],"date-time":"2009-02-25T10:55:38Z","timestamp":1235559338000},"page":"1-40","source":"Crossref","is-referenced-by-count":12,"title":["CRITICAL POINTS OF PAIRS OF VARIETIES OF ALGEBRAS"],"prefix":"10.1142","volume":"19","author":[{"given":"PIERRE","family":"GILLIBERT","sequence":"first","affiliation":[{"name":"LMNO, CNRS UMR 6139, D\u00e9partement de Math\u00e9matiques, BP 5186, Universit\u00e9 de Caen, Campus 2, 14032 Caen cedex, France"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.3792\/pia\/1195573786"},{"key":"rf2","volume-title":"Universal Algebra","author":"Gr\u00e4tzer G.","year":"1968"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1982.103.475"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-005-1931-3"},{"key":"rf5","doi-asserted-by":"crossref","first-page":"71","DOI":"10.4064\/cm-83-1-71-84","volume":"83","author":"Plo\u0161\u010dica M.","journal-title":"Colloq. Math."},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-8641(02)00259-6"},{"key":"rf7","doi-asserted-by":"crossref","first-page":"269","DOI":"10.4064\/cm-76-2-269-278","volume":"76","author":"Plo\u0161\u010dica M.","journal-title":"Colloq. Math."},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/s000120200011"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196706003049"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2007.05.016"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196709004932","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T13:35:01Z","timestamp":1565184901000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196709004932"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,2]]},"references-count":10,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2009,2]]}},"alternative-id":["10.1142\/S0218196709004932"],"URL":"https:\/\/doi.org\/10.1142\/s0218196709004932","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,2]]}}}