{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T03:07:34Z","timestamp":1772852854013,"version":"3.50.1"},"reference-count":14,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2002,8]]},"abstract":"<jats:p> A commutative residuated lattice, is an ordered algebraic structure [Formula: see text], where (L, \u00b7, e) is a commutative monoid, (L, \u2227, \u2228) is a lattice, and the operation \u2192 satisfies the equivalences [Formula: see text] for a, b, c \u220a L. The class of all commutative residuated lattices, denoted by [Formula: see text], is a finitely based variety of algebras. Historically speaking, our study draws primary inspiration from the work of M. Ward and R. P. Dilworth appearing in a series of important papers [9, 10, 19\u201322]. In the ensuing decades special examples of commutative, residuated lattices have received considerable attention, but we believe that this is the first time that a comprehensive theory on the structure of residuated lattices has been presented from the viewpoint of universal algebra. In particular, we show that [Formula: see text] is an \"ideal variety\" in the sense that its congruences correspond to order-convex subalgebras. As a consequence of the general theory, we present an equational basis for the subvariety [Formula: see text] generated by all commutative, residuated chains. We conclude the paper by proving that the congruence lattice of each member of [Formula: see text] is an algebraic, distributive lattice whose meet-prime elements form a root-system (dual tree). This result, together with the main results in [12, 18], will be used in a future publication to analyze the structure of finite members of [Formula: see text]. A comprehensive study of, not necessarily commutative, residuated lattices is presented in [4]. <\/jats:p>","DOI":"10.1142\/s0218196702001048","type":"journal-article","created":{"date-parts":[[2002,10,1]],"date-time":"2002-10-01T20:07:50Z","timestamp":1033502870000},"page":"509-524","source":"Crossref","is-referenced-by-count":87,"title":["THE STRUCTURE OF COMMUTATIVE RESIDUATED LATTICES"],"prefix":"10.1142","volume":"12","author":[{"given":"JAMES B.","family":"HART","sequence":"first","affiliation":[{"name":"Box 34, Middle Tennessee State  University, Murfreesboro TN 37132, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"LORI","family":"RAFTER","sequence":"additional","affiliation":[{"name":"Department of Mathematics,  Vanderbilt University, Nashville TN 37235, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CONSTANTINE","family":"TSINAKIS","sequence":"additional","affiliation":[{"name":"Department of Mathematics,  Vanderbilt University, Nashville TN 37235, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"p_5","doi-asserted-by":"crossref","first-page":"109","DOI":"10.24033\/bsmf.1618","volume":"93","author":"Blyth T. 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(Szeged)"},{"key":"p_17","first-page":"129","author":"Monteiro A.","year":"1954","journal-title":"Buenos Aires) ("},{"key":"p_18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01190439"},{"key":"p_19","doi-asserted-by":"publisher","DOI":"10.2307\/1968634"},{"key":"p_20","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-37-00351-X"},{"key":"p_21","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.24.3.162"},{"key":"p_22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1939-1501995-3"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196702001048","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T21:56:54Z","timestamp":1565128614000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196702001048"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,8]]},"references-count":14,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2002,8]]}},"alternative-id":["10.1142\/S0218196702001048"],"URL":"https:\/\/doi.org\/10.1142\/s0218196702001048","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,8]]}}}