{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T12:28:27Z","timestamp":1773145707564,"version":"3.50.1"},"reference-count":42,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":832,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2011,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member<jats:bold>A<\/jats:bold>of a<jats:italic>\u03c4<\/jats:italic>-regular variety<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001365_inline1\"\/>the lattice of congruences of<jats:bold>A<\/jats:bold>is isomorphic to the lattice of deductive filters on<jats:bold>A<\/jats:bold>of the<jats:italic>\u03c4<\/jats:italic>-assertional logic of<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001365_inline1\"\/>. Moreover, if<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001365_inline1\"\/>has a constant 1 in its type and is 1-subtractive, the deductive filters on<jats:bold>A<\/jats:bold>\u2208<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001365_inline1\"\/>of the 1-assertional logic of<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001365_inline1\"\/>coincide with the<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001365_inline1\"\/>-ideals of<jats:bold>A<\/jats:bold>in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation.<\/jats:p><jats:p>However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and<jats:italic>\u03c4<\/jats:italic>-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics.<\/jats:p>","DOI":"10.2178\/jsl\/1318338848","type":"journal-article","created":{"date-parts":[[2011,10,11]],"date-time":"2011-10-11T14:32:16Z","timestamp":1318343536000},"page":"1261-1286","source":"Crossref","is-referenced-by-count":13,"title":["Quasi-subtractive varieties"],"prefix":"10.1017","volume":"76","author":[{"given":"Tomasz","family":"Kowalski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesco","family":"Paoli","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matthew","family":"Spinks","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200001365_ref026","doi-asserted-by":"publisher","DOI":"10.1007\/s000120050088"},{"key":"S0022481200001365_ref022","doi-asserted-by":"publisher","DOI":"10.1007\/BF01190718"},{"key":"S0022481200001365_ref037","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-008-9113-x"},{"key":"S0022481200001365_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BF01182089"},{"key":"S0022481200001365_ref035","first-page":"53","article-title":"On some properties of quasi-MV algebras and quasi-MV algebras. Part I","volume":"44","author":"Paoli","year":"2008","journal-title":"Reports on Mathematical Logic"},{"key":"S0022481200001365_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-007-0185-8"},{"key":"S0022481200001365_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01234106"},{"key":"S0022481200001365_ref021","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2004.07.002"},{"key":"S0022481200001365_ref019","doi-asserted-by":"publisher","DOI":"10.1023\/A:1024621922509"},{"key":"S0022481200001365_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8130-3"},{"key":"S0022481200001365_ref034","doi-asserted-by":"publisher","DOI":"10.1007\/BF01462258"},{"key":"S0022481200001365_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF01188178"},{"key":"S0022481200001365_ref041","unstructured":"van Alten C. J. , An algebraic study of residuated ordered monoids and logics without exchange and contraction, Ph.D. thesis, University of Natal, 1998."},{"key":"S0022481200001365_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/BF02584057"},{"key":"S0022481200001365_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/s000120050020"},{"key":"S0022481200001365_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/s000120050059"},{"key":"S0022481200001365_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19980440304"},{"key":"S0022481200001365_ref005","doi-asserted-by":"publisher","DOI":"10.1023\/A:1024673906579"},{"key":"S0022481200001365_ref006","doi-asserted-by":"crossref","DOI":"10.1090\/memo\/0396","volume-title":"Algebraizable logics","author":"Blok","year":"1989"},{"key":"S0022481200001365_ref036","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(88)90021-8"},{"key":"S0022481200001365_ref008","first-page":"167","volume-title":"Models, algebras and proofs","author":"Blok","year":"1999"},{"key":"S0022481200001365_ref016","volume-title":"Theory of lattice ordered groups","author":"Darnel","year":"1995"},{"key":"S0022481200001365_ref033","volume-title":"Algebras, lattices, varieties","volume":"1","author":"Mc Kenzie","year":"1987"},{"key":"S0022481200001365_ref009","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196708004627"},{"key":"S0022481200001365_ref017","doi-asserted-by":"crossref","first-page":"197","DOI":"10.21136\/CMJ.1987.102149","article-title":"Arithmelicity at 0","volume":"37","author":"Duda","year":"1987","journal-title":"Czechoslovak Mathematical Journal"},{"key":"S0022481200001365_ref028","first-page":"161","article-title":"On some properties of quasi-MV algebras and quasi-MV algebras. Part III","volume":"45","author":"Kowalski","year":"2010","journal-title":"Reports on Mathematical Logic"},{"key":"S0022481200001365_ref018","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19760220128"},{"key":"S0022481200001365_ref020","volume-title":"Residuated lattices: An algebraic glimpse on substructurai logics","volume":"151","author":"Galatos","year":"2007"},{"key":"S0022481200001365_ref023","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1969-095-x"},{"key":"S0022481200001365_ref024","doi-asserted-by":"publisher","DOI":"10.1007\/BF01191491"},{"key":"S0022481200001365_ref025","doi-asserted-by":"publisher","DOI":"10.1023\/B:STUD.0000037130.29400.97"},{"key":"S0022481200001365_ref027","doi-asserted-by":"publisher","DOI":"10.1023\/B:STUD.0000037129.58589.0c"},{"key":"S0022481200001365_ref029","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-011-0137-0"},{"key":"S0022481200001365_ref014","doi-asserted-by":"crossref","first-page":"283","DOI":"10.21136\/MB.2005.134099","article-title":"Normalization of MV algebras","volume":"130","author":"Chajda","year":"2005","journal-title":"Mathematica Bohemica"},{"key":"S0022481200001365_ref030","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-006-7202-2"},{"key":"S0022481200001365_ref031","first-page":"3","article-title":"On the general theory of algebraic systems","volume":"35","author":"Mal'cev","year":"1954","journal-title":"Matematicheskii Sbornik (N.S.)"},{"key":"S0022481200001365_ref032","first-page":"211","volume-title":"Logic and algebra: Proceedings of the Magari conference","author":"Mc Kenzie","year":"1996"},{"key":"S0022481200001365_ref039","doi-asserted-by":"publisher","DOI":"10.1007\/BF01236518"},{"key":"S0022481200001365_ref012","first-page":"35","volume-title":"General algebra and discrete mathematics","author":"Chajda","year":"1995"},{"key":"S0022481200001365_ref038","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-008-9138-1"},{"key":"S0022481200001365_ref040","doi-asserted-by":"publisher","DOI":"10.1007\/BF02121118"},{"key":"S0022481200001365_ref042","doi-asserted-by":"crossref","DOI":"10.1515\/9783112733332","volume-title":"Discriminator algebras","author":"Werner","year":"1978"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200001365","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T18:32:13Z","timestamp":1741804333000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200001365\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,12]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2011,12]]}},"alternative-id":["S0022481200001365"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1318338848","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,12]]}}}