{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T04:07:39Z","timestamp":1759032459351},"reference-count":25,"publisher":"Oxford University Press (OUP)","issue":"5","license":[{"start":{"date-parts":[[2021,7,5]],"date-time":"2021-07-05T00:00:00Z","timestamp":1625443200000},"content-version":"vor","delay-in-days":3,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,9,15]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson\u2019s constructive logic with strong negation, is the axiomatic extension of the substructural logic $FL_{ew}$ (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. $FL_{ew}$-algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication\u2013negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.<\/jats:p>","DOI":"10.1093\/jigpal\/jzab023","type":"journal-article","created":{"date-parts":[[2021,6,10]],"date-time":"2021-06-10T11:27:52Z","timestamp":1623324472000},"page":"807-839","source":"Crossref","is-referenced-by-count":7,"title":["Fragments of Quasi-Nelson: The Algebraizable Core"],"prefix":"10.1093","volume":"30","author":[{"given":"Umberto","family":"Rivieccio","sequence":"first","affiliation":[{"name":"Departamento de Inform\u00e1tica e Matem\u00e1tica Aplicada , Universidade Federal do Rio Grande do Norte, Campus Universit\u00e1rio, Lagoa Nova 59072-970, Natal (RN), Brazil"}]}],"member":"286","published-online":{"date-parts":[[2021,7,2]]},"reference":[{"key":"2022092120484915100_ref1","doi-asserted-by":"crossref","DOI":"10.1090\/memo\/0396","article-title":"Algebraizable logics","volume-title":"Memoirs of the American Mathematical Society","author":"Blok","year":"1989"},{"key":"2022092120484915100_ref2","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-8130-3","volume-title":"A Course in Universal Algebra","author":"Burris","year":"1981"},{"key":"2022092120484915100_ref3","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s11083-006-9032-2","article-title":"Algebras defined from ordered sets and the varieties they generate","volume":"23","author":"Berman","year":"2006","journal-title":"Order"},{"key":"2022092120484915100_ref4","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1007\/BF01195727","article-title":"On the structure of varieties with equationally definable principal congruences III","volume":"32","author":"Blok","year":"1994","journal-title":"Algebra Universalis"},{"key":"2022092120484915100_ref5","first-page":"463","article-title":"Representation and duality for Hilbert algebras","volume":"7","author":"Celani","year":"2009","journal-title":"Cent. Eur. J. Math."},{"key":"2022092120484915100_ref6","volume-title":"Abstract Algebraic Logic: An Introductory Textbook","author":"Font","year":"2016"},{"key":"2022092120484915100_ref7","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1016\/j.apal.2019.05.003","article-title":"Categories of models of R-mingle","volume":"170","author":"Fussner","year":"2019","journal-title":"Annals of Pure and Applied Logic"},{"key":"2022092120484915100_ref8","volume-title":"Residuated Lattices: An Algebraic Glimpse at Substructural Logics","author":"Galatos","year":"2007"},{"key":"2022092120484915100_ref9","doi-asserted-by":"crossref","first-page":"3189","DOI":"10.1090\/S0002-9947-2014-06072-8","article-title":"Idempotent residuated structures: some category equivalences and their applications","volume":"367","author":"Galatos","year":"2015","journal-title":"Transactions of the American Mathematical Society"},{"key":"2022092120484915100_ref10","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-0348-7633-9","volume-title":"General Lattice Theory","author":"Gr\u00e4tzer","year":"1978"},{"key":"2022092120484915100_ref11","first-page":"450","article-title":"Algebraic semantics for quasi-Nelson logic","volume-title":"Logic, Language, Information, and Computation. Proc. WoLLIC 2019","author":"Liang","year":"2019"},{"key":"2022092120484915100_ref12","volume-title":"An Algebraic Study of Modal Operators on Heyting Algebras With Applications to Topology and Sheafification","author":"Macnab","year":"1976"},{"key":"2022092120484915100_ref13","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/BF02483860","article-title":"Modal operators on Heyting algebras","volume":"12","author":"Macnab","year":"1981","journal-title":"Algebra Universalis"},{"key":"2022092120484915100_ref14","doi-asserted-by":"crossref","first-page":"107","DOI":"10.21146\/2074-1472-2021-27-1-107-123","article-title":"Negation and implication in quasi-Nelson logic","volume":"27","author":"Nascimento","year":"2021","journal-title":"Logical Investigations"},{"key":"2022092120484915100_ref15","doi-asserted-by":"crossref","first-page":"16","DOI":"10.2307\/2268973","article-title":"Constructible falsity","volume":"14","author":"Nelson","year":"1949","journal-title":"Journal of Symbolic Logic"},{"key":"2022092120484915100_ref16","volume-title":"An Algebraic Approach to Non-Classical Logics","author":"Rasiowa","year":"1974"},{"key":"2022092120484915100_ref17","first-page":"499","article-title":"Fragments of quasi-Nelson: two negations","volume":"7","author":"Rivieccio","year":"2020","journal-title":"Journal of Applied Logic"},{"key":"2022092120484915100_ref18","doi-asserted-by":"crossref","first-page":"8685","DOI":"10.1007\/s00500-020-04885-w","article-title":"Representation of De Morgan and (semi-)Kleene lattices","volume":"24","author":"Rivieccio","year":"2020","journal-title":"Soft Computing"},{"key":"2022092120484915100_ref19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/FUZZ48607.2020.9177641","article-title":"On the representation of (weak) nilpotent minimum algebras","volume-title":"2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)","author":"Rivieccio","year":"2020"},{"key":"2022092120484915100_ref20","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129521000049","article-title":"Quasi-Nelson algebras and fragments","author":"Rivieccio","year":"2021","journal-title":"Mathematical Structures in Computer Science"},{"key":"2022092120484915100_ref21","first-page":"634","article-title":"Two dualities for weakly pseudo-complemented quasi-Kleene algebras","volume-title":"Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2020. Communications in Computer and Information Science","author":"Rivieccio","year":"2020"},{"key":"2022092120484915100_ref22","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.entcs.2019.07.011","article-title":"Quasi-Nelson algebras","volume":"344","author":"Rivieccio","year":"2019","journal-title":"Electronic Notes in Theoretical Computer Science"},{"key":"2022092120484915100_ref23","first-page":"133","article-title":"Quasi-Nelson; or, non-involutive Nelson algebras","volume-title":"Trends in Logic","author":"Rivieccio","year":"2020"},{"key":"2022092120484915100_ref24","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1007\/s00500-018-3588-9","article-title":"Compatibly involutive residuated lattices and the Nelson identity","volume":"23","author":"Spinks","year":"2019","journal-title":"Soft Computing"},{"key":"2022092120484915100_ref25","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s11225-008-9113-x","article-title":"Constructive logic with strong negation is a substructural logic. I","volume":"88","author":"Spinks","year":"2008","journal-title":"Studia Logica"}],"container-title":["Logic Journal of the IGPL"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jigpal\/article-pdf\/30\/5\/807\/45878657\/jzab023.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jigpal\/article-pdf\/30\/5\/807\/45878657\/jzab023.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T21:06:01Z","timestamp":1663794361000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jigpal\/article\/30\/5\/807\/6311966"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,2]]},"references-count":25,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,7,2]]},"published-print":{"date-parts":[[2022,9,15]]}},"URL":"https:\/\/doi.org\/10.1093\/jigpal\/jzab023","relation":{},"ISSN":["1367-0751","1368-9894"],"issn-type":[{"value":"1367-0751","type":"print"},{"value":"1368-9894","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2022,10]]},"published":{"date-parts":[[2021,7,2]]}}}