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In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by \u0391$\\textrm{lg} (\\textrm{PWK})$. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest\u2019s logic of paradox, $\\textrm{PWK}$ has only one proper nontrivial extension apart from classical logic: $\\textrm{PWK}_{\\textrm{E}}\\textrm{,}$ PWK logic plus explosion. This $6$-valued logic, unlike $\\textrm{PWK} $, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models.<\/jats:p>","DOI":"10.1093\/jigpal\/jzaa024","type":"journal-article","created":{"date-parts":[[2020,6,15]],"date-time":"2020-06-15T15:11:07Z","timestamp":1592233867000},"page":"798-822","source":"Crossref","is-referenced-by-count":8,"title":["Extensions of paraconsistent weak Kleene logic"],"prefix":"10.1093","volume":"29","author":[{"given":"Francesco","family":"Paoli","sequence":"first","affiliation":[{"name":"Department of Pedagogy, Psychology, Philosophy, University of Cagliari, Cagliary 09125, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michele","family":"Pra Baldi","sequence":"additional","affiliation":[{"name":"Department of Pedagogy, Psychology, Philosophy, University of Cagliari, Cagliary 09125, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,7,27]]},"reference":[{"key":"2021091710534934400_ref1","doi-asserted-by":"crossref","first-page":"207","DOI":"10.4064\/fm-68-2-207-214","article-title":"A representation theorem for distributive quasilattices","volume":"68","author":"Balbes","year":"1970","journal-title":"Fundamenta Mathematicae"},{"key":"2021091710534934400_ref2","doi-asserted-by":"crossref","first-page":"136","DOI":"10.26686\/ajl.v13i6.3976","article-title":"A new interpretation of weak Kleene logic","volume":"13","author":"Beall","year":"2016","journal-title":"Australasian Journal of Logic"},{"key":"2021091710534934400_ref3","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/s00012-015-0323-6","article-title":"Commutative idempotent groupoids and the constraint satisfaction problem","volume":"73","author":"Bergman","year":"2015","journal-title":"Algebra 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