{"id":222,"date":"2011-10-11T15:29:33","date_gmt":"2011-10-11T13:29:33","guid":{"rendered":"http:\/\/logica.dmi.unisa.it\/lucaspada\/?p=222"},"modified":"2017-10-05T09:49:54","modified_gmt":"2017-10-05T08:49:54","slug":"duality-projectivity-and-unification-in-lukasiewicz-logic-and-mv-algebras","status":"publish","type":"post","link":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/222-duality-projectivity-and-unification-in-lukasiewicz-logic-and-mv-algebras\/","title":{"rendered":"Duality, projectivity, and unification in \u0141ukasiewicz logic and MV-algebras"},"content":{"rendered":"<p>We prove that the unification type of Lukasiewicz infinite-valued propositional logic and of its equivalent algebraic semantics, the variety of MV-algebras,is nullary. The proof rests upon Ghilardi&#8217;s algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally,\u00a0a \u00a0homotopy-theoretic argument that exploits \u00a0lifts of continuous maps to the universal covering space of the circle. We\u00a0discuss the background to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra &#8211; a fundamental result that, albeit known to specialists, seems to appear in print here for the first time.<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-content\/uploads\/uniMV-final-improved.pdf\">Duality, projectivity, and unification in \u0141ukasiewicz logic and MV-algebras<\/a><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We prove that the unification type of Lukasiewicz infinite-valued propositional logic and of its equivalent algebraic semantics, the variety of MV-algebras,is nullary. The proof rests upon Ghilardi&#8217;s algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17],"tags":[49,48,51,28,52,37,47,44,50],"class_list":["post-222","post","type-post","status-publish","format-standard","hentry","category-preprint","tag-covering-space","tag-fundamental-group","tag-lifts","tag-lukasiewicz-logic","tag-projective-mv-algebra","tag-rational-polyhedra","tag-retractions","tag-unification","tag-universal-cover"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/222","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/comments?post=222"}],"version-history":[{"count":8,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/222\/revisions"}],"predecessor-version":[{"id":1157,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/222\/revisions\/1157"}],"wp:attachment":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/media?parent=222"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/categories?post=222"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/tags?post=222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}