{"id":62108,"date":"2026-01-06T18:54:37","date_gmt":"2026-01-06T16:54:37","guid":{"rendered":"https:\/\/nvd-nanu.org.ua\/?p=62108"},"modified":"2026-01-06T18:54:37","modified_gmt":"2026-01-06T16:54:37","slug":"algebraic-unification-of-first-degree-entailment-in-relevance-and-linear-intuitionism","status":"publish","type":"post","link":"https:\/\/nvd-nanu.org.ua\/en\/algebraic-unification-of-first-degree-entailment-in-relevance-and-linear-intuitionism\/","title":{"rendered":"Algebraic unification of first degree entailment in relevance and linear intuitionism"},"content":{"rendered":"","protected":false},"excerpt":{"rendered":"","protected":false},"author":5,"featured_media":50129,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"vydavnytstvo":[],"vydavets":[],"vyd_vydannia":[252],"vyd_periodyka":[],"inshyi_zasnovnyk":[],"katehoriia_periodyka":[],"knyzhkovyi_proekt":[],"misto":[],"mova":[301],"mova_periodyka":[],"periodychnist":[],"rozpovsiudzhennia":[],"rik_vydannia":[947],"rik_zasnuvannia":[],"rubryka":[285],"typ_vydannia":[260],"typ_periodyka":[],"ustanova_nan_ukrainy":[],"fakhovist":[],"tsytuvannia":[],"class_list":["post-62108","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-books","vyd_vydannia-monohrafii","mova-anhliiska","rik_vydannia-947","rubryka-matematyka","typ_vydannia-elektronne"],"acf":{"\u0440\u0456\u043a_\u0432\u0438\u0434\u0430\u043d\u043d\u044f":947,"\u0442\u0438\u043f_\u0432\u0438\u0434\u0430\u043d\u043d\u044f":260,"\u0440\u0443\u0431\u0440\u0438\u043a\u0430":285,"\u0432\u0438\u0434_\u0432\u0438\u0434\u0430\u043d\u043d\u044f":252,"\u0430\u0432\u0442\u043e\u0440\u0438":"A.P. 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Such is said to be [quasi-strong|] \u00aba [nearly] Morgan\/Kleene lattice\u00bb iff every element of it [\u00abexceeded by\u00bb exceeding its negation] exceeds its double negation, any Morgan\/Kleene lattice being so. Applying the technique of characteristic functions of prime filters as homomorphisms from lattices onto the two-element chain one and their functional pro\u0002ducts, we prove that the variety of MS lattices is the quasi-variety enerated by a six-element one having an equational disjunctive system and expanding the direct product of the three- and two-element chain lattices, in which case subdirectly-irreducible MS lattices are exactly isomor\u0002phic copies of nine non-one-element pair-wise-non-isomorphic subalgebras of the six-element generating MS lattice, and so we get a 29-element non-chain distributive lattice of varieties of MS lattices subsuming the four-\/three-element chain one of Morgan\/Stone lattices\/algebras (viz., constant-free versions of De Morgan algebras)\/(more precisely, their term-wise definition\u0002ally equivalent constant-free versions, called Stone lattices). Among other things, we provide an REDPC scheme for MS lattices. Laying a special emphasis onto the universal\/[quasi-]equational unbounded approximation of MS algebras (viz., the greatest iversal\/[quasi]equational class of MS lattices without members with both bounds but expandable to no MS algebra) with members being exactly quasi-strong MS lattices, we find a 29\/75-element non-chain distributive\/non\u0002distributive lattice of quasi-varieties of quasi-strong\/nearly MS\/Kleene lattices \u00absubsuming the fifteen-element one of the [quasi-]equational join (viz., the [quasi-]variety generated by the un\u0002ion) of Morgan and Stone lattices, in its turn, subsuming the eight-element non-chain distributive one of those of the variety of Morgan lattices found earlier\u00bb\/. On the other hand, that of nearly Morgan lattices is proved to have a countable decreasing chain with non-finitely-axiomatizable intersection.","summary":"\u0420\u0435\u0448\u0456\u0442\u043a\u0438 \u041c\u043e\u0440\u0433\u0430\u043d\u0430\/\u041a\u043b\u0456\u043d\u0456-\u0421\u0442\u043e\u0443\u043d\u0430 (MS) \u0430\u043a\u0441\u0456\u043e\u043c\u0430\u0442\u0438\u0437\u0443\u044e\u0442\u044c\u0441\u044f \u043f\u043e\u0441\u0442\u0456\u0439\u043d\u043e \u0432\u0456\u043b\u044c\u043d\u0438\u043c\u0438 \u0442\u043e\u0002\u0442\u043e\u0436\u043d\u043e\u0441\u0442\u044f\u043c\u0438 \u0437 \u0442\u0438\u0445, \u0449\u043e \u0430\u043a\u0441\u0456\u043e\u043c\u0430\u0442\u0438\u0437\u0443\u044e\u0442\u044c \u0410\u043b\u0433\u0435\u0431\u0440\u0438 \u041c\u043e\u0440\u0433\u0430\u043d\u0430-\u0421\u0442\u043e\u0443\u043d\u0430 (MS) \u0456 \u0442\u043e\u0442\u043e\u0436\u043d\u0456\u0441\u0442\u044e \u041a\u043b\u0456\u0002\u043d\u0456, \u0449\u043e \u0430\u043a\u0441\u0456\u043e\u043c\u0430\u0442\u0438\u0437\u0443\u044e\u0442\u044c \u0440\u0435\u0448\u0456\u0442\u043a\u0438 \u041a\u043b\u0456\u043d\u0456 \u0432\u0456\u0434\u043d\u043e\u0441\u043d\u043e \u0440\u0435\u0448\u0456\u0442\u043e\u043a \u0414\u0435 \u041c\u043e\u0440\u0433\u0430\u043d\u0430. \u041a\u0430\u0436\u0435\u043c\u043e, \u0449\u043e \u0446\u0435 [\u043a\u0432\u0430\u0437\u0456\u0441\u0438\u043b\u044c\u043d\u0430|]\u00ab[\u043c\u0430\u0439\u0436\u0435] \u0440\u0435\u0448\u0456\u0442\u043a\u0430 \u041c\u043e\u0440\u0433\u0430\u043d\u0430\/\u041a\u043b\u0456\u043d\u0456\u00bb \u044f\u043a\u0449\u043e \u043a\u043e\u0436\u0435\u043d \u0439\u043e\u0433\u043e \u0435\u043b\u0435\u043c\u0435\u043d\u0442, \u0449\u043e \u043f\u0435\u0440\u0435\u0432\u0438\u0449\u0443\u0454 \u0441\u0432\u043e\u0454 \u043f\u043e\u0434\u0432\u0456\u0439\u043d\u0435 \u0437\u0430\u043f\u0435\u0440\u0435\u0447\u0435\u043d\u043d\u044f, \u044f\u043a \u0431\u0443\u0434\u044c-\u044f\u043a\u0430 \u0440\u0435\u0448\u0456\u0442\u043a\u0430 \u041c\u043e\u0440\u0433\u0430\u043d\u0430\/\u041a\u043b\u0456\u043d\u0456. \u0412\u0438\u043a\u043b\u0430\u0434\u0435\u043d\u043e \u0437\u0430\u0441\u0442\u043e\u0441\u0443\u0432\u0430\u043d\u043d\u044f \u0442\u0435\u0445\u043d\u0456\u043a\u0438 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u0447\u043d\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0456\u0439 \u043f\u0440\u043e\u0441\u0442\u0438\u0445 \u0444\u0456\u043b\u044c\u0442\u0440\u0456\u0432 \u044f\u043a \u0433\u043e\u043c\u043e\u043c\u043e\u0440\u0444\u0456\u0437\u043c\u0456\u0432 \u0437 \u0440\u0435\u0448\u0456\u0442\u043e\u043a \u043d\u0430 \u0434\u0432\u043e\u0435\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u0438\u0439 \u043b\u0430\u043d\u0446\u044e\u0433 \u0456 \u0457\u0445 \u0444\u0443\u043d\u043a\u0446\u0456\u043e\u043d\u0430\u043b\u044c\u043d\u0456 \u043f\u0440\u043e\u0434\u0443\u043a\u0442\u0438, \u043c\u0438 \u0434\u043e\u0432\u043e\u0434\u0438\u043c\u043e, \u0449\u043e \u0440\u0456\u0437\u043d\u043e\u043c\u0430\u043d\u0456\u0442\u0442\u044f\u043c MS\u0002\u0440\u0435\u0448\u0456\u0442\u043e\u043a \u0454 \u043a\u0432\u0430\u0437\u0456\u043c\u043d\u043e\u0433\u043e\u0432\u0438\u0434, \u043f\u043e\u0440\u043e\u0434\u0436\u0435\u043d\u0438\u0439 \u0448\u0435\u0441\u0442\u0438\u0435\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e\u044e MS-\u0440\u0435\u0448\u0456\u0442\u043a\u043e\u044e \u0437 \u0434\u0438\u0437\u2019\u044e\u043d\u043a\u0442\u0438\u0432\u043d\u043e\u044e \u0441\u0438\u0441\u0442\u0435\u043c\u043e\u044e \u0456 \u0440\u043e\u0437\u043a\u043b\u0430\u0434\u0430\u043d\u043d\u044f\u043c \u043f\u0440\u044f\u043c\u043e\u0433\u043e \u0434\u043e\u0431\u0443\u0442\u043a\u0443 \u043d\u0430 \u0442\u0440\u0438- \u0442\u0430 \u0434\u0432\u043e\u0435\u043b\u0435\u043c\u0435\u043d\u0442\u043b\u0430\u043d\u0446\u044e\u0433\u043e\u0432\u0456 \u0440\u0435\u0448\u0456\u0442\u043a\u0438, \u0432 \u0446\u044c\u043e\u043c\u0443 \u0432\u0438\u043f\u0430\u0434\u043a\u0443 \u043f\u0456\u0434\u043f\u0440\u044f\u043c\u043e\u043d\u0435\u0437\u0432\u0456\u0434\u043d\u0456 MS-\u0440\u0435\u0448\u0456\u0442\u043a\u0438 \u0454 \u0442\u043e\u0447\u043d\u043e \u0456\u0437\u043e\u043c\u043e\u0440\u0444\u043d\u0438\u043c\u0438 \u043a\u043e\u043f\u0456\u044f\u043c\u0438 \u0434\u0435\u0432\u2019\u044f\u0442\u0438 \u043d\u0435\u043e\u0434\u043d\u043e\u0435\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u0438\u0445 \u043f\u043e\u043f\u0430\u0440\u043d\u043e-\u043d\u0435\u0456\u0437\u043e\u043c\u043e\u0440\u0444\u043d\u0438\u0445 \u043f\u0456\u0434\u0430\u043b\u0433\u0435\u0431\u0440 \u0448\u0435\u0441\u0442\u0438\u0435\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u043e\u0457 \u043f\u043e\u0440\u043e\u0434\u0436\u0443\u044e\u0447\u043e\u0457 MS\u0002\u0440\u0435\u0448\u0456\u0442\u043a\u0438, \u0456 \u0442\u043e\u043c\u0443 \u043e\u0442\u0440\u0438\u043c\u0443\u0454\u043c\u043e 29-\u0435\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u0443 \u043d\u0435\u043b\u0430\u043d\u0446\u044e\u0433\u043e\u0432\u0443 \u0440\u043e\u0437\u043f\u043e\u0434\u0456\u043b\u044c\u043d\u0443 \u0440\u0435\u0448\u0456\u0442\u043a\u0443 \u0440\u0456\u0437\u043d\u043e\u0432\u0438\u0434\u0456\u0432 MS\u0002\u0440\u0435\u0448\u0456\u0442\u043e\u043a, \u0449\u043e \u0432\u043a\u043b\u044e\u0447\u0430\u044e\u0442\u044c \u0447\u043e\u0442\u0438\u0440\u0438-\/\u0442\u0440\u044c\u043e\u0445\u0435\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u0443 \u043b\u0430\u043d\u0446\u044e\u0433\u043e\u0432\u0443 \u0440\u0435\u0448\u0456\u0442\u043a\u0443\/\u0430\u043b\u0433\u0435\u0431\u0440\u0443 \u041c\u043e\u0440\u0433\u0430\u043d\u0430\/\u0421\u0442\u043e\u0443\u043d\u0430. \u0421\u0435\u0440\u0435\u0434 \u0456\u043d\u0448\u043e\u0433\u043e, \u043d\u0430\u0432\u0435\u0434\u0435\u043d\u043e \u0441\u0445\u0435\u043c\u0443 REDPC \u0434\u043b\u044f MS-\u0440\u0435\u0448\u0456\u0442\u043e\u043a. \u0420\u043e\u0431\u043b\u044f\u0447\u0438 \u043e\u0441\u043e\u0431\u043b\u0438\u0432\u0438\u0439 \u0430\u043a\u0446\u0435\u043d\u0442 \r\n\u043d\u0430 \u0443\u043d\u0456\u0432\u0435\u0440\u0441\u0430\u043b\u044c\u043d\u0435\/[\u043a\u0432\u0430\u0437\u0456-]\u0435\u043a\u0432\u0430\u0446\u0456\u0439\u043d\u0435 \u043d\u0435\u043e\u0431\u043c\u0435\u0436\u0435\u043d\u0435 \u043d\u0430\u0431\u043b\u0438\u0436\u0435\u043d\u043d\u044f \u0430\u043b\u0433\u0435\u0431\u0440 MS (\u0442\u043e\u0431\u0442\u043e, \u043d\u0430\u0439\u0431\u0456\u043b\u044c\u0448\u0438\u0439 \u0443\u043d\u0456\u0432\u0435\u0440\u0441\u0430\u043b\u044c\u043d\u0438\u0439\/[\u043a\u0432\u0430\u0437\u0456-]\u0435\u043a\u0432\u0430\u0446\u0456\u0439\u043d\u0438\u0439 \u043a\u043b\u0430\u0441 MS-\u0440\u0435\u0448\u0456\u0442\u043e\u043a \u0431\u0435\u0437 \u0447\u043b\u0435\u043d\u0456\u0432 \u0437 \u043e\u0431\u043e\u043c\u0430 \u043c\u0435\u0436\u0430\u043c\u0438, \u0430\u043b\u0435 \u0437 \u043c\u043e\u0436\u043b\u0438\u0432\u0456\u0441\u0442\u044e \u0440\u043e\u0437\u0448\u0438\u0440\u0435\u043d\u043d\u044f \u0431\u0435\u0437 \u0430\u043b\u0433\u0435\u0431\u0440\u0438 MS) \u0437 \u0447\u043b\u0435\u043d\u0430\u043c\u0438, \u044f\u043a\u0456 \u0454 \u0442\u043e\u0447\u043d\u043e \u043a\u0432\u0430\u0437\u0456\u0441\u0438\u043b\u044c\u043d\u0438\u043c\u0438 \u0433\u0440\u0430\u0442\u0430\u043c\u0438 MS, \u0437\u043d\u0430\u0439\u0434\u0435\u043d\u043e 29\/75-\u0435\u043b\u0435\u043c\u0435\u043d\u0442\u043d\u0443 \u043d\u0435\u043b\u0430\u043d\u0446\u044e\u0433\u043e\u0432\u0443 \u0440\u043e\u0437\u043f\u043e\u0434\u0456\u043b\u044c\u043d\u0443\/\u043d\u0435\u0440\u043e\u0437\u043f\u043e\u0434\u0456\u043b\u044c\u043d\u0443 \u0440\u0435\u0448\u0456\u0442\u043a\u0443 \u043a\u0432\u0430\u0437\u0456\u0440\u0456\u0437\u043d\u043e\u0432\u0438\u0434\u0456\u0432 \u043a\u0432\u0430\u0437\u0456\u0441\u0438\u043b\u044c\u043d\u0438\u0445\/\u043c\u0430\u0439\u0436\u0435 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