{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T00:11:38Z","timestamp":1699920698480},"reference-count":19,"publisher":"Wiley","issue":"6","license":[{"start":{"date-parts":[[2010,11,9]],"date-time":"2010-11-09T00:00:00Z","timestamp":1289260800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2010,12]]},"abstract":"Abstract<\/jats:title>A quasivariety is said to be implicative<\/jats:italic> if it is generated by a class of algebras with equationally\u2010definable implication of equalities. Implicative finitely\u2010generated quasivarieties appear naturally within logic, for instance, as equivalent quasivarieties of Gentzen\u2010style calculi for finitely\u2010valued propositional logics with equality determinant (cf. [17], [18, Subsection 7.5] and Section A). Furthermore, any discriminator quasivariety is implicative. We prove that, for any implicative locally\u2010finite quasivariety \u211a and any skeleton S of the class of all finite \u211a\u2010simple members of \u211a, the image of the first component of a natural Galois connection between the dual poset of subquasivarieties of \u211a and the poset of all sets of finite subsets of S is the closure system of all U<\/jats:italic>S<\/jats:sub>\u2010ideals<\/jats:italic> of the poset \u3008S, \u2aaf\u3009, where \u2aaf is the embeddability relation and U<\/jats:italic>S<\/jats:sub> is the up\u2010set on S constituted by all members of S having a one\u2010element subalgebra, with closure basis determined by the sets of all principal and non\u2010empty finitely\u2010generated up\u2010sets on S. It is also shown that the first component of the Galois connection under consideration is injective if and only if, for each finite sequence magnified image<\/jats:alt-text><\/jats:graphic><\/jats:boxed-text> of members of S and any subdirect product \ud835\udd05 of magnified image<\/jats:alt-text><\/jats:graphic><\/jats:boxed-text>, there is a covering C<\/jats:italic> of the image of magnified image<\/jats:alt-text><\/jats:graphic><\/jats:boxed-text> such that, for each P<\/jats:italic> \u2208 \ud835\udc9e, \u220f P<\/jats:italic> is embeddable into \ud835\udd05. This condition (even with one\u2010element coverings) holds, for instance, when \u211a is discriminator or when \u211a has a set of binary terms satisfying semilattice identities while the direct product of any finite T<\/jats:italic> \u2286 S is generated by corresponding semilattice zeros. We present several examples of quasivarieties which are equivalent to Gentzenstyle calculi for certain well\u2010known finitely\u2010valued logics with equality determinant, satisfy the latter sufficient condition of injectivity but are not congruence\u2010permutable, some of them having non\u2010simple relatively\u2010simple algebras. Moreover, we give an example of a quasivariety equivalent to Gentzen\u2010style calculi for some wellknown finitely\u2010valued logics with equality determinant, for which the first component of the Galois connection involved is not injective. We also apply our general elaboration to some discriminator quasivarieties, one of which is equivalent to no Gentzen\u2010style calculus associated with any finitely\u2010valued logic having an equality determinant while is equivalent to an extension of such a calculus (\u00a9 2010 WILEY\u2010VCH Verlag GmbH & Co. KGaA, Weinheim)<\/jats:p>","DOI":"10.1002\/malq.200810161","type":"journal-article","created":{"date-parts":[[2010,11,9]],"date-time":"2010-11-09T08:34:48Z","timestamp":1289291688000},"page":"643-658","source":"Crossref","is-referenced-by-count":1,"title":["Subquasivarieties of implicative locally\u2010finite quasivarieties"],"prefix":"10.1002","volume":"56","author":[{"given":"Alexej P.","family":"Pynko","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,11,9]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093891610"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1006\/aima.2000.1962"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.2307\/2271149"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01111252"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8640.1988.tb00280.x"},{"key":"e_1_2_1_7_2","first-page":"41","article-title":"O pewnym, tr\u00f3jwarto\u015bciowym systemie rachunku zda\u0144","volume":"57","author":"Ha\u0142kowska K.","year":"1988","journal-title":"Acta Universitatis Wratislaviensis. Prace Filozoficzne"},{"key":"e_1_2_1_8_2","unstructured":"S. C.Kleene Introduction to Metamathematics (D. Van Nostrand Company 1952)."},{"key":"e_1_2_1_9_2","unstructured":"A. I.Mal'cev To a general theory of algebraic systems (in Russian). Mathematical Collection (New Seria)35 (77) 3\u201320 (1954)"},{"key":"e_1_2_1_10_2","unstructured":"A. I.Mal'cev Algebraic systems (Springer\u2010Verlag 1965)."},{"key":"e_1_2_1_11_2","unstructured":"V. M.Popov Sequential formulations of some paraconsistent logical systems (in Russian). In: Syntactic and semantic investigations of non\u2010extensional logics (V. A. Smirnov ed) pp. 285\u2013289 (Nauka 1989)."},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00258428"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(99)00007-2"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(98)00058-X"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1080\/11663081.1999.10510958"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(00)00173-4"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/s001530100115"},{"key":"e_1_2_1_18_2","first-page":"23","article-title":"Sequential calculi for many\u2010valued logics with equality determinant","volume":"33","author":"Pynko A. 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