Abstract
Let q(K) denote the least quasivariety containing a given class K of algebraic structures. Mal'cev [3] has proved that q(K) = ISP r(K)(1). Another description of q(K) is given in Grätzer and Lakser [2], that is, q(K) = ISPP u(K)2. We give here other proofs of these results. The method which enables us to do that is borrowed from prepositional logics (cf. [1]).
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References
J. Czelakowski, Model-theoretic Methods in Methodology of Propositional Calculi, The Polish Academy of Sciences, Institute of Philosophy and Sociology, Warszawa 1980.
G. Grätzer and H. Lakser, A note on the implicational class generated by a class of structures, Canadian Mathematical Bulletin 16 (1973), pp. 603–605.
A. I. Mal'cev. Several remarks on quasivarieties of algebraic systems (Russian), Algebra i Logika 5 (1966), pp. 3–9.
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Czelakowski, J., Dziobiak, W. Another proof that ISP r(K) is the least quasivariety containing K . Stud Logica 41, 343–345 (1982). https://doi.org/10.1007/BF00403333
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DOI: https://doi.org/10.1007/BF00403333