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Completeness theorems for universal and implicational logics of algebras via congruences


Author: Robert W. Quackenbush
Journal: Proc. Amer. Math. Soc. 103 (1988), 1015-1021
MSC: Primary 03C05; Secondary 08C10, 08C15
DOI: https://doi.org/10.1090/S0002-9939-1988-0954975-7
MathSciNet review: 954975
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Abstract: In this paper, simple algebraic proofs are given for the completeness theorems for the implicational and universal logics of algebras. The proofs are obtained by examining congruences, $ \theta $, on the algebra of terms, $ F(\omega )$, such that $ F(\omega )/\theta $ belongs to the given class of algebras. Thus, they are direct analogs of G. Birkhoff's proof of the completeness theorem for equational logic.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0954975-7
Keywords: Completeness theorem, quasivariety, universal class of algebras
Article copyright: © Copyright 1988 American Mathematical Society