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Strictly finite schema axiomatization of quasi-polyadic algebras. (English) Zbl 0751.03033

Algebraic logic, Pap. Colloq., Budap./Hung. 1988, Colloq. Math. Soc. János Bolyai 54, 539-571 (1991).
[For the entire collection see Zbl 0741.00041.]
J. D. Monk proved [J. Symb. Log. 34, 331-343 (1969; Zbl 0181.300)] that the class of representable \(CA\)’s of finite dimension is not finitely axiomatizable. In an accompanying paper [ibid. 34, 344-352 (1969; Zbl 0181.300)], J. S. Johnson did the same for quasipolyadic algebras (\(QPA\)’s). Moreover, Monk also showed that, for \(\alpha\) infinite, the class of representable \(CA_ \alpha\)’s is not axiomatizable by a finite number of equation schemata. The authors of the paper under review discuss why the expected analogue of this latter result has not been, and could not has been, stated for \(QPA\)’s. They conclude that quasipolyadic algebras must be given another kind of definition more suitable to Monk’s methods. To this end, finitary polyadic algebras (\(FPA\)’s) and finitary polyadic equality algebras (\(FQEA\)’s) are introduced. Roughly, an \(FPA\) (resp. \(FPEA\)) is a Pinter’s quantifier algebra (resp. quantifier algebra with equality [see C. C. Pinter, Notre Dame J. Formal Log. 14, 361-366 (1973; Zbl 0245.02053)]) expanded by a family of unary operations each of which permutes two variables. For \(\alpha>2\), the classes \(FPA_ \alpha\) and \(QPA_ \alpha\), as well as \(FPEA_ \alpha\) and \(QPEA_ \alpha\), turn out to be term-definitionally equivalent; the elaborated proof of these equivalences make the bulk of the paper. The principal result (the proof of which is not so detailed) of the paper states that, indeed, neither the class of representable \(FPA_ \alpha\)’s nor that of representable \(FPEA_ \alpha\)’s is finite schema axiomatizable when \(\alpha>2\).

MSC:

03G15 Cylindric and polyadic algebras; relation algebras
08B99 Varieties

Software:

QPA
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