The cardinality of subdirectly indecomposable systems in quasivarieties. (English. Russian original) Zbl 0602.08006
Algebra Logic 25, 1-34 (1986); translation from Algebra Logika 25, No. 1, 3-50 (1986).
The structural characterization of the residually small (r.s.) varieties obtained by W. Taylor [Algebra Univers. 2, 33-53 (1972; Zbl 0263.08005)] is generalized to quasivarieties. Using this generalization the author proves that the quasivarieties of abelian groups and of unars are r.s.
The syntactic characterization of the r.s. varieties, which is contained in Taylor’s paper mentioned, is generalized to quasivarieties. Thus it is given a negative answer to a question of J. T. Baldwin and J. D. Berman [Algebra Univers. 5, 379-389 (1975; Zbl 0348.08002)].
Let \(\gamma\) be a cardinal. A quasivariety K is said to be residually \(<\gamma\) iff \(| A| <\gamma\) for any subdirectly K-indecomposable system. K is said to be residually finite (countable) iff it is residually \(<\omega\) \((<\omega_ 1)\). The author finds a syntactic characterization of the residually \(<\gamma\) quasivarieties for any \(\gamma\). Thus he solves the problem of syntactic characterization of the residually finite and the residually countable varieties posed by Baldwin and Berman.
The Hanf number for subdirect indecomposability in quasivarieties is found. In assuming GCH there is given a characterization of the spectra of powers of subdirectly indecomposable systems in quasivarieties of uncountable signature. These results are closely related to some results of R. McKenzie and S. Shelah [Proc. Symp. Pure Math. 25, 53- 74 (1974; Zbl 0316.02057)].
The theorem of R. Magari [Ann. Univ. Ferrara, Nov. Ser. 14, 1-4 (1969; Zbl 0247.08016)] on existence of prime algebras in varieties is generalized to universally axiomatized classes. In conclusion the author insists that the algebraic characterization of quasivarieties (lemma 1.2) was originally indicated (”apparently”) by the author and V. I. Tumanov in 1982.
The syntactic characterization of the r.s. varieties, which is contained in Taylor’s paper mentioned, is generalized to quasivarieties. Thus it is given a negative answer to a question of J. T. Baldwin and J. D. Berman [Algebra Univers. 5, 379-389 (1975; Zbl 0348.08002)].
Let \(\gamma\) be a cardinal. A quasivariety K is said to be residually \(<\gamma\) iff \(| A| <\gamma\) for any subdirectly K-indecomposable system. K is said to be residually finite (countable) iff it is residually \(<\omega\) \((<\omega_ 1)\). The author finds a syntactic characterization of the residually \(<\gamma\) quasivarieties for any \(\gamma\). Thus he solves the problem of syntactic characterization of the residually finite and the residually countable varieties posed by Baldwin and Berman.
The Hanf number for subdirect indecomposability in quasivarieties is found. In assuming GCH there is given a characterization of the spectra of powers of subdirectly indecomposable systems in quasivarieties of uncountable signature. These results are closely related to some results of R. McKenzie and S. Shelah [Proc. Symp. Pure Math. 25, 53- 74 (1974; Zbl 0316.02057)].
The theorem of R. Magari [Ann. Univ. Ferrara, Nov. Ser. 14, 1-4 (1969; Zbl 0247.08016)] on existence of prime algebras in varieties is generalized to universally axiomatized classes. In conclusion the author insists that the algebraic characterization of quasivarieties (lemma 1.2) was originally indicated (”apparently”) by the author and V. I. Tumanov in 1982.
Reviewer: S.R.Kogalovskij
Keywords:
quasivarieties of abelian groups; syntactic characterization; r.s. varieties; residually finite; residually countable varieties; Hanf number; subdirect indecomposability in quasivarieties; GCH; spectra of powers of subdirectly indecomposable systems in quasivarieties; prime algebrasReferences:
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