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Pseudovarieties, generalized varieties and similarly described classes. (English) Zbl 0548.08007

For a class of algebras, H(K), S(K), P(K), \(P_ f(K)\), Pow(K) and E(K) denote respectively the classes of homomorphic images, subalgebras, direct products, finite direct products, direct powers, and elementary subalgebras of members of K. The classical theorem of G. Birkhoff [Proc. Camb. Philos. Soc. 31, 433-454 (1935; Zbl 0013.00105)] shows that K is a variety if and only if \(K=HSP(K)\). This paper is concerned with the relationship between pseudovarieties, which are classes of finite algebras closed under H, S and \(P_ f\), and generalised varieties, which are classes of algebras satisfying one of the following four equivalent conditions: 1. K is closed under H, S, \(P_ f\) and Pow; 2. \(K=HSP_ fPow(K)\); 3. K is the union of some directed family of varieties; 4. there exists a filter F over E such that for all algebras A, \(A\in K\leftrightarrow Id(A)\in F\) (where Id(A) denotes the set of identities true in A). In fact, it is shown that a pseudovariety consists precisely of the finite members of some generalised variety. Relationships between these and similar classes of algebras and systems of identities are also investigated and a table of results provided.
Reviewer: Sh.Oates-Williams

MSC:

08C15 Quasivarieties
08B99 Varieties
03C05 Equational classes, universal algebra in model theory

Citations:

Zbl 0013.00105
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References:

[1] Chang, C. C.; Keisler, H. J., Model Theory (1973), North-Holland: North-Holland Amsterdam · Zbl 0276.02032
[2] Eilenberg, S.; Schützenberger, M. P., On pseudovarieties, Adv. in Math., 19, 413-418 (1976) · Zbl 0351.20035
[3] Grätzer, G., Universal Algebra (1968), Van Nostrand: Van Nostrand Princeton, N.J · Zbl 0182.34201
[4] Shelah, S., Every two elementarily equivalent models have isomorphic ultrapowers, Israel J. Math., 10, 224-233 (1971) · Zbl 0224.02045
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