×
Dziobiak, W.; Ježek, J.; Maróti, M.

Minimal varieties and quasivarieties of semilattices with one automorphism. (English) Zbl 1171.08002

Semigroup Forum 78, No. 2, 253-261 (2009).
The authors describe all minimal quasivarieties and all minimal varieties in the variety \(S\) of semilattices with one automorphism (considered as algebras with one binary and two unary operations). They remark that the lattice of varieties contained in \(S\) is uncountable.
Reviewer: Alexander Ivanovich Budkin (Barnaul)

Cited in 1 Review
Cited in 3 Documents

MSC:

08C15 Quasivarieties
06A12 Semilattices
08B15 Lattices of varieties

Keywords:

semilattice; automorphism; variety; quasivariety
PDF BibTeX XML Cite
\textit{W. Dziobiak} et al., Semigroup Forum 78, No. 2, 253--261 (2009; Zbl 1171.08002)
Full Text: DOI

References:

[1] Adams, M.E., Adaricheva, K.A., Dziobiak, W., Kravchenko, A.V.: Open questions related to the problem of Birkhoff and Maltsev. Stud. Log. 78, 357–378 (2004) · Zbl 1105.08007
[2] Gorbunov, V.A.: Algebraic Theory of Quasivarieties. Siberian School of Algebra and Logic. Consultants Bureau, New York (1998)
[3] Ježek, J.: On atoms in lattices of primitive classes. Comment. Math. Univ. Carol. 11, 515–532 (1970) · Zbl 0211.03002
[4] Ježek, J.: Subdirectly irreducible semilattices with an automorphism. Semigroup Forum 43, 178–186 (1991) · Zbl 0770.08004
[5] Kalicki, J.: The number of equationally complete classes of equations. Indag. Math. 17, 660–662 (1955) · Zbl 0073.24601
[6] Kalicki, J., Scott, D.: Equational completeness of abstract algebras. Indag. Math. 17, 650–659 (1955) · Zbl 0073.24501
[7] Kearnes, K., Szendrei, A.: A characterization of minimal locally finite varieties. Trans. Am. Math. Soc. 349, 1749–1768 (1997) · Zbl 0871.08005
[8] Mal’cev, A.I.: Algebraic Systems. Nauka, Moscow (1970) (in Russian)
[9] McKenzie, R., McNulty, G., Taylor, W.: Algebras, Lattices, Varieties, vol. I. Wadsworth & Brooks/Cole, Monterey (1987) · Zbl 0611.08001
[10] Szendrei, A.: A survey on strictly simple algebras and minimal varieties. In: Universal Algebra and Quasigroup Theory, Jadwisin, 1989. Res. Exp. Math., vol. 19, pp. 209–239. Heldermann, Berlin (1992) · Zbl 0769.08001
[11] Shafaat, A.: On implicational completeness. J. Can. Math. Soc. 26, 761–768 (1974) · Zbl 0295.08003
[12] Shafaat, A.: The number of proper minimal quasivarieties of groupoids. Proc. Am. Math. Soc. 49, 54–58 (1975) · Zbl 0307.08005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.