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Primitive lattice varieties

Peter Jipsen

Department of Mathematics, Chapman University, Orange, CA 92866, USA

E-mail Address: [email protected]

Corresponding author.

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and

J. B. Nation

Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA

E-mail Address: [email protected]

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https://doi.org/10.1142/S021819672250031XCited by:0

Abstract

A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition (W). For example, if every finite subdirectly irreducible lattice in a locally finite variety $𝒱$ satisfies Whitman’s condition (W), then $𝒱$ is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are $2^{ℵ_{0}}$ such varieties. Some lattices that fail (W) also generate primitive varieties. But if I is a (W)-failure interval in a finite subdirectly irreducible lattice $L$ , and $L [I]$ denotes the lattice with I doubled, then $𝕍 (L [I])$ is never primitive.

Communicated by K. Kearnes

Keywords:

AMSC: 06B20, 08B15

References

1. M. Adams and W. Dziobiak , Q-universal quasivarieties of algebras, Proc. Amer. Math. Soc. 120 (1994) 1053–1059. ISI, Google Scholar
2. C. Bergman , Structural completeness in algebra and logic, in Algebraic Logic, Colloquia Mathematica Societatis János Bolyai, Vol. 54, eds. I. N. H. AndrékaJ. D. Monk (North-Holland, Amsterdam, 1991), pp. 59–73, https://faculty.sites.iastate.edu/cbergman/files/inline-files/StructComp.pdf. Google Scholar
3. B. A. Davey and B. Sands , An application of Whitman’s condition to lattices with no infinite chains, Algebra Universalis 7 (1977) 171–178. Crossref, Google Scholar
4. A. Day , A simple solution to the word problem for lattices, Canad. Math. Bull. 13 (1970) 253–254. Crossref, Google Scholar
5. A. Day , Doubling constructions in lattice theory, Canad. J. Math. 44 (1992) 252–269. Crossref, Google Scholar
6. R. Freese, J. Ježek and J. Nation , Free Lattices, Mathematical Surveys and Monographs, Vol. 42 (American Mathematical Society, Providence, 1995). Crossref, Google Scholar
7. R. Freese and J. Nation , Covers in free lattices, Trans. Amer. Math. Soc. 288 (1985) 1–43. Crossref, Google Scholar
8. V. Gorbunov , Covers in lattices of quasivarieties and independent axiomatizability, Algebra Logic 16 (1977) 340–369. Crossref, Google Scholar
9. V. Gorbunov , Algebraic Theory of Quasivarieties (Plenum, New York, 1998). Google Scholar
10. G. Grätzer and H. Lakser , The lattice of quasivarieties of lattices, Algebra Universalis 9 (1979) 102–115. Crossref, Google Scholar
11. J. Hyndman and J. Nation , The Lattice of Subquasivarieties of a Locally Finite Quasivariety, CMS Books in Mathematics (Springer, New York, 2018). Crossref, Google Scholar
12. J. Ježek and V. S. K , Primitive lattices, Czechoslovak Math J. 29 (1979) 595–634. Crossref, Google Scholar
13. P. Jipsen, Finite s. i. lattices, https://github.com/jipsen/Finite-s.i.-lattices (2021). Google Scholar
14. P. Jipsen and H. Rose , Varieties of lattices (Chap. 1), in Lattice Theory: Special Topics and Applications, Vol. 2, eds. G. GrätzerF. Wehrung (Birkhäuser, Cham, 2016), pp. 1–26. Crossref, Google Scholar
15. B. Jónsson and J. Nation , A report on sublattices of a free lattice, in Universal Algebra and Lattice Theory, Contributions to Universal Algebra, Colloquia Mathematics Societatis János Bolyai, Vol. 17 (North-Holland, 1977), pp. 223–257. Crossref, Google Scholar
16. B. Jónsson and I. Rival , Lattice varieties covering the smallest nonmodular variety, Pacific J. Math. 82 (1979) 463–478. Crossref, ISI, Google Scholar
17. R. McKenzie , Equational bases and non-modular lattice varieties, Trans. Amer. Math. Soc. 174 (1972) 1–43. Crossref, ISI, Google Scholar
18. J. Nation , Some varieties of semidistributive lattices, in Proceedings of the Charleston Conference on Lattice Theory and Universal Algebra (Springer, 1985), pp. 198–223. Crossref, Google Scholar
19. J. Nation , A counterexample to the finite height conjecture, Order 13 (1996) 1–9. Crossref, Google Scholar
20. V. Slavík , A note on subquasivarieties of some varieties of lattices, Comment. Math. Univ. Carolin. 16 (1975) 173–181. Google Scholar
21. P. M. Whitman , Free lattices, Ann. of Math. (2) 42 (1941) 325–330. Crossref, Google Scholar

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Vol. 32, No. 04

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History

Received 20 February 2021

Accepted 18 January 2022

Published: 28 March 2022

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Primitive lattice varieties

Abstract

References

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