Every 𝑚-permutable variety satisfies the congruence identity 𝛼𝛽_{ℎ}=𝛼𝛾_{ℎ}

Lipparini, Paolo

doi:10.1090/S0002-9939-07-09337-9

Every $m$-permutable variety satisfies the congruence identity $\alpha \beta _h= \alpha \gamma _h$
HTML articles powered by AMS MathViewer

by Paolo Lipparini

Proc. Amer. Math. Soc. 136 (2008), 1137-1144

DOI: https://doi.org/10.1090/S0002-9939-07-09337-9

Published electronically: December 5, 2007

PDF | Request permission

Abstract:

It is known that congruence lattices of algebras in $m$-permutable varieties satisfy non-trivial identities; however, the identities discovered so far are rather artificial and seem to have little intrinsic interest. We show here that every $m$-permutable variety satisfies the well-known and well-studied congruence identity $\alpha \beta _h= \alpha \gamma _h$. We also get a new condition equivalent to $m$-permutability.

References

Gábor Czédli, A characterization for congruence semidistributivity, Universal algebra and lattice theory (Puebla, 1982) Lecture Notes in Math., vol. 1004, Springer, Berlin, 1983, pp. 104–110. MR 716177, DOI 10.1007/BFb0063432
Joachim Hagemann and A. Mitschke, On $n$-permutable congruences, Algebra Universalis 3 (1973), 8–12. MR 330010, DOI 10.1007/BF02945100
David Hobby and Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. MR 958685, DOI 10.1090/conm/076
Bjarni Jónsson, Congruence varieties, Algebra Universalis 10 (1980), no. 3, 355–394. MR 564122, DOI 10.1007/BF02482916
Bjarni Jónsson and Ivan Rival, Lattice varieties covering the smallest nonmodular variety, Pacific J. Math. 82 (1979), no. 2, 463–478. MR 551703
Keith A. Kearnes, Congruence join semidistributivity is equivalent to a congruence identity, Algebra Universalis 46 (2001), no. 3, 373–387. MR 1857204, DOI 10.1007/PL00000351
K. Kearnes, personal communication, December 2004.
P. Lipparini, $n$-permutable varieties satisfy nontrivial congruence identities, Algebra Universalis 33 (1995), no. 2, 159–168. MR 1318980, DOI 10.1007/BF01190927
Paolo Lipparini, Congruence identities satisfied in $n$-permutable varieties, Boll. Un. Mat. Ital. B (7) 8 (1994), no. 4, 851–868 (English, with Italian summary). MR 1315822
Paolo Lipparini, A non-trivial congruence implication between identities weaker than modularity, Acta Sci. Math. (Szeged) 68 (2002), no. 3-4, 593–609. MR 1954538
Paolo Lipparini, Commutator theory without join-distributivity, Trans. Amer. Math. Soc. 346 (1994), no. 1, 177–202. MR 1257643, DOI 10.1090/S0002-9947-1994-1257643-7
P. Lipparini, A congruence identity satisfied by $m$-permutable varieties, arXiv: math.GM/0508548 (2005).
Walter Taylor, Some applications of the term condition, Algebra Universalis 14 (1982), no. 1, 11–24. MR 634412, DOI 10.1007/BF02483903