On congruences of G -sets

Boris M. Vernikov

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 3, page 603-613
  • ISSN: 0010-2628

Abstract

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We describe G -sets whose congruences satisfy some natural lattice or multiplicative restrictions. In particular, we determine G -sets with distributive, arguesian, modular, upper or lower semimodular congruence lattice as well as congruence n -permutable G -sets for n = 2 , 2 . 5 , 3 .

How to cite

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Vernikov, Boris M.. "On congruences of $G$-sets." Commentationes Mathematicae Universitatis Carolinae 38.3 (1997): 603-613. <http://eudml.org/doc/248081>.

@article{Vernikov1997,
abstract = {We describe $G$-sets whose congruences satisfy some natural lattice or multiplicative restrictions. In particular, we determine $G$-sets with distributive, arguesian, modular, upper or lower semimodular congruence lattice as well as congruence $n$-permutable $G$-sets for $n=2,2.5,3$.},
author = {Vernikov, Boris M.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$G$-set; congruence lattice; congruence distributivity; congruence modularity; congruence $n$-permutability; -set; congruence lattice; semimodularity; orbits; modularity; distributivity; -permutability; group action},
language = {eng},
number = {3},
pages = {603-613},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On congruences of $G$-sets},
url = {http://eudml.org/doc/248081},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Vernikov, Boris M.
TI - On congruences of $G$-sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 3
SP - 603
EP - 613
AB - We describe $G$-sets whose congruences satisfy some natural lattice or multiplicative restrictions. In particular, we determine $G$-sets with distributive, arguesian, modular, upper or lower semimodular congruence lattice as well as congruence $n$-permutable $G$-sets for $n=2,2.5,3$.
LA - eng
KW - $G$-set; congruence lattice; congruence distributivity; congruence modularity; congruence $n$-permutability; -set; congruence lattice; semimodularity; orbits; modularity; distributivity; -permutability; group action
UR - http://eudml.org/doc/248081
ER -

References

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  1. Berman J., On the congruence lattices of unary algebras, Proc. Amer. Math. Soc. 36 (1972), 34-38. (1972) Zbl0249.08002MR0309833
  2. Crawley P., Dilworth R.P., Algebraic Theory of Lattices, Prentice-Hall, 1973. Zbl0494.06001
  3. Egorova D.P., Unars with the congruence lattice of a special kind, in: V.V. Vagner (ed.), Investigations in Algebra, Saratov State University, Saratov 5 (1977), pp.3-19 Russian. MR0491412
  4. Egorova D.P., The congruence lattice of unary algebra, in: V.N. Salii (ed.), Ordered Sets and Lattices, Saratov State University, Saratov 5 (1978), pp.11-44 Russian. MR0547294
  5. Egorova D.P., Skornjakov L.A., On the congruence lattice of unary algebra, in: V.N. Salii (ed.), Ordered Sets and Lattices, Saratov State University, Saratov 4 (1977), pp.28-40 Russian. MR0480280
  6. McKenzie R.N., McNulty G.F., Taylor W.F., Algebras. Lattices. Varieties, Vol. I, Wadsworth & Brooks/Cole, Monterey, 1987. Zbl0611.08001MR0883644
  7. Skornjakov L.A., Unars, Colloq. Math. Soc. J. Bolyai 29 (1982), 735-743. (1982) Zbl0504.08006MR0660907
  8. Vernikov B.M., Distributivity, modularity, and related conditions in lattices of overcommutative semigroup varieties, Proc. Int. Conf. ``Semigroups and their Applications, including Semigroup Rings'', accepted. Zbl0970.20037
  9. Vernikov B.M., Volkov M.V., Commutative semigroup varieties with modular subvariety lattices, in: J. Rhodes (ed.), Monoids and Semigroups with Applications, World Scientific, Singapore, 1991, pp. 233-253. Zbl0797.20047MR1142380
  10. Vernikov B.M., Volkov M.V., Structure of lattices of nilpotent semigroup varieties, in: C. Bonzini, A. Cherubini and C. Tibiletti (eds.), Semigroups. Algebraic Theory and Applications to Formal Languages and Codes, World Scientific, Singapore, 1993, pp. 297-299. Zbl0813.20064MR1647267
  11. Vernikov B.M., Volkov M.V., Lattices of nilpotent semigroup varieties. II, Izv. Ural State University, Ser. Matem., Mechan., accepted Russian. 
  12. Vernikov B.M., Volkov M.V., Semimodular varieties of semigroups, to appear. MR1001692
  13. Vernikov B.M., Volkov M.V., Permutability of fully invariant congruences on relatively free semigroups, Acta Sci. Math. (Szeged), submitted. Zbl0889.20031
  14. Volkov M.V., Commutative semigroup varieties with distributive subvariety lattices, Contrib. to General Algebra 7 (1991), 351-359. (1991) Zbl0762.20022MR1143098
  15. Volkov M.V., Semigroup varieties with commuting fully invariant congruences on free objects, Contemp. Math. 131 part 3 (1992), 295-316. (1992) Zbl0768.20025MR1175889
  16. Volkov M.V., Young diagrams and the structure of the lattice of overcommutative semigroup varieties, in: P.M. Higgins (ed.), Transformation Semigroups, Proc. Int. Conf. held at the Univ. Essex, Univ. Essex, Colchester, 1994, pp.99-110. Zbl0880.20042MR1491912
  17. Volkov M.V., Identities in the Lattice of Semigroup Varieties, Doctor Sci. Dissertation, St. Petersburg, 1994 Russian. 

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