On modular elements of the lattice of semigroup varieties

Boris M. Vernikov

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 4, page 595-606
  • ISSN: 0010-2628

Abstract

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A semigroup variety is called modular if it is a modular element of the lattice of all semigroup varieties. We obtain a strong necessary condition for a semigroup variety to be modular. In particular, we prove that every modular nil-variety may be given by 0-reduced identities and substitutive identities only. (An identity u = v is called substitutive if the words u and v depend on the same letters and v may be obtained from u by renaming of letters.) We completely determine all commutative modular varieties and obtain an essential information about modular varieties satisfying a permutable identity.

How to cite

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Vernikov, Boris M.. "On modular elements of the lattice of semigroup varieties." Commentationes Mathematicae Universitatis Carolinae 48.4 (2007): 595-606. <http://eudml.org/doc/250215>.

@article{Vernikov2007,
abstract = {A semigroup variety is called modular if it is a modular element of the lattice of all semigroup varieties. We obtain a strong necessary condition for a semigroup variety to be modular. In particular, we prove that every modular nil-variety may be given by 0-reduced identities and substitutive identities only. (An identity $u=v$ is called substitutive if the words $u$ and $v$ depend on the same letters and $v$ may be obtained from $u$ by renaming of letters.) We completely determine all commutative modular varieties and obtain an essential information about modular varieties satisfying a permutable identity.},
author = {Vernikov, Boris M.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semigroup; variety; nil-variety; 0-reduced identity; substitutive identity; permutable identity; lattice of subvarieties; modular element of a lattice; upper-modular element of a lattice; modular semigroup varieties; lattices of varieties; modular elements},
language = {eng},
number = {4},
pages = {595-606},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On modular elements of the lattice of semigroup varieties},
url = {http://eudml.org/doc/250215},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Vernikov, Boris M.
TI - On modular elements of the lattice of semigroup varieties
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 4
SP - 595
EP - 606
AB - A semigroup variety is called modular if it is a modular element of the lattice of all semigroup varieties. We obtain a strong necessary condition for a semigroup variety to be modular. In particular, we prove that every modular nil-variety may be given by 0-reduced identities and substitutive identities only. (An identity $u=v$ is called substitutive if the words $u$ and $v$ depend on the same letters and $v$ may be obtained from $u$ by renaming of letters.) We completely determine all commutative modular varieties and obtain an essential information about modular varieties satisfying a permutable identity.
LA - eng
KW - semigroup; variety; nil-variety; 0-reduced identity; substitutive identity; permutable identity; lattice of subvarieties; modular element of a lattice; upper-modular element of a lattice; modular semigroup varieties; lattices of varieties; modular elements
UR - http://eudml.org/doc/250215
ER -

References

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  1. Aĭzenštat A.Ya., On some sublattices of the lattice of semigroup varieties, in E.S. Lyapin (ed.), Contemp. Algebra, Leningrad State Pedagogical Institute, Leningrad, 1 (1974), 3-15 (Russian). MR0412306
  2. Evans T., The lattice of semigroup varieties, Semigroup Forum 2 (1971), 1-43. (1971) Zbl0225.20043MR0284528
  3. Ježek J., The lattice of equational theories. Part I: Modular elements, Czechoslovak Math. J. 31 (1981), 127-152. (1981) MR0604120
  4. Ježek J., McKenzie R.N., Definability in the lattice of equational theories of semigroups, Semigroup Forum 46 (1993), 199-245. (1993) MR1200214
  5. Mel'nik I.I., On varieties of Ø m e g a -algebras, in V.V. Vagner (ed.), Investig. on Algebra, Saratov State University, Saratov, 1 (1969), 32-40 (Russian). MR0253966
  6. Mel'nik I.I., On varieties and lattices of varieties of semigroups, in V.V. Vagner (ed.), Investig. on Algebra, Saratov State University, Saratov, 2 (1970), 47-57 (Russian). Zbl0244.20071MR0412307
  7. Pollák Gy., On the consequences of permutation identities, Acta Sci. Math. (Szeged) 34 (1973), 323-333. (1973) MR0322084
  8. Saliĭ V.N., Equationally normal varieties of semigroups, Izv. Vysš. Učebn. Zaved. Matematika no. 5 (1969), 61-68 (Russian). (1969) MR0263956
  9. Vernikov B.M., Special elements of the lattice of overcommutative varieties of semigroups, Mat. Zametki 70 (2001), 670-678 (Russian); English translation: Math. Notes 70 (2001), 608-615. (2001) Zbl1036.20052MR1882341
  10. Vernikov B.M., Upper-modular elements of the lattice of semigroup varieties, Algebra Universalis, to appear. Zbl1161.08002MR2470588
  11. Vernikov B.M., Lower-modular elements of the lattice of semigroup varieties, Semigroup Forum, to appear. Zbl1143.20039MR2353282
  12. Vernikov B.M., Volkov M.V., Lattices of nilpotent semigroup varieties, L.N. Shevrin (ed.), Algebraic Systems and their Varieties, Ural State University, Sverdlovsk (1988), pp.53-65 (Russian). Zbl0813.20064
  13. Vernikov B.M., Volkov M.V., Commuting fully invariant congruences on free semigroups, Contributions to General Algebra, 12 (Vienna, 1999), pp.391-417, Heyn, Klagenfurt, 2000. Zbl0978.20028MR1777677
  14. Vernikov B.M., Volkov M.V., Modular elements of the lattice of semigroup varieties II, Contributions to General Algebra, 17, pp.173-190, Heyn, Klagenfurt, 2006. Zbl1108.20058MR2237815
  15. Volkov M.V., Commutative semigroup varieties with distributive subvariety lattices, Contributions to General Algebra, 7, pp.351-359, Hölder-Pichler-Tempsky, Vienna, 1991. Zbl0762.20022MR1143098
  16. Volkov M.V., Semigroup varieties with commuting fully invariant congruences on free objects, Contemp. Math., 131, Part 3, pp.295-316, American Mathematical Society, Providence, 1992. Zbl0768.20025MR1175889
  17. Volkov M.V., Modular elements of the lattice of semigroup varieties, Contributions to General Algebra, 16, pp.275-288, Heyn, Klagenfurt, 2005. Zbl1108.20058MR2166965

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