Abstract.
We completely determine all commutative semigroup varieties that are upper-modular elements of the lattice of all semigroup varieties. It is verified that if a semigroup variety is an upper-modular element of this lattice and different from the variety of all semigroups then it is a periodic variety and every nilsemigroup in the variety is commutative and satisfies the identity x2y = xy2.
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Dedicated to George Grätzer and E. Tamás Schmidt on their 70th birthdays
The work was supported by the Russian Foundation for Basic Research (grant No. 06-01-00613).
Received December 30, 2006; accepted in final form February 20, 2008.
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Vernikov, B.M. Upper-modular elements of the lattice of semigroup varieties. Algebra univers. 59, 405–428 (2008). https://doi.org/10.1007/s00012-008-2108-7
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DOI: https://doi.org/10.1007/s00012-008-2108-7
Keywords and phrases:
- semigroup
- variety
- lattice of subvarieties
- commutative semigroup
- upper-modular element
- distributive lattice
- periodic variety
- nil-variety