Geodesic
On modular elements of the lattice of semigroup varieties
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 4, pp. 595-606 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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A semigroup variety is called {\it 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 {\it 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.
A semigroup variety is called {\it 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 {\it 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.
Classification : 08B15, 20M07
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 
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     author = {Vernikov, Boris M.},
     title = {On modular elements of the lattice of semigroup varieties},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {595--606},
     year = {2007},
     volume = {48},
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     zbl = {1174.20324},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2007_48_4_a3/}
}
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Vernikov, Boris M. On modular elements of the lattice of semigroup varieties. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 4, pp. 595-606. http://geodesic.mathdoc.fr/item/CMUC_2007_48_4_a3/