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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
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
Summary:
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.
References:
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