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Structure of lattices of nilpotent semigroup varieties. (English) Zbl 0813.20064

Bonzini, C. (ed.) et al., Semigroups: algebraic theory and applications to formal languages and codes. Proceedings of the international conference on semigroups, held in Luino, Italy, from 22nd to 27th June, 1992. Singapore: World Scientific. 297-299 (1993).
Let \(G\) be a group acting on a set \(M\). A pair \((\rho,N)\) is called coherent if \(\rho\) is a congruence on the \(G\)-set \(M\), \(N\) is a \(G\)- subset of \(M\) and \(N\) is a \(\rho\)-class. For any congruence \(\rho\), \((\rho,\emptyset)\) is considered coherent. The set \(CP(M)\) is a lattice under inclusion and contains \(\text{Con}(M)\) as a sublattice. Let \(W_ n^ k\) be the set of all words of length \(n\) on exactly \(k\) letters, \(k \leq n\). We can consider \(W_ n^ k\) as an \(S_ n\)-set where \(S_ n\) is the group of permutations of \(n\) letters. Let \(\text{Int}_ n\) (\(\text{Int}_ n^ k\)) be the interval of all nilpotent semigroup varieties \(V\) such that (a) all words of length greater than \(n\) (and all words of length \(n\) depending on less than \(k\) letters) are 0 in \(V\) and (b) all words of length less than \(n\) (and all words of length \(n\) depending on more than \(k\)-letters) are isoterms relative to \(V\). The main result in the paper is Theorem: The interval \(\text{Int}_ n^ k\) is dually isomorphic to the lattice \(CP(W_ n^ k)\). The interval \(\text{Int}_ n\) is a subdirect product of the intervals \(\text{Int}_ n^ k\), \(k = 1,2,\dots,n\).
For the entire collection see [Zbl 0799.00023].

MSC:

20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties
08A30 Subalgebras, congruence relations
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