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Modular elements of the lattice of semigroup varieties. II. (English) Zbl 1108.20058

Dorfer, G. (ed.) et al., Proceedings of the 70th workshop on general algebra “70. Arbeitstagung Allgemeine Algebra”, Vienna, Austria, May 26–29, 2005. Klagenfurt: Verlag Johannes Heyn (ISBN 3-7084-0194-8/pbk). Contributions to General Algebra 17, 173-190 (2006).
In contrast with the lattice of group varieties, the lattice \(L\) of semigroup varieties is not modular. It is of interest to examine the extent to which the ‘modular implication’ \(x\leq z\rightarrow (x\vee y)\wedge z=x\vee(y\wedge z)\) holds in \(L\). An element \(a\) of \(L\) is ‘modular’ if this implication holds whenever \(a\) plays the role of \(y\). Likewise it is ‘lower- [upper-] modular’ if it holds whenever \(a\) plays the role of \(x\) [resp. \(z\)].
In part I [Contrib. Gen. Algebra 16, 275-288 (2005; Zbl 1090.20030)], the second author answered the question of which semigroup varieties are both modular and lower-modular in \(L\). The authors now answer the dual question. Apart from the variety of all semigroups, a variety is both modular and upper-modular in \(L\) if and only if it is contained in the join of the variety of semilattices with the variety defined by the identities \(x^2y=0\), \(xy=yx\). As part of the proof, they prove that a variety of nilsemigroups is upper-modular in \(L\) if and only if it satisfies \(x^2y=xy^2\), \(xy=yx\).
For the entire collection see [Zbl 1089.08001].

MSC:

20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties
20M05 Free semigroups, generators and relations, word problems

Citations:

Zbl 1090.20030
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