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The lattice of varieties of *-regular band monoids. (English) Zbl 0767.20023

A \(*\)-regular band is a band equipped with an involution \(x\to x^*\) that satisfies \(xx^*x = x\). In this short note, the author finds all varieties of *-regular band monoids, in a similar fashion to her description [ibid. 33, 187-198 (1986; Zbl 0591.20060)] of all varieties of band monoids. The lattice is obtained from the natural numbers (under the usual order) by adjoining an identity at the top and a four-element diamond at the bottom.

MSC:

20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties

Citations:

Zbl 0591.20060
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References:

[1] Adair, C. L.,Bands with Involution, J. Algebra75 (1982), 297–314. · Zbl 0501.20040 · doi:10.1016/0021-8693(82)90042-4
[2] Birjukov, A. P.,The lattice of varieties of idempotent semigroups, All-Union Colloquium on General Algebra, Riga (1967), 16–18.
[3] Fennemore, C.,All varieties of bands, Math. Nachr.48 (1971), 237–262. · Zbl 0194.02801 · doi:10.1002/mana.19710480118
[4] Gerhard, J. A.,The lattice of equational classes of idempotent semigroups, J. Algebra15 (1970), 195–224. · Zbl 0194.02701 · doi:10.1016/0021-8693(70)90073-6
[5] Wismath, S. L.,The Lattices of Varieties and Pseudovarieties of Band Monoids, Semigroup Forum33 (1986), 187–198. · Zbl 0591.20060 · doi:10.1007/BF02573192
[6] Wismath, S. L.,The Lattices of Varieties and Pseudovarieties of Band Monoids, M.Sc. Thesis, Simon Fraser University, 1983. · Zbl 0591.20060
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