Hypersubstitutions for the variety of star bands (Q2758076)

A star-band is an algebra of type \(\langle 2,1\rangle\) which is a band under the binary operation and satisfies the following laws: \(x^{**}=x\) (involution law), \((xy)^*=y^*x^*\) (product law), \(xx^*x=x\) (absorption law). The lattice of all star-band varieties is countable and distributive, it was completely determined by \textit{C. L. Adair} [J. Algebra 75, 297-314 (1982; Zbl 0501.20040)]; see also \textit{M. Petrich} [Semigroup Forum 59, 141-151 (1999; Zbl 0946.20032)].NEWLINENEWLINENEWLINEIn the article under review the author applies the theory of hypersubstitutions and hypervarieties to varieties of star-bands. The normal form for hypersubstitutions for the variety \(StB\) of all star-bands is determined. Furthermore, the author finds the monoid of all star-band-proper hypersubstitutions. One more result states that none of the five identities defining a basis for \(StB\) (namely, the associative, idempotent, involution, product and absorption laws) is a hyperidentity for \(StB\). For each of these five identities the author determines a list of all star-band varieties in which this identity holds as hyperidentity: it is the variety of normal star-bands and its subvarieties only for the associative law, the trivial variety only for the product law, the variety of semilattice star-bands and the trivial variety only for the idempotent, involution and absorption laws. Finally, it is verified that the variety \(StB\) does have some hyperidentities of any arity.