On semigroups having \(n^ 2\) essentially \(n\)-ary polynomials (Q1802258)

If \(\mathcal V\) is a variety of algebras and \(n\) is a nonnegative integer, then \(p_ n({\mathcal V})\) denotes the number of essentially \(n\)-ary term operations of an algebra generating \(\mathcal V\). The authors consider the following question about an arbitrary semigroup variety \(\mathcal V\). When is it true that \(p_ n({\mathcal V}) = n^ 2\) for every nonnegative integer \(n\)? A conjecture, which the authors attribute to G. Grätzer and A. Kisielewicz, is that the only such variety is the variety of all normal bands. In this paper they prove that the conjecture is correct.