Finite semigroups with slowly growing \(p_n\)-sequences (Q1866814)

The \(p_n\)-sequence of an algebra \({\mathcal A}\) is a sequence of cardinalities \(p_n(A)\) of all essentially \(n\)-ary term operations of \({\mathcal A}\). It is polynomially bounded if there exist a positive constant \(c\) and an natural number \(r\) such that \(p_n(A)\leq cn^r\) holds for \(n\geq 1\). The paper contains a characterization of finite semigroups having polynomially bounded \(p_n\)-sequences in terms of semigroup identities. In addition, the authors supply an effective procedure for deciding whether a finite semigroup has polynomially bounded \(p_n\)-sequences.