On the minimal extension of the sequence \(\langle 0,1,1,7\rangle \) (Q1272125)

The \(p_n\)-sequence of an algebra \(A\) is the sequence \(\langle p_0,p_1,p_2,\ldots \rangle \) of cardinalities of the sets of essentially \(n\)-ary operations over \(A\). Having the 4-tuple \(\langle 0,1,1,7\rangle \) there is the question whether there exists a \(p_n\)-sequence beginning with this 4-tuple and having the minimal possible cardinalities \(p_i\). The authors show that \(\langle 0,1,1,7\rangle \) has this minimal extension property in the class of all algebras with a nonassociative binary operation.