Problem 17 of Grätzer and Kisielewicz (Q1272252)

If \(A\) is an algebra and \(n\in N\), then \(p_n(A)\) denotes the number of \(n\)-ary operations of \(A\) depending on all \(n\) variables. Let \({\mathcal C}\) be the class of algebras of type (2,2) in which the two basic operations are both commutative and idempotent. Problem 17 of Grätzer and Kisielewicz asks whether there is an \(A\in {\mathcal C}\) with \(p_2(A)=2\) and for all \(B\in {\mathcal C},\) if \(p_2(B)=2\) then \(p_n(B)\geq p_n(A)\) for all \(n\in N.\) J. Dudek has shown that if the answer is yes, then \(A\) can be taken to be the two-element lattice \(D_2\) and that to answer the question it suffices to compare the \(p_n\)-sequence of \(D_2\) with the \(p_n\)-sequence of two very specific four-element members of \({\mathcal C}.\) One of them is \(N_2.\) Using a computer, the authors show \(p_4(N_2)=p_4(D_2)\) but \(p_5(N_2)=2586<6894=p_5(D_2)\), which solves Problem 17 negatively.