On idempotent algebras with \(p_ n({\mathfrak A})=2n\) (Q1090353)

Let \({\mathfrak A}=(A,F)\) be an algebra. The polynomial \(f(x_ 1,x_ 2,...,x_ n)\) in \({\mathfrak A}\) is said to be essentially n-ary if it depends on each variable \(x_ 1,x_ 2,...x_ n\). The number of essentially n-ary polynomials in \({\mathfrak A}\) is denoted by \(p_ n({\mathfrak A})\). If every polynomial in \({\mathfrak A}\) satisfies the condition \(f(x,x,...,x)=x\), the algebra \({\mathfrak A}\) is called idempotent. The main result states that \(p_ n({\mathfrak A})=2n\) for all \(n\geq 2\) in an idempotent algebra \({\mathfrak A}\) iff \({\mathfrak A}\) is polynomially equivalent to an algebra (B\(\times C,\cdot,\circ)\) of type (2,2) where \((b_ 1,c_ 1)\cdot (b_ 2,c_ 2)=(b_ 1,c_ 2)\) and \((b_ 1,c_ 1)\circ (b_ 2,c_ 2)=(b_ 1,c_ 1+c_ 2)\) for some semilattice operation \(+\) on C.