On algebras with unique essentially \(n\)-ary operations (Q922568)

For a universal algebra \(\underline A=(A,F)\) and a positive integer \(n>0\), let \(p_ n(\underline A)\) denote the number of essentially n-ary polynomials over \(\underline A\) (in the sense of Grätzer's book ``Universal algebra'') and \(p_ 0(\underline A)\) the number of constant unary polynomials over \(\underline A\). Let B denote the variety of commutative semigroups in which the identities \(xx=yy\) and \(xxy=y\) hold and C the variety of commutative semigroups in which \(xx=yy\) and \(xxy=xx\) hold. Furthermore, let \(C_{\omega}\) be the class of all algebras \(\underline A\) in C such that no identity of the form \(x_ 1x_ 2....x_{k+1}=xx\) \((k>0)\) holds in \(\underline A\). For any nontrivial algebra \(\underline A\) in B we have \(p_ n(\underline A)=1\) for all \(n\geq 0\), and the same holds for any algebra \(\underline A\) in \(C_{\omega}\). The author shows that the classes B and \(C_{\omega}\) can be characterized by the condition \(p_ n=1\) in the following sense: If \(\underline A=(A,f)\) is an algebra with exactly one fundamental operation f and \(p_ n(\underline A)=1\) for all \(n\geq 0\), then \(\underline A\) is polynomially equivalent either to a nontrivial algebra in B or to a member of \(C_{\omega}\).