\(p_n\)-sequences of algebras with one fundamental operation
The \(p_n\)-sequence of an algebra is the sequence \((p_0,p_1,\dots)\), where \(p_n\) is the number of distinct essentially \(n\)-ary term operations of the algebra if \(n>0\), and \(p_0\) is the number of distinct constant unary term operations. It is known that in general almost every sequence \((p_0,p_1,\dots)\) satisfying \(p_1>0\) can be represented as the \(p_n\)-sequence of a suitable algebra; see \textit{G.~Grätzer}, \textit{J.~Płonka} and \textit{A.~Sekanina} [Colloq. Math. 22, 9--11 (1970; Zbl 0211.32102)]. The authors show that the situation is quite different for algebras with only one fundamental operation. Namely, they prove that in this case the inequality \(p_1\geq p_0\) must hold. Moreover, if \(2<p_0<\aleph_0\), then \(p_1>2p_0\).
scientific article; zbMATH DE number 5129543