A note on large minimally free algebras
Let A be a (finitary) universal algebra. A is called \(\kappa\)-free (\(\kappa\) a cardinal) provided there exists a subset \(X\subseteq A\) of cardinality \(\kappa\) such that every function f: \(X\to A\) extends uniquely to an endomorphism \(\phi\) : \(A\to A\). So 0-free is the same as rigid. \textit{I. Kříž} and \textit{A. Pultr} [Algebra Univ. 21, 46-53 (1985; Zbl 0597.08010)] have shown that every variety containing arbitrarily large rigid algebras must also contain arbitrarily large \(\kappa\)-free algebras for any fixed \(\kappa\). The author provides a short and elegant proof of this result (in fact, of a more general theorem) by a clever modification of Birkhoff's well-known construction of the free algebra on \(\kappa\) generators relative to the variety generated by A. This approach is completely different from that of Kříž and Pultr.
scientific article; zbMATH DE number 4134124
scientific article