All creatures great and small (Q2821662)

A clone on a set \(X\) is a set of finitary operations \(f:X^n\to X\) which contains all the projections and is closed under composition, or equivalently, a clone is the set of all term functions on some universal algebra over \(X\). The family of all clones forms a complete lattice \(\text{Cl}(X)\). The greatest element of this lattice is the clone consisting of all finitary operations on \(X\). For \(|X|=1\) the lattice \(\text{Cl}(X)\) is trivial; for \(|X|=2\) the lattice \(\text{Cl}(X)\) is countable; and for \(|X|\geq3\) the lattice \(\text{Cl}(X)\) is uncountable. It is also known that the lattice \(\text{Cl}(X)\) on any finite set \(X\) is dually atomic. NEWLINENEWLINENEWLINEIn [Trans. Am. Math. Soc. 357, No. 9, 3525--3551 (2005; Zbl 1081.08006)], the authors proved that assuming the continuum hypothesis the lattice \(\text{Cl}(X)\) is not dually atomic for countable infinite sets \(X\). In the paper under review, the authors prove that if \(|X|=\lambda\) and \(2^\lambda=\lambda^+\) the lattice of clones on the set \(X\) is not dually atomic.