Jónsson Jónsson–Tarski algebras

By studying the variety of Jónsson–Tarski algebras, we demonstrate two obstacles to the existence of large Jónsson algebras in certain varieties. First, if an algebra J in a language L has cardinality greater than \(|L|^+\) and a distributive subalgebra lattice, then it must have a proper subalgebra of size |J|. Second, if an algebra J in a language L satisfies \({{\,\textrm{cf}\,}}(|J|) > 2^{|L|^+}\) and lies in a residually small variety, then it again must have a proper subalgebra of size |J|. We apply the first result to show that Jónsson algebras in the variety of Jónsson–Tarski algebras cannot have cardinality greater than \(\aleph _1\). We also construct \(2^{\aleph _1}\) many pairwise nonisomorphic Jónsson algebras in this variety, thus proving that for some varieties the maximum possible number of Jónsson algebras can be achieved.