J\'onsson J\'onsson-Tarski Algebras

By studying the variety of J\'{o}nsson-Tarski algebras, we demonstrate two obstacles to the existence of large J\'{o}nsson 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 $\text{cf}(|J|) > 2^{|L|^+}$ 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 J\'{o}nsson algebras in the variety of J\'{o}nsson-Tarski algebras cannot have cardinality greater than $\aleph_1$. We also construct $2^{\aleph_1}$ many pairwise nonisomorphic J\'{o}nsson algebras in this variety, thus proving that for some varieties the maximum possible number of J\'{o}nsson algebras can be achieved.