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 $\mathrm{\mid }L{\mathrm{\mid }}^{+}$ and a distributive subalgebra lattice, then it must have a proper subalgebra of size $\mathrm{\mid }J\mathrm{\mid }$. Second, if an algebra $J$ in a language $L$ satisfies $\text{cf}(\mathrm{\mid }J\mathrm{\mid })>2^{\mathrm{\mid }L{\mathrm{\mid }}^{+}}$ and lies in a residually small variety, then it again must have a proper subalgebra of size $\mathrm{\mid }J\mathrm{\mid }$. 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 ${\mathrm{\aleph }}_1$. We also construct $2^{{\mathrm{\aleph }}_1}$ 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.