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 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 ℵ 1 . We also construct 2 ℵ 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.