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arXiv · 2202.02460

Jónsson Jónsson-Tarski Algebras

Jordan DuBeau
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Abstract

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 $\text{cf}(|J|) > 2^{|L|^+}$ 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 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.

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Jordan DuBeau. 2022-09-10. Jónsson Jónsson-Tarski Algebras. https://arxiv.org/abs/2202.02460

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