J\'onsson J\'onsson-Tarski Algebras

TL;DR

This paper investigates the size and diversity of Jónsson-Tarski algebras, establishing limitations on their cardinalities and constructing many nonisomorphic examples, thereby advancing understanding of their structural properties.

Contribution

It provides new results on the maximum size of Jónsson algebras in Jónsson-Tarski varieties and constructs a large family of nonisomorphic such algebras.

Findings

Jónsson algebras in this variety cannot exceed size ℵ₁.

Constructed 2^{ℵ₁} nonisomorphic Jónsson algebras.

Identified conditions preventing large Jónsson algebras in certain varieties.

Abstract

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.