Jónsson Jónsson–Tarski algebras

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 o...

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Veröffentlicht in:Algebra universalis 2023-08, Vol.84 (3), Article 26
1. Verfasser: DuBeau, Jordan
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description 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.
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title Jónsson Jónsson–Tarski algebras
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