Jónsson Jónsson-Tarski Algebras

2022/02/05 by Jordan DuBeau, DuBeau, Jordan
Computer Science · #Advanced Algebra and Logic #semigroups and automata theory #Logic, programming, and type systems

paper · pdf · doi:10.48550/arxiv.2202.02460

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 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 ℵ₁. We also construct 2^ℵ₁ many pairwise nonisomorphic Jónsson algebras in this variety, thus proving that for some varieties the maximum possible number of Jónsson algebras can be achieved.

Related

• An uncountable Jónsson algebra in a minimal variety
• Jónsson groups of various cardinalities
• Jonssons's theorem in non-modular varieties
• Adapted algebras for the Berenstein-Zelevinsky conjecture
• J-ternary algebras, structurable algebras, and Lie superalgebras