论文标题
JónssonJónsson-Tarski代数
Jónsson Jónsson-Tarski Algebras
论文作者
论文摘要
通过研究Jónsson-Tarski代数的种类，我们证明了在某些品种中存在大型Jónsson代数存在的两个障碍。首先，如果语言中的代数$ j $具有大于$ | l |^+$的基数和一个分布式subalgebra lattice，则必须具有适当的尺寸$ | j | $的子代理。其次，如果语言中的代数$ j $满足$ \ text {cf}（| j |）> 2^{| l |^+} $，并且位于残留的小品种中，那么它必须再次具有适当的subalgebra尺寸$ | j | $。我们采用第一个结果表明，Jónsson代数在各种Jónsson-Tarski代数中的基数不能大于$ \ aleph_1 $。我们还构建了$ 2^{\ aleph_1} $在此品种中许多成对的非构形jónsson代数，因此证明，对于某些品种，可以实现最大可能的jónsson代数数量。
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.
