It was proved by the authors that the quasivariety of quasi-Stone algebras \(\mathbf {Q}_{\mathbf {1,2}}\) is finite-to-finite universal relative to the quasivariety \(\mathbf {Q}_{\mathbf {2,1}}\) contained in \(\mathbf {Q}_{\mathbf {1,2}}\). In this paper, we prove that \(\mathbf {Q}_{\mathbf {1,2}}\) is not Q-universal. This provides a positive answer to the following long standing open question: Is there a quasivariety that is relatively finite-to-finite universal but is not Q-universal?

中文翻译：

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一个相对有限到有限的普遍但不是 Q 普遍的拟变量

作者证明了准斯通代数的拟变异性\(\mathbf {Q}_{\mathbf {1,2}}\)相对于拟变异性\(\mathbf {Q }_{\mathbf {2,1}}\)包含在\(\mathbf {Q}_{\mathbf {1,2}}\)中。在本文中，我们证明了\(\mathbf {Q}_{\mathbf {1,2}}\)不是 Q-universal。这为以下长期悬而未决的问题提供了积极的答案：是否存在相对有限到有限普遍但不是 Q 普遍的准变量？