Universal varieties of quasi-Stone algebras

Abstract

The lattice of varieties of quasi-Stone algebras ordered by inclusion is an \({\omega+1}\) chain. It is shown that the variety \({\mathbf{Q_{2,2}}}\) (of height 13) is finite-to-finite universal (in the sense of Hedrlin and Pultr). Further, it is shown that this is sharp; namely, the variety \({\mathbf{Q_{3,1}}}\) (of height 12) is not finite-to-finite universal and, hence, no proper subvariety of \({\mathbf{Q_{2,2}}}\) is finite-to-finite universal. In fact, every proper subvariety of \({\mathbf{Q_{2,2}}}\) fails to be universal. However, \({\mathbf{Q_{1,2}}}\) (the variety of height 9) is shown to be finite-tofinite universal relative to \({\mathbf{Q_{2,1}}}\) (the variety of height 8). This too is sharp; namely, no proper subvariety of \({\mathbf{Q_{1,2}}}\) is finite-to-finite relatively universal. Consequences of these facts are discussed.