Covers of some varietal products
The author studies the location of cover pairs in the lattice of interpretability types of varieties. Recently the existence of a minimum variety strictly above each primal variety was proved. Now the meet of two primal varieties is considered. Let \(\mathcal P_n\) be the variety generated by the \(n\)-element primal algebra \(\mathcal P_n\), \(\mathcal X\) the variety with basic operations \(0\), \(1\) and \(\cdot \) satisfying \(0\cdot x\approx 0\), \(1\cdot x\approx x\). It is shown that if \(\mathcal V\) is a variety above \(\mathcal X\) and \(n\) is not in the spectrum of \(\mathcal V\) then \(\mathcal V\wedge \mathcal P_n\) has a minimum element above it in the interval \([\mathcal V\wedge \mathcal P_n,\mathcal V]\). Further the existence of a proper class of cover pairs, elements with any finite number of covers, and elements with denumerably many covers are shown. Finally, the non-existence of a 3-element cover chain with 3 distinct spectra is proved.
lattice of interpretability types of varieties
scientific article; zbMATH DE number 1226198