Abstract
We generalize D. Kelly's and K. A. Nauryzbaev's results of 1-variable and 2-variable equational compactness of complete distributive lattices satisfying the infinite distributive law and its dual (“bi-frames”) to objects similar to monadic algebras (which we will callprojection algebras). This will lead us in particular to an example of bi-frame that is not 3-variable equationally compact, even forcountable equation systems, thus solving a problem of G. Gratzer. This example is realized as a certain complete sublattice of the complete Boolean algebra of regular open subsets of some Polish space.
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