On algebras and varieties with semilattice reducts (Q1802261)

\textit{W. Blok} [ibid. 27, 299-303 (1990; Zbl 0728.08010)] proved how some results of \textit{R. Lewin} [ibid. 24, 149-166 (1987; Zbl 0614.06009)], on interpretations in the variety of Heyting algebras, could be derived from a lemma concerning finite algebras having a special property with respect to a reduct in the variety \({\mathcal {SL}}_ 1\) of all semilattices with neutral element. The first aim of this note is to extend this lemma to all such algebras -- called by Blok semilattices with filter-preserving operations. As a consequence, we obtain that the lattice of subvarieties of Heyting algebras is dually embeddable in the lattice of interpretability types of varieties. In Algebra, logic and number theory, Proc. 15th Port.-Span. Meet. Math., Evora/Port. 1990, Vol. I, 37-42 (1991; Zbl 0743.06004), the author gave a characterization of algebras with a reduct in \({\mathcal {SL}}\) having a property stronger than the one considered by Blok; namely, the property \((H)\) of having a congruence lattice isomorphic to the lattice of all filters of a semilattice reduct, under an isomorphism which maps each congruence \(\theta\) to a \(\theta\)- class. We now characterize those varieties \({\mathcal V}\) having the \(H\)- property with respect to an interpretation of \({\mathcal {SL}}_ 1\) in \({\mathcal V}\).