On interpretations of varieties with semilattice reduct (Q804615)

A variety V is interpretable in a variety W (V\(\leq W)\) if for every fundamental V-operation \(f_ t\), \(t\in T\), there exists a W-term \(\alpha_ t\) such that for every algebra \(A\in W\) the algebra \((A;\alpha_ t,t\in T)\in V\). The assignment I: \(f_ t\mapsto \alpha_ t\), \(t\in T\), is called an interpretation of V in W. The paper delivers a simplified approach of results of R. Levin and of W. Taylor on interpretations of distributive lattices in Heyting algebras and of n- element linearly ordered Heyting algebra in \(n+1\)-element linearly ordered Heyting algebras. Among other results the following is shown for varieties V, W of Heyting algebras which are generated by three finite members. If \(V\leq W\) and W properly contains the variety of Boolean algebras then there is exactly one interpretation of V in W. If \(V\nleq W\) then V is not interpretable in W.