Interpolation and amalgamation properties in varieties of equivalential algebras (Q1821777)

Zucker and Renardel de Lavalette studied the interpolation property in fragments of intuitionistic propositional logic. The equivalential fragment of this logic determines a variety E, which can also be axiomatized by the following identities due to Kabziński and Wroński: \((a\leftrightarrow a)\leftrightarrow b=b\), \(a\leftrightarrow (a\leftrightarrow (b\leftrightarrow c))=(a\leftrightarrow b)\leftrightarrow (a\leftrightarrow c)\), ((a\(\leftrightarrow b)\leftrightarrow ((a\leftrightarrow c)\leftrightarrow c))\leftrightarrow ((a\leftrightarrow c)\leftrightarrow c)=a\leftrightarrow b\). The author proves that both the amalgamation and the interpolation property fail for E. It is also proved that the only nontrivial subvariety of E having both properties is the subvariety generated by a two-element chain.