Interpolation and amalgamation properties in varieties of equivalential algebras

Abstract

Important positive as well as negative results on interpolation property in fragments of the intuitionistic propositional logic (INT) were obtained by J. I. Zucker in [6]. He proved that the interpolation theorem holds in purely implicational fragment of INT. He also gave an example of a fragment of INT for which interpolation fails. This fragment is determined by the constant falsum (⊥), well known connectives: implication (→) and conjunction (∧), and by a ternary connective δ defined as follows: δ (p, q, r)= df (p∧q)∨(ℸp∧r). Extending this result of J. I. Zucker, G. R. Renardel de Lavalette proved in [5] that there are continuously many fragments of INT without the interpolation property. This paper is meant to continue the research mentioned above. To be more precise, its aim is to answer questions concerning interpolation and amalgamation properties in varieties of equivalential algebras, particularly in the variety determined by the purely equivalential fragment of INT.