We show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig’s Interpolation Theorem holds but a strong form of this theorem does not hold. Translating these results into Algebraic Logic we obtain a finitely axiomatizable subvariety of finite dimensional Representable Cylindric Algebras that has the Strong Amalgamation Property but does not have the Superamalgamation Property. This settles a conjecture of Pigozzi .
"On weak and strong interpolation in algebraic logics." J. Symbolic Logic 71 (1) 104 - 118, March 2006. https://doi.org/10.2178/jsl/1140641164