The uniqueness condition for the double pushout transformation of algebras

The double pushout approach to the algebraic graph transformation of hypergraphs was invented 30 years ago and it has been generalized since then to more general objects, like for instance relational systems or total and partial unary algebras. We have recently introduced the double pushout transformation of partial and partly total algebras over an arbitrary signature. In this paper we study the uniqueness condition for these rewriting formalisms, which turns out to be given by a suitable generalization to partial algebras of the well known congruence extension property for total algebras.