Galois connection for multiple-output operations

It is a classical result from universal algebra that the notions of polymorphisms and invariants provide a Galois connection between suitably closed classes (clones) of finitary operations f : B n → B, and classes (coclones) of relations r ⊆ B k. We will present a generalization of this duality to classes of (multi-valued, partial) functions f : B n → B m , employing invariants valued in partially ordered monoids instead of relations. In particular, our setup encompasses the case of permutations f : B n → B n , motivated by problems in reversible computing.