Approximate Hagemann–Mitschke Co-operations

We show that varietal techniques based on the existence of operations of a certain arity can be extended to n-permutable categories with binary coproducts. This is achieved via what we call approximate Hagemann–Mitschke co-operations, a generalisation of the notion of approximate Mal’tsev co-operation [2]. In particular, we extend characterisation theorems for n-permutable varieties due to J. Hagemann and A. Mitschke [8, 9] to regular categories with binary coproducts