Remove the permutation assumption on Phi

Remove the assumption that every component function in every tuple belonging to the collection Φ is a permutation of its corresponding alphabet in the reductions relating Φ-triviality for unary, binary, and arbitrary-arity polymorphisms and in the connection with impossibility domains under unanimity.

Background

The reduction theorem and the theorem relating Φ-triviality to impossibility domains assume that each component function of every tuple in Φ is a permutation. This hypothesis is used to control the structure of unary polymorphisms and to invert the component functions.

The authors explicitly identify removing this assumption as an open question, seeking versions of these results for more general collections Φ that may contain non-permutations.