Reflexive digraphs with near unanimity polymorphisms

In this paper we prove that if a finite reflexive digraph admits Gumm operations, then it also admits a near unanimity operation. This is a generalization of similar results obtained earlier for posets and symmetric reflexive digraphs by the second author and his collaborators. In the special case of reflexive digraphs our new result confirms a conjecture of Valeriote that states that any finite relational structure of finite signature that admits Gumm operations also admits an edge operation. We also prove that every finite reflexive digraph that admits a near unanimity operation admits totally symmetric idempotent operations of all arities. Finally, the aforementioned results yield a polynomial-time algorithm to decide whether a finite reflexive digraph admits a near unanimity operation.