The structure of polynomial operations associated with smooth digraphs

Abstract

With every digraph we associate an algebra whose fundamental operations are the polymorphisms of the digraph. In a 2012 paper, the second and third authors proved that the digraph of endomorphisms of any finite connected reflexive digraph is connected, provided that the algebra associated with the digraph lies in a congruence join-semidistributive over modular variety. In the same paper, this connectivity result led to a proof of the statement that if the algebra associated with a finite reflexive digraph generates a congruence modular variety, then the digraph has a near-unanimity polymorphism. A digraph is smooth if it has no sinks and no sources. Smooth digraphs of algebraic length 1 are a broad generalization of reflexive digraphs. In a 2009 paper, Barto et al. proved that every finite smooth digraph of algebraic length 1 whose associated algebra lies in a congruence meet-semidistributive over modular variety has a loop edge. This is a powerful theorem that has nice applications in algebra and computer science. In the present paper, we prove that the digraph of unary polynomial operations of the algebra associated with a finite smooth connected digraph of algebraic length 1 is connected, provided that the algebra lies in a congruence join-semidistributive over modular variety. This generalizes our connectivity result mentioned above and implies a restricted version of the result of Barto et al. in the congruence join-semidistributive over modular case. We also give a characterization of locally finite idempotent congruence join-semidistributive over modular varieties via smooth compatible digraphs of algebraic length 1. It remains as an open question whether the congruence modularity of the variety generated by the algebra associated with a finite smooth digraph of algebraic length 1 implies the existence of a near-unanimity polymorphism of the digraph.