The structure of polynomial operations associated with smooth digraphs. (Q485112)

Let \(G\) be a finite connected smooth (i.e., having neither sources nor sinks) digraph of algebraic length 1, and let \(\mathrm{Alg}(G)\) be the algebra on the vertex set of \(G\) with the polymorphisms of \(G\) as fundamental operations; assume that \(\mathrm{Alg}(G)\) generates a congruence join-semidistributive over modular variety. Then the digraph of all unary polynomial operations of \(\mathrm{Alg}(G)\) is connected. This generalizes a theorem of \textit{M. Maróti} and \textit{L. Zádori} [Discrete Math. 312, No. 15, 2316-2328 (2012; Zbl 1245.05059)]. Moreover, \(G\) must have a loop edge. This conclusion resembles an analogous result of \textit{L. Barto} et al. [SIAM J. Comput. 38, No. 5, 1782-1802 (2009; Zbl 1191.68460)].