Reflexive Graphs with Near Unanimity but no Semilattice Polymorphisms

Abstract

We show that every generator, in a certain set of generators for the variety of reflexive near unanimity graphs, admits a semilattice polymorphism. We then find a retract of a product of such graphs (paths, in fact) that has no semilattice polymorphism. This verifies for reflexive graphs that the variety of graphs with semilattice polymorpisms does not contain the variety of graphs with near-unanimity, or even $3$-ary near-unanimity polymorphisms.