Near-unanimity polymorphisms on reflexive graphs have many nice characterizations. Semilattice polymorphisms, another main weak near-unanimity polymorphisms, is less studied.

We define several types of semi-lattice polymorphisms on reflexive graphs based on how the edges of the graph are situated in relation to semi-lattice ordering.

Considering the classes of reflexive graphs that admit semi-lattice polymorphisms with various combinations of these properties, we get a hierarchy of graph classes that in a natural way generalize chordal graphs.

We look at various aspects of this hierarchy with the goal of characterizing both the classes of reflexive graphs admitting semi-lattice polymorphisms of various types, and the intersection of these classes with the classes of graphs admitting near-unanimity polymorphisms.