sources of $\operatorname{TSI}$ polymorphisms, are near-unanimity ($\operatorname{NU}$) polymorphisms, and semilattice ($\operatorname{SL}$) polymorphisms. While $\operatorname{NU}$ polymorphisms have been well studied, and the classes of graphs admitting them have several nice characterisations, no such study had been attempted for $\operatorname{SL}$ polymorphisms until [@HS], where we looked at the family of reflexive graphs admitting $\operatorname{SL}$ polymorphisms.

A (meet) semilattice ordering on the vertices of a graph defines a $2$-ary operation $\vee$ on the vertex set. If this operation is a polymorphism of the graph then it is an [*$\operatorname{SL}$ polymorphism*]{} of the graph. The problem of characterising reflexive graphs admitting $\operatorname{SL}$ polymorphsims was difficult, and we restricted our attention to those graphs $G$ that admit $\operatorname{SL}$ polymorphisms for which the Hasse diagram of the semilattice ordering is a tree, and a subgraph of $G$. We showed that the class of such graphs extends the class of chordal graphs. We were unable to say much in the case that the semilattice was not a tree. The other extreme is when the ordering is a lattice– the Hasse diagram is not a tree except in the trivial case that the lattice it is a chain. This leads us to consider lattice polymorphisms.

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