Thanks! Is there any relationship? Some of the graphical examples [1] in your first link look like they could have a relationship to lattices of the second link.

[1] https://en.wikipedia.org/wiki/Lattice_(order)#Examples

The common generalization of semilattices and groups are inverse semigroups which are one way of capturing the idea of partial, or local, symmetries.

https://en.wikipedia.org/wiki/Inverse_semigroup

I always find that name-clash very annoying.