Trees, as in connected graphs without cycles, have an interesting property. Given any two nodes, there is a unique path connecting them. Therefore, if we have some map from the set of paths to the set of nodes, we define a magma on a given tree.
What are some nice compatibilities we can prescribe between the tree and its magma? There are two that are obvious to me.
We say a map from paths to nodes, or by extension a tree magma defined with it, respects nodes if a path always evaluates to a node it passes through.
We say a tree magma respects paths if the product of the path and is equal to the path , or equationally: . A tree magma satisfying both compatibilities is said to be arboreal.
I think we can connect this to semilattices somehow, but it is not as simple as I thought. Further musing required.