Laws between the singleton law and the constant law?

Stream: Equational

[Harald Husum]

Does there exist (equational) laws strictly between the singleton law and the constant law? If so, how many operations are needed to express them?

[Terence Tao]

All non-trivial constant magmas satisfy exactly the same set of laws (those laws with at least one application of the operation on both sides, as well as the trivial law x=x), so the answer is "no".

[Harald Husum]

That implies one can divide the edges in the Hasse diagram into two classes - let me call them atomic and non-atomic. The edge between the singleton law and the constant law can be called atomic because it won't be replaced by a more complex structure if we were to look into laws with more operations. Other edges clearly hide further equivalence classes and can be considered non-atomic.

I'm sure this observation has been made already and that there is better terminology for this.

In general, do we know which edges are atomic or not? Are there further examples of atomic ones, presumably on the "strong" side of the Hasse diagram?

[Harald Husum]

Instead of atomicity, I guess one can talk about "completeness". 2 ⇒ 46 is a connection in a cut-set of the Hasse diagram that is "complete" in the sense that there is nothing more to discover as one generalizes the project to more operations.

If this "complete" cut-set is nontrivial, it is in it self an interesting structure to keep track of. My intuition on this is poor, but I'm imagining that such a subset could grow as the number of operations under consideration grows, albeit at a slower rate than the overall Hasse diagram.