And the fact that it being a monoid is actually irrelevant to the solution. It is a lattice with an artificially created lower bound [1], which does induce a mo

And the fact that it being a monoid is actually irrelevant to the solution. It is a lattice with an artificially created lower bound [1], which does induce a monoid structure via its join operation, but by thinking about it as a monoid, you abstract out the one relevant property (ordering!) that you actually care about. [1] Assuming that there aren't any duplicate elements, in which case it technically isn't either a lattice or a monoid, unless you consider equivalence classes modulo attribute equality.