> For instance you say you want a "monoidal partial order which also an upper semilattice with arbitrary joins". But if I click on the link you gave, we find that an upper semilattice means a partial order with all non-empty finite joins. So once we see what an upper semilattice is, we realise that it is redundant in the above sentence. You need a monoidal partial order which has arbitrary joins.
By the way, as you hinted, there's a difference between the order theorist's definition of 'semilattice' and the category theorist's.
An order theorist says an upper semilattice, or join-semilattice, is a poset with joins of all _non-empty_ finite sets. A category theorist will dislike that italicized exception, and say a join-semilattice is a poset with joins of _all_ finite sets. An order theorist will call this a join-semilattice _with a bottom element \\(\bot\\) (namely the join of the empty set).
Unsurprisingly, I am with the category theorists on this one!
The same disagreement afflicts the definition of lower semilattice or meet-semilattice, and of lattice.
My definition of 'quantale' in [Lecture 63](https://forum.azimuthproject.org/discussion/2295/lecture-63-chapter-4-composing-enriched-profunctors/p1) is also nonstandard - neither the usual one, nor Fong and Spivak's! My quantales are all skeletal (posets rather than preorders), because I see no great advantage in discussing isomorphic-but-not-equal objects in what we're doing now. That's a choice of convenience, not principle. And my quantales are all unital - because I prefer monoids to semigroups. That's a choice of principle!