Matthew, you keep editing your comments when people comment on them. This makes it difficult to make sense of other people's comments. It also makes it hard to follow the conversation and see how it is developing.

Let me make a further comment. You introduce some seemingly new jargon "upper semilattice" but don't tell us what it is, instead you link to a random part of a wikipedia page. Be explicit, tell us what this means, otherwise herein lies a good place for errors to creep in. Similarly tell us why you need this new jargon. There's a danger of mixing terminology from different areas, namely category theory type monoidal preorder language and lattice theory in that they are based on different perspectives, although they might be equivalent, there's plenty of scope for confusing things. It's fine to say a monoidal preorder with a widget is the same as sprocket semi-lattice or something if that helps clarify, but usually you would need to prove such a thing.

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.

And when you say "Let me suggest a new definition", I think you mean to say that you are citing an established definition, rather than coming up with a new one of your own.