Given a complete bounded lattice $L$, what do we know about the possibility of defining an addition operation $+$ that, broadly speaking, behaves like arithmetical addition? By this, I mean that as well as being associative, commutative, unital etc, it also allows us to add equal elements to get something strictly larger (this rules out the join operation). Of course, if $L$ is not linearly ordered, then we can't assume the existence of unique successors to define addition in the usual way.

I'm particularly interested in the case where $L$ isn't linearly ordered, and for my purposes, it can work if $+$ is only a partial function that isn't necessarily defined on all elements. Sorry for the vague question, but any ideas or pointers to relevant literature would be much appreciated. ${}{}$