Addition Operations in Complete Lattices

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. ${}{}$

• Well, if it's bounded (so it has a top element), you'll never get your "strictly larger" requirement. You should probably rethink what you want exactly (that "etc" bothers me as well). – Todd Trimble May 30 '16 at 19:38
• Thanks Todd. Sorry the question was unclear. Of course, I assume that the 'greater than' requirement only applies to non maximal elements. I guess this implies that L would have to be dense in some intuitive sense. – King Kong May 30 '16 at 21:26
• A reasonable notion of an “addition operation” may be to make $L$ an interval in an (abelian) l-group. In which case the lattice better be distributive, but that’s likely not a sufficient condition. – Emil Jeřábek May 30 '16 at 21:39
• Thanks for the suggestion Emil. If the idea is that $L$ can be represented as an interval in the lattice of subgroups of an Abelian group (assuming distributivity), but what then is the natural notion of addition in this context (i.e. how do we add subgroups)? – King Kong Jun 6 '16 at 9:24
• For distributive lattices, check out Rota's idea of a valuation ring. See Rota, G-C. Proceeding University of Houston Lattice Theory Conference, 1973, pp 574-628. – Walter Bruce Sinclair Jul 20 '16 at 8:26