Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question: http://mathoverflow.net/questions/24723/chain-hierarchy-of-monoids. My application domain is reasoning about modifiers of modifiers in software product line engineering, thus lacking established mathematical background. I find it easier to adapt existing results, even if the application domain is significantly different, so any help would be appreciated.