Rich language and refinements of cheap-talk equilibria

I formalize the notion of equilibrium language, define the relation of having a richer language, and show the existence of equilibria with maximally rich language. I also study the stability of equilibria with respect to inflows of messages that can enrich their language, and provide examples of implausible cheap-talk equilibria, not refined away by the existing refinements, which can be refined away on the grounds of stability properties.