Further Progress on the Algebra of Concurrency

I've further developed the algebra of concurrency, to the point where it now very much resembles vector space algebra. The primary difference is that I have meets and joins instead of sums. Differences remain, and the axioms for all these constructs are simply those from set algebra.

It comes as little surprise, given the vector space algebra similarities, but I think the semantic definitions of the resulting algebra are better given using metric spaces. It is known that a metric distance metric exists for any context-free grammar; a memory state can be described in a manner similar to a vector, which means that we can derive a distance metric between states as well (it should be obvious from this that the construct is closed under product spaces). I should be able to use known results from metric spaces to derive that outer product spaces are likewise metric, and to justify all the rules. Recursively-defined states describe a monotonically decreasing sequence of open sets (a Cauchy sequence), which is known to converge to a unique fixed-point.

I'll be working to completely formalize this in the near future.