Let G be a formal grammar of any Chomsky type (as defined by the production rules) and let the production rules establish the order among words of this grammar: words a and b are a<=b if a can be generated by b by the immediate use of production rule.
My question is: does the language (set of words) with such order relation forms the lattice? If this such order does not exist (maybe I have not worked out exactly) or if such order cannot form a lattice, then - is there different order on set of words that can form the lattice nevertheless? Or maybe set of words forms a different structure, not the lattice?
The context of my question: there is nice theory of complex systems in physics, but the state of each element of the complex system (e.g. particle, physically modeled agent (e.g. predator/prey in the Euclidean space)) is described by the vector in the phase space R^n. I would like to consider the complex systems whose elements are knowledge based - e.g. each element is the agent having knowledge base. Knowledge base if set of formulas - effectively - each knowledge base is the long single conjunction of all some formulas. So - some word in some grammar describes the state of each element of such complex systems. What is what is - I would like to find mathematical structure that describes the set of words and consider the functions on such structure, e.g. functions can describe the metric on the state space: how far one state (word) is from another state (word). Such functions, of course, rely on the semantics of the language (e.g. how many semantical steps should be made to go from one word/state to another word/state, possibly goal state) and not on the syntax (e.g. syntactical editing distance). My hypothesis is that set of grammatically correct words can form the lattice and there is whole theory of functions on the lattice and from that theory I can take my functions/metrics to describe the state space of the element of the complex system and, possibly, the environment as well.
So - is the set of grammatically correct words a lattice? And is there already some work in the complex systems research on the complex systems which uses languages as state spaces and not the usual vector spaces as the state spaces? Google gives nothing, but maybe I am used just bad keywords. Thanks!