Can Language Homomorphisms Map Truth Across Different Symbolic Systems?

[mXSCNT]

Hey, I was thinking of this generalization of homomorphisms. You have a language L_1 = (A, B) where A is a set of symbols and B is a set of sequences of symbols in A. Given languages L_1 = (A, B) and L_2 = (C, D) a function f: A \rightarrow C is defined to be a homomorphism of languages if, given any sequence a_1 a_2 ... a_n \in B, we have f(a_1) f(a_2) ... f(a_n) \in D.

This is seen as a generalization. For example, if A is a set of group elements, and B is a set of sentences of the form "a_1 a_2 a_3" (the multiplication table, a_1 * a_2 = a_3), then every homomorphism of languages from A to the alphabet of another language set up like this corresponds to a group homomorphism.

Another example: if A is a set of ring elements together with special elements * + =, and B is a set of sentences of the form "a_1 * a_2 = a_3" (the multiplication table), unioned with a set of sentences of the form "a_1 + a_2 = a_3" (the addition table), then homomorphisms of languages that map * to * and + to + and = to = correspond to ring homomorphisms.

To look at it in general, A is a set of objects, and B is a set of true statements about elements of A. A homomorphism of languages f : A \rightarrow C induces a map from statements in B to corresponding statements in D that "maps truth to truth" (doesn't generate false statements, i.e. strings outside of D).What do you think? Have you heard of this?

 

[economicsnerd]

If I'm not mistaken, this is related to the notion of a homomorphism as it appears in model theory.

 

[mXSCNT]

If I'm not mistaken, this is related to the notion of a homomorphism as it appears in model theory.

Yeah, it is just like that. Thanks!

Another note. f : A \rightarrow C induces a map h : B \rightarrow D. Specifically, h takes in a string in B and maps each character in the string to another single character, such that the result is always in D. I wonder if we can generalize further, to other kinds of functions h. For instance, maybe there is something interesting if we consider functions h : B \rightarrow D where the mapping of h is given by an arbitrary finite state transducer (with outputs always in D). What parts of the structure of B would such a mapping preserve?

 

[Stephen Tashi]

I think of the idea of "homomorphism" as implicitly involving the ideas of "kernel" and "quotient". If we have a mapping between "things" that "preserves" operations but the things don't necessarily have an implementation of an "identity" thing, is there a name for such a mapping other than "homomorphism"? (I suppose one could resort to the "functor" of category theory.)