Hey, I was thinking of this generalization of homomorphisms. You have a language [itex]L_1 = (A, B)[/itex] where [itex]A[/itex] is a set of symbols and [itex]B[/itex] is a set of sequences of symbols in [itex]A[/itex]. Given languages [itex]L_1 = (A, B)[/itex] and [itex]L_2 = (C, D)[/itex] a function [itex]f: A \rightarrow C[/itex] is defined to be a homomorphism of languages if, given any sequence [itex]a_1 a_2 ... a_n \in B[/itex], we have [itex]f(a_1) f(a_2) ... f(a_n) \in D[/itex].
This is seen as a generalization. For example, if [itex]A[/itex] is a set of group elements, and [itex]B[/itex] is a set of sentences of the form "[itex]a_1 a_2 a_3[/itex]" (the multiplication table, [itex]a_1 * a_2 = a_3[/itex]), then every homomorphism of languages from [itex]A[/itex] to the alphabet of another language set up like this corresponds to a group homomorphism.
Another example: if [itex]A[/itex] is a set of ring elements together with special elements * + =, and [itex]B[/itex] is a set of sentences of the form "[itex]a_1 * a_2 = a_3[/itex]" (the multiplication table), unioned with a set of sentences of the form "[itex]a_1 + a_2 = a_3[/itex]" (the addition table), then homomorphisms of languages that map * to * and + to + and = to = correspond to ring homomorphisms.
To look at it in general, [itex]A[/itex] is a set of objects, and [itex]B[/itex] is a set of true statements about elements of [itex]A[/itex]. A homomorphism of languages [itex]f : A \rightarrow C[/itex] induces a map from statements in [itex]B[/itex] to corresponding statements in [itex]D[/itex] that "maps truth to truth" (doesn't generate false statements, i.e. strings outside of [itex]D[/itex]).
What do you think? Have you heard of this?
This is seen as a generalization. For example, if [itex]A[/itex] is a set of group elements, and [itex]B[/itex] is a set of sentences of the form "[itex]a_1 a_2 a_3[/itex]" (the multiplication table, [itex]a_1 * a_2 = a_3[/itex]), then every homomorphism of languages from [itex]A[/itex] to the alphabet of another language set up like this corresponds to a group homomorphism.
Another example: if [itex]A[/itex] is a set of ring elements together with special elements * + =, and [itex]B[/itex] is a set of sentences of the form "[itex]a_1 * a_2 = a_3[/itex]" (the multiplication table), unioned with a set of sentences of the form "[itex]a_1 + a_2 = a_3[/itex]" (the addition table), then homomorphisms of languages that map * to * and + to + and = to = correspond to ring homomorphisms.
To look at it in general, [itex]A[/itex] is a set of objects, and [itex]B[/itex] is a set of true statements about elements of [itex]A[/itex]. A homomorphism of languages [itex]f : A \rightarrow C[/itex] induces a map from statements in [itex]B[/itex] to corresponding statements in [itex]D[/itex] that "maps truth to truth" (doesn't generate false statements, i.e. strings outside of [itex]D[/itex]).
What do you think? Have you heard of this?
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