What is an L-Embedding? For isomorphisms, not for sleeping.
What is an L-Embedding?
Simply put, an L-Embedding for some language L from M to N is a 1-1 mapping η: M → N that preserves the interpretations of the symbols of L. Where M and N are L-structures and M and N are their respective underlying sets or universes. The specific conditions for “interpretation preserving” are as follows:
η(fm(a1,…,aj)) = fn(η(a1),…,η(aj)) for each f in F (the set of function symbols of L) and a1…aj in M
(a1,…a) belongs to Rm if and only if (η(a1),…,η(an)) belongs to Rn for each R in R (the set of relation symbols of L)
cn = η(cm) for each constant symbol c in C (the set of constant symbols of L)
The subtle thing to keep in mind here is that there aren’t two categories to keep in mind, there are three. We cannot forget about L. The objects, relations, and processes of M and N are correlated to symbols in L and that’s how the mapping η can be interpretation preserving: It does not matter if you go straight from L to N or take a detour through M and arrive at N by way of the mapping η, the result will be the same.
If M is a subset of N then M is called a submodel of N, and N is an extension of M.
Finally for isomorphism. In the context of the model theory an isomorphism is a bijective L-Embedding. Bijective means that it’s one to one and onto, which can be thought of as being one to one in both directions. Every element in the domain is mapped to one and only one element in the codomain, and every element in the codomain is the image of one and only one element in the domain. If there is an isomorphism η: M → M from a model M to itself, then the η is called an automorphism.
