I read the following sentence (from these notes on logic):
If $\mathcal A \subseteq \mathcal B$, then the inclusion $ a \to a : A \to B$ is an embedding $\mathcal A \to \mathcal B$
where $\mathcal{A} = (A, (R^{\mathcal A})_{R \in L^r}, (F^{\mathcal A})_{F \in L^f} )$, $\mathcal{B} = (B, (R^{\mathcal B})_{R \in L^r}, (F^{\mathcal B})_{F \in L^f} )$ are different L-structures.
I was trying to understand what that specific sentence meant. The things that confuse me the most are:
- what inclusion means in this context.
- what the notation $ a \to a : A \to B$ means.
usually $ x \to f(x) $ means x is mapped to f(x) e.g. $x \to e^x$. Then word inclusion from googling seems to just mean subset. Thus I'm confused, is what we are considering mapping the whole set A to B or elements from A to elements of B? What is an embedding in this context? I know embeddings in general are injective strong homomorphisms, but right now its unclear what the homomorphism we are talking about and what sets we are talking about and what relations/functions we are talking about. What are we mapping from what to what and where does this fit with the given sentence?