Canonical expansion of a structure

Question

Fixed a signature $\tau = (I,J,K,ar)$ and a $\tau -structure$ $A$, the canonical expansion of $A$ by a $B$ (where $B$ is a subset of the universe of $A$) is the structure where the elements of $B$ are new constant symbols. I understand the concept formally but I cannot find out a concrete example. About the reduction of a structure in a smaller language, I can visualize the concept as a forgetful functor e.g. from a ring to a grou. Is there a way to visualize the canonical expansion likewise?

Answer 1

Answering the question in the comments: "What's the use?"

Let's say $M$ is a $\tau$-structure, and $A\subseteq M$. Then we can form a new signature $\tau_A$ by adding a new constant symbol for each element of $A$ and view $M$ is a $\tau_A$-structure $M_A$.

The basic reason to do this is that whenever $\varphi(x_1,\dots,x_n)$ is a $\tau$-formula and $a_1,\dots,a_n$ is a tuple from $A$, we can view $\varphi(a_1,\dots,a_n)$ as a $\tau_A$-sentence. So if we let $T_A = \text{Th}_{\tau_A}(M_A)$ and $T = \text{Th}_\tau(M)$, then $T_A$ contains $T$ together with the full type / elementary diagram of $A$: $$T_A = T\cup \{\varphi(\overline{a})\mid \overline{a}\in A\text{ and }M\models \varphi(\overline{a})\}.$$ In other words, a model of $T_A$ is a model $N\models T$, together with a specified subset realizing the type of $A$, or equivalently a specified partial elementary map $A\to N$. And a $\tau_A$-elementary embedding $N\to N'$ between two models of $T_A$ is a $\tau$-elementary embedding which preserves the copy of $A$.

A typical application is this: Let's say you want to use a compactness argument to find an elementary extension of a structure $M$ with certain properties. It's not enough to find a model $N\models T$ satisfying these properties, since there might not be an elementary embedding of $M$ into $N$. But if we expand the language by constants for $M$ (taking $A = M$ in the above discussion), then every model of $T_M$ has a specified elementary embedding of $M$, so it's enough to use your compactness argument to find a model of $T_M$ satisfying your properties.

You ask about a "functorial" view. Like any expansion, there will be a reduct / "forgetful" functor from the category of models of $T_A$ to the category of models of $T$. Unlike in universal algebra (groups, rings, etc.), where free constructions abound, there is unlikely to be a nice functor in the other direction, from models of $T$ to models of $T_A$. But if you like category theory, you can think about the category of models of $T_A$ as being a vast generalization of the category of pointed sets: An object of this category is an object of the base category (sets / models of $T$) with extra structure (a choice of base point / a choice of copy of $A$), though not every object in the base category admits such structure (e.g. the empty set / a structure which doesn't realize the type of $A$). And the morphisms are restricted to those which preserve the extra structure.

Alternatively, the category of substructures of models of $T_A$ and partial elementary maps is the coslice category over $A$ of the category of substructures of models of $T$ and partial elementary maps.

For a concrete example of this point about restricting morphisms: In the field language, $\mathbb{C}$ has an automorphism sending $i$ to $-i$. In fact, the subfield fixed by all automorphisms of $\mathbb{C}$ is just $\mathbb{Q}$. If we add a constant symbol for $i$ to the language, then any automorphism of $\mathbb{C}_{\{i\}}$ fixes $i$, and the subfield fixed by all automorphisms of this structure is $\mathbb{Q}[i]$.