Formal definition of the union of two structures over the same set

Question

Let $S$ be a set, and consider two structures (in the sense of model theory) over $S$. How does one formally define the union of the two structures? Let me give some motivating examples. The union of $(\mathbb{N},+)$ and $(\mathbb{N},0)$ is the structure $(\mathbb{N},+,0)$. The union of $(\mathbb{N},+)$ with itself is the structure $(\mathbb{N},+)$. The union of $(\mathbb{N},+)$ with $(\mathbb{N},+,*)$ is $(\mathbb{N},+,*)$. Based on these examples, can someone give a formal definition of the union of two structures over the same base set?

Answer 1

Based on your examples, the "union" is not well-defined for arbitrary pairs of structures with the same underlying set. For example, let $L$ be the language with a single constant symbol $c$. Let $A$ be the structure with underlying set $\{0,1\}$ and $c^A = 0$, and let $B$ be the structure with underlying set $\{0,1\}$ and $c^B = 1$. What should the "union" of $A$ and $B$ be?

We need to require more than that the two structures have the same underlying set - the two structures should have equal reducts to the intersection of their languages.

Definition: Let $L_1$ and $L_2$ be two languages. Let $L_\cap = L_1\cap L_2$, and let $L_\cup = L_1\cup L_2$. Let $A$ be an $L_1$-structure, and let $B$ be an $L_2$ structure, and assume that $A|_{L_\cap} = B|_{L_\cap}$ (in particular, $A$ and $B$ have the same underlying set). Then the "union" of $A$ and $B$ is the unique $L_\cup$-structure $C$ such that $C|_{L_1} = A$ and $C|_{L_2} = B$.