I have a question about a notation appeared in "Elements of Finite Model Theory" by Libkin.
Let $\mathfrak{A}$ be a structure that consists of a universe set $A$ and constants and relations, and let $a \in A$ its element.
What does mean the tuple $(\mathfrak{A},a)$? It seems to be another structure, but what's the difference between $\mathfrak{A}$ and the tuple?
For example, this notation appears in Lemma 2.3, Lemma 3.7, etc.
The idea is to expand the signature to have a new symbol that names the element $a$. Here are more details:
Suppose that $\mathfrak{A}$ is an $\sigma$-structure for some signature $\sigma$ (I believe that this is the terminology used in this book). Let $c$ be a new constant symbol that does not already appear in $\sigma$. Then $(\mathfrak{A}, a)$ is the $\sigma \cup \{c\}$ structure with universe $A$, where everything in $\sigma$ is interpreted just as it was in $\mathfrak{A}$ and the new symbol $c$ is interpreted as $a$.