I am currently reading a book, Tame Topology and O-minimal Structures by van den Dries. He defines a structure on a nonempty set $R$ to be a sequence $\mathcal{S}=(\mathcal{S}_m)_{m\in\mathbb{N}}$ such that for each $m\geq 0$, the following hold:
- $\mathcal{S}_m$ is a boolean algebra of subsets of $R^m$.
- If $A\in\mathcal{S_m}$, then $R\times A$ and $A\times R$ belong to $\mathcal{S}_{m+1}$.
- $\{(x_1,...,x_m)\in R^m|\,x_1=x_m\}\in\mathcal{S}_m$
- If $A\in\mathcal{S}_{m+1}$ then $\pi(A)\in\mathcal{S}_m$, where $\pi:R^{m+1}\to R^m$ is the projection map on the first $m$ coordinates.
From what I can tell, he defines a structure by its definable subsets. That is, a structure $\mathcal{S}$ on a nonempty set $R$ is nothing more than the structure $R$ along with some $n$-ary relation symbols. Following this interpretation, the fact that $\mathcal{S}_m$ is a boolean algebra on subsets of $R^m$ arises naturally. We can make other definable subsets of $R^m$ via the logical connectives, $\wedge,\vee$ which naturally correspond to unions/intersections of sets in $\mathcal{S}_m$, which are well-defined operations on a collection equipped with a boolean algebra. The second property merely captures that the set defined by $\phi(x,y)$ is equivalent to that defined by $\phi(x)$, if $\phi$ is a formula in only $x$ say, with $x=(x_1,...,x_n)$. Finally, the fourth of these properties captures the action of the existential quantifier $\exists$ on a formula which defines some set.
However, I fail to see what property is being captured by the third statement. What do we achieve by mandating that the set of $x$ so $x_1=x_m$ is definable in $R^m$?
Property 3 corresponds to the fact that $=$ is a primitive logical symbol, so $x_1 = x_m$ is always a formula.
You can get equalities between variables other than the first and the last by starting with an instance of 3 and then applying 2 repeatedly.