nLab
structure (model theory)

A structure on a set $X$ consists of a collection

$${𝒮}_n\subseteq P(X^n)$$\mathcal{S}_n \subseteq P(X^n)

for each natural number $n$ such that

• ${𝒮}_n$ is a Boolean subalgebra of $P(X^n)$,

• If $A\in {𝒮}_n$, then $A\times X$ and $X\times A$ belong to ${𝒮}_{n+1}$,

• The set $\Delta =\{(x_1,\dots ,x_n):{\forall }_{1\le i\le n}{x}_i=x_1\}$ belongs to ${𝒮}_n$,

• If $\pi :X^{n+1}\to X^n$ denotes the projection onto the first $n$ coordinates, then the image $\pi (A)$ belongs to ${𝒮}_n$ whenever $A\in {𝒮}_{n+1}$.

Morally, a structure on a set is the collection of subsets of a finite power of $X$ which are definable with respect to a first-order language that has been interpreted in $X$ (such an interpretation being called a structure of the language). Indeed, the definable sets of such a language do form a structure in the present sense. Conversely, to any structure in this sense, we may introduce a relational language whose $n$-ary relation symbols are named by the elements of ${𝒮}_n$, and then the tautological structure of this language on $X$, where each relation is interpreted as the set that names it, is a structure of this language.

Each structure $𝒮$ on $X$ induces a bicategory of relations: the objects are natural numbers, and 1-cells $m\to n$ are triples $(m,n,R)$ where $R\in {𝒮}_{m+n}$, ordered by inclusion. (It is indeed not difficult to show that 1-cells are closed under set-theoretic relational composition.)