A \emph{signature}$\ssign$ is a set $S$ of \emph{symbols}\footnote{We only consider relational signatures.}, together with an \emph{arity function}, which we denote by $\ar:S\rightarrow\nat$. Abusing notations, we will often write $R\in\ssign$ instead of $R\in S$.
\paragraph{Structures} A \emph{structure}$w$ over a signature $\ssign$ is given as a \emph{domain}$D$ together with a function which maps any symbol $R$ of $\ssign$ to a set denoted $R^w$ such that $R^w\subseteqD^r$, with $r=\ar(R)$.
\paragraph{Structures} A \emph{structure}$A$ over a signature $\ssign$ is given as a \emph{domain}$\dom_A$ together with an \emph{interpretation function} which maps any symbol $R$ of $\ssign$ to a set denoted $\inter_A(R)$ such that $\inter_A(R)-\subseteq\dom(A)^r$, with $r=\ar(R)$.
\paragraph{Operations} The \emph{disjoint union} of two structures over
\paragraph{Operations}
Any structure over a signature can be seen as a structure over a larger signature where additional symbols are interpreted as empty sets.
The \emph{disjoint union} (or coproduct) of two structures $A,B$ over the signature $\ssign$ is a structure $A+B$ over signature $\ssign+\set{\lefto,\righto}$ where $\lefto, \righto$ are unary symbols. The domain of $A+B$ is $\dom_A+\dom_B=(\dom_A{\times}\set{1})\cup(\dom_B{\times}\set{2})$. The interpretation of a symbol $R$ is $\inter_A(R)+\inter_B(R)$. The interpretation of $\lefto$ is $\dom_A\times\set{1}$ and the interpretation of $\righto$ is $\dom_B\times\set{2}$.
The \emph{product} of two structures $A,B$ over a signature $\ssign$ is a structure $A\times B$ over signature $\ssign\times\set{\lefto,\righto}$ where $(R,\lefto)$ and $(R, \righto)$ have the same arity as $R$. The domain of $A\times B$ is $\dom_A\times\dom_B$. The interpretation of a symbol $(R,\lefto)$ is $\inter_A(R)\times\dom_B$ and the interpretation of $(R,\righto)$ is $\dom_A\times\inter_B(R)$.
The \emph{ordered coproduct} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigoplus_{i\leq n} A_i$ over the signature $\ssign+\set{\sqsubseteq}$, where $\sqsubseteq$ has arity $2$. The domain of $\bigoplus_{i\leq n} A_i$ is $\bigoplus_{i\leq n}\dom_{A_i}=\bigcup_{i\leq n}\dom_{A_i}{\times}\set{i}$. The interpretation of a symbol $R$ is $\bigoplus_{i\leq n}\inter_{A_i}(R)$. The interpretation of $\sumorder$ is $\bigcup_{i\leq j\leq n}(\dom_{A_i}{\times}\set{i})\times(\dom_{A_j}{\times}\set{j})$.
The \emph{ordered product} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigotimes_{i\leq n} A_i$ over the signature $\ssign\times\set{\sqsubseteq}$. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n}\dom_{A_i}$. The interpretation of a symbol $R$ is $\bigoplus_{i\leq n}\inter_{A_i}(R)$. The interpretation of $\sumorder$ is $\bigcup_{i\leq j\leq n}(\dom_{A_i}{\times}\set{i})\times(\dom_{A_j}{\times}\set{j})$.
