m

main.tex

@@ -113,9 +113,10 @@ The \emph{product} of two structures $A,B$ over a signature $\ssign$ is a struct
 
 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 $\ssign^{1} + \ssign^{>1}\times\set{\sqsubseteq}$, where $\ssign^{1}$, $\ssign^{>1}$ denote the symbols of arity one and at least two, respectively. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n} \dom_{A_i}$.
-The interpretation of a symbol $R$ of arity $1$ is $\bigotimes_{i\leq n} \inter_{A_i}(R)$. The interpretation of $(R,\sumorder)$ is: $$\bigcup_{j\leq n}\quad \bigotimes_{i<j}\Delta_{i,k} \times\inter_{A_j}\times\bigotimes_{j<i\leq n}\dom_{A_i}$$
-where $\Delta_{i,k}=\set{(a,\ldots,a)\in \dom_{A_i}^k\mid\ a\in \dom_{A_i}}$ is the $k$-fold diagonal of $\dom_{A_i}$.
+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 $\ssign + \set{\sim}$, where $\sim$ has arity $2$ and holds for tuples that are equal over all coordinates except one. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n} \dom_{A_i}$.
+The interpretation of a symbol $R$ of arity $k$ is $\set{(\bar a_1,\dots,\bar a_k)\mid\ \forall i\leq n,\ (\bar a_1[i],\dots,\bar a_k[i])\in  \inter_{A_i}(R)}$. The interpretation of $\sim$ is $ \set{(\bar a,\bar b)\mid \exists ! i\leq n,\ \bar a[i] \neq \bar b[i]}$
+
+
 The \emph{powerset structure} of a structure $A$ over signature $\ssign$ is a structure $\pow A$ over signature $\ssign\uplus \set{\subseteq}$, where $\subseteq$ is a binary symbol. The domain of $\pow A$ is $\pow{\dom_A}$. The interpretation of a symbol $R$ is the set $\set{(\set{x_1},\dots,\set{x_k})\mid\ (x_1,\ldots,x_n)\in \inter_A(R)}$.
 The interpretation of $\subseteq$ is the actual set inclusion $\set{(x,y)\mid\ x\subseteq y\subseteq \dom_A}$.
 \paragraph{Word models}