If \$$X\$$ and \$$Y\$$ are sets with preorders, you can define a preorder on \$$X \sqcup Y\$$ by \$$a \leq b\$$ iff \$$a \leq b\$$ in \$$X\$$ or \$$a \leq b\$$ in \$$Y\$$. This should be the coproduct in the category of posets. This is like putting \$$X\$$ and \$$Y\$$ right next to each other.

Another preorder on \$$X \sqcup Y\$$ is given by taking the one above and adding in \$$a \leq b\$$ whenever \$$a \in X\$$ and \$$b \in Y\$$. This is like putting \$$Y\$$ on top of \$$X\$$.