Definition (Column property). We say that a rational matrix

$$A=\left(\begin{array}{cccc}\hfill \uparrow \hfill & \hfill \uparrow \hfill & \hfill \cdots \hfill & \hfill \uparrow \hfill \\ \hfill c_1\hfill & \hfill c_2\hfill & \hfill \cdots \hfill & \hfill c_n\hfill \\ \hfill \downarrow \hfill & \hfill \downarrow \hfill & \hfill \cdots \hfill & \hfill \downarrow \hfill \end{array}\right)$$

has the column property (CP) if there exists a partition of $[n]=B_1\bigsqcup B_2\cdots \bigsqcup B_r$ such that:

• (1) ${\sum }_{i\in B_1}{c}_i=0$
• (2) ${\sum }_{i\in B_t}{c}_i\in \mathrm{span}⟨c_j:j\in B_1\cup B_2\cup \cdots B_{t-1}⟩$ (note that it doesn’t make a difference whether the span is the $ℝ$-linear or $\mathbb{Q}$-linear span)