2 Regularity of 1-Sum
Let \(R\) be a magma containing zero (we will use \(R = \mathbb {Z}_{2}\) and \(R = \mathbb {Q}\)). Let \(B_{\ell } \in R^{X_{\ell } \times Y_{\ell }}\) and \(B_{r} \in R^{X_{r} \times Y_{r}}\) be matrices where \(X_{\ell }, Y_{\ell }, X_{r}, Y_{r}\) are pairwise disjoint sets. The \(1\)-sum \(B = B_{\ell } \oplus _{1} B_{r}\) of \(B_{\ell }\) and \(B_{r}\) is
A matroid \(M\) is a \(1\)-sum of matroids \(M_{\ell }\) and \(M_{r}\) if there exist standard \(\mathbb {Z}_{2}\) representation matrices \(B_{\ell }\), \(B_{r}\), and \(B\) (for \(M_{\ell }\), \(M_{r}\), and \(M\), respectively) of the form given in Definition 35.
Let \(A\) be a square matrix of the form \(A = \begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{bmatrix}\). Then \(\det A = \det A_{11} \cdot \det A_{22}\).
This result is proved in Mathlib.
Let \(B_{\ell }\) and \(B_{r}\) from Definition 35 be TU matrices (over \(\mathbb {Q}\)). Then \(B = B_{\ell } \oplus _{1} B_{r}\) is TU.
We prove that \(B\) is TU by Definition 3. To this end, let \(T\) be a square submatrix of \(B\). Our goal is to show that \(\det T \in \{ 0, \pm 1\} \).
Let \(T_{\ell }\) and \(T_{r}\) denote the submatrices in the intersection of \(T\) with \(B_{\ell }\) and \(B_{r}\), respectively. Then \(T\) has the form
First, suppose that \(T_{\ell }\) and \(T_{r}\) are square. Then \(\det T = \det T_{\ell } \cdot \det T_{r}\) by Lemma 37. Moreover, \(\det T_{\ell }, \det T_{r} \in \{ 0, \pm 1\} \), since \(T_{\ell }\) and \(T_{r}\) are square submatrices of TU matrices \(B_{\ell }\) and \(B_{r}\), respectively. Thus, \(\det T \in \{ 0, \pm 1\} \), as desired.
Without loss of generality we may assume that \(T_{\ell }\) has fewer rows than columns. Otherwise we can transpose all matrices and use the same proof, since TUness and determinants are preserved under transposition. Thus, \(T\) can be represented in the form
where \(T_{11}\) contains \(T_{\ell }\) and some zero rows, \(T_{22}\) is a submatrix of \(T_{r}\), and \(T_{12}\) contains the rest of the rows of \(T_{r}\) (not contained in \(T_{22}\)) and some zero rows. By Lemma 37, we have \(\det T = \det T_{11} \cdot \det T_{22}\). Since \(T_{11}\) contains at least one zero row, \(\det T_{11} = 0\). Thus, \(\det T = 0 \in \{ 0, \pm 1\} \), as desired.
Let \(M\) be a \(1\)-sum of regular matroids \(M_{\ell }\) and \(M_{r}\). Then \(M\) is also regular.
Let \(B_{\ell }\), \(B_{r}\), and \(B\) be standard \(\mathbb {Z}_{2}\) representation matrices from Definition 36. Since \(M_{\ell }\) and \(M_{r}\) are regular, by Lemma 34, \(B_{\ell }\) and \(B_{r}\) have TU signings \(B_{\ell }'\) and \(B_{r}'\), respectively. Then \(B' = B_{\ell }' \oplus _{1} B_{r}'\) is a TU signing of \(B\). Indeed, \(B'\) is TU by Lemma 38, and a direct calculation shows that \(B'\) is a signing of \(B\). Thus, \(M\) is regular by Lemma 34.