 The foregoing discussion motivates the following. Call projections \(P\) and \(Q\) orthogonal, and write \(P\binbot Q\) iff \(P \le Q'\). Note that \(P\binbot Q\) iff \(PQ = QP = 0\). If \(P\) and \(Q\) are orthogonal projections, then their join is simply their sum; traditionally, this is denoted \(P\oplus Q\). We denote the identity mapping on \(\mathbf{H}\) by \(\mathbf{1}\).