Given a [[field (mathematics)|field]] ''F'' there are two [[embedding]]s of ''F'' into the [[projective line|projective line]] P(''F''): ''z'' → [''z'' : 1] and ''z'' → [1 : ''z'']. These embeddings overlap except for [0:1] and [1:0]. The parallel operator relates the addition operation between the embeddings. In fact, the [[homography|homographies]] on the projective line are represented by 2 x 2 matrices M(2,''F''), and the field operations (+ and ×) are extended to homographies. Each embedding has its addition ''a'' + ''b'' represented by the following [[matrix multiplication]]s in M(2,''A''):
:<math>\begin{align}
\begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix}
