Given a [[ringfield (mathematics)|ringfield]] ''AF'' (for example, [[square matrices]]) there are two [[embedding]]s of ''AF'' into the [[projective line over a ring|projective line over ''A'']] P(''AF''): ''z'' → [''z'' : 1] and ''z'' → [1 : ''z'']. These embeddings overlap onexcept thefor [[group0:1] ofand units[1:0]] of ''A''. The parallel operator relates the addition operation between the embeddings. In fact, the [[homography|homographies]] on the projective ringline are represented by 2 x 2 matrices M(2,''AF''), and the ringfield 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}
Line 251:
\end{align}</math>
The two matrix products show that there are two subgroups of M(2,''AF'') isomorphic to (''AF'',+), the additive group of ''AF''. Depending on which embedding is used, one operation is +, the other is <math>\parallel.</math>
