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Illustration

This is a picture of the correspondence (viii) and (ix) between critical pairs and S-polynomials and the four ways in which they can occur, as described in the above proof.

possible overlaps		possible matches
of rules		of polynomials
$l_1 \to r_1$ and $l_2 \to r_2$		and

$\displaystyle l_1=u_2l_2v_2 \hspace{0.5cm} \xymatrix{\ar@{-}@/^2pc/[rrr]\vert{r... ...\ar@{-}[r]\vert{l_2} & \ar@{-}[r]\vert{v_2} & \\ }\hspace{0.5cm} l_1=u_2l_2v_2$

$\displaystyle (r_1,u_2r_2v_2) \makebox[6cm]{} u_2r_2v_2-r_1$

$\displaystyle u_1l_1v_1=l_2 \hspace{0.5cm} \xymatrix{\ar@{-}@/_2pc/[rrr]\vert{r... ...\ar@{-}[r]\vert{l_1} & \ar@{-}[r]\vert{v_1} & \\ }\hspace{0.5cm} u_1l_1v_1=l_2$

$\displaystyle (u_1r_1v_1,r_2) \makebox[6cm]{} r_2-u_1r_1v_1$

$\displaystyle l_1v_1=u_2l_2 \hspace{0.5cm} \xymatrix{\ar@{-}@/^2pc/[rr]\vert{r_... ...\vert{r_2} \ar@{-}[r] & \ar@{-}[r]\vert{v_1} &\\ }\hspace{0.5cm} l_1v_1=u_2l_2$

$\displaystyle (r_1v_1,u_2r_2) \makebox[6cm]{} u_2r_2-r_1v_1$

$\displaystyle u_1l_1=l_2v_2 \hspace{0.5cm} \xymatrix{\ar@{-}@/_2pc/[rr]\vert{r_... ...\vert{r_1} \ar@{-}[r] & \ar@{-}[r]\vert{v_2} &\\ }\hspace{0.5cm} u_1l_1=l_2v_2$

$\displaystyle (u_1r_1,r_2v_2) \makebox[6cm]{} r_2v_2-u_1v_1$

Author: A. Heyworth, tel: +44 (0)116 252 3884
Last updated: 2001-05-08
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Department of Mathematics & Computer Science

Illustration