Changes

Notice that, with rooted trees like these, drawing the arrows is optional, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, ''up'' being the same as ''away from the root''.

Notice that, with rooted trees like these, drawing the arrows is optional, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, ''up'' being the same as ''away from the root''.

We have already seen various forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.</math>  For the sake of comparison, let's record the planar and dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.</math>

We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.</math>  For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.</math>

First the plane-embedded maps:

First the plane-embedded maps:

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Next the planar maps and their dual trees superimposed:

Next the plane-embedded maps and their dual trees superimposed:

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