Changes

The outermost region of the plane-embedded graph is singled out for special consideration and the corresponding node of the dual graph is referred to as its ''root node''.  By way of graphical convention in the present text, the root node is indicated by means of a horizontal strike-through.

The outermost region of the plane-embedded graph is singled out for special consideration and the corresponding node of the dual graph is referred to as its ''root node''.  By way of graphical convention in the present text, the root node is indicated by means of a horizontal strike-through.

Extracting the dual graph from its composite matrix gives the following picture:

Extracting the dual graph from its composite matrix, we get this picture:

{| align="center" cellpadding="10"

{| align="center" cellpadding="10"

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 "(( )) = ".  For the sake of comparison, let's record the planar and dual forms of the axiom that is formulated in string form as "( )( ) = ( )".

We have already seen various forms of the axiom that is formulated in string form as "&nbsp;<math>((~))~=</math>&nbsp;&nbsp;&nbsp;&nbsp;".  For the sake of comparison, let's record the planar and dual forms of the axiom that is formulated in string form as "<math>(~)(~)~=~(~)</math>".

First the planar form:

First the plane-embedded maps:

{| align="center" cellpadding="10"

| [[Image:Logical_Graph_Figure_7_Visible_Frame.jpg|500px]]

|}

` ` ` | ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` ` ` | ` ` ` | ` ` `

` ` ` | ` ` ` | ` ` ` | ` ` ` | ` ` ` = ` ` ` | ` ` ` | ` ` `

` ` ` | ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` ` ` | ` ` ` | ` ` `

Next the planar and dual forms superimposed:

Next the planar maps and their dual trees superimposed:

{| align="center" cellpadding="10"

| [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]]

|}

` ` ` | ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` ` ` | ` ` ` | ` ` `

` ` ` | ` o ` | ` ` ` | ` o ` | ` ` ` = ` ` ` | ` o ` | ` ` `

` ` ` | ` `\` | ` ` ` | `/` ` | ` ` ` ` ` ` ` | ` | ` | ` ` `

` ` ` o-----\-o ` ` ` o-/-----o ` ` ` ` ` ` ` o---|---o ` ` `

Finally the dual form by itself:

Finally the dual trees by themselves:

{| align="center" cellpadding="10"

| [[Image:Logical_Graph_Figure_9_Visible_Frame.jpg|500px]]

|}

` ` ` ` ` `\` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `

` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `

` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `

` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `

==Categories of structured individuals==

==Categories of structured individuals==