Gits & Gambits

[Jon Awbrey]

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Gambit Gits -- Yes, You Brits, I Watch 'Red Dwarf' --

The construction of gambits that I have given yields
a family of rooted trees, and when I say that, I am
speaking a dialect of graph theory where the default
references of all terminology for species of graphs
is first and foremost to their unlabeled varieties.

In many ways this family of gambits, that has its generations
and knows its rightful and proper successions as given in the
protocol that I give again below, is very like the arithmetic
of a mathematical system for which we now seek the associated
algebra.  And as you might suspect, I am going to go about it
in slightly peculiar way.  Select a finite set of gambits for
your personal repertory, and pretend that you do not know any
other possible use, value, worth, whatever for these selected
gambits but to mark up the rest of their family dominion with
the traces of their unresolved identities, like those Kilrois
who desire no better mark in life but to put their individual
stamps inanely all about their encompassing realm of graffiti.
The palette of 'reserved gambits' that now go to make up your
personal repertoire you may now treat as distinctive 'labels',
to wit, individually developed 'characters', in the usual way.
The stage being set in this way you may now consider yourself
free in your imagination to label gambits as you were wont to
formerly, since it can always be justified in the end just as
if it were in the beginning.  With all of that in mind I will
go on to treat of labeled gambits, assuming some potential if
peculiar constitution of their markèd natures within the neck
of the woods where bare borne and free gambits per se do play.

I have to break for a bit of sustenance here ...

Jon Awbrey

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Recursion Scheme for Gambits (By Default Unlabeled)

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|   Definition                                                                  |
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|                                       |                                       |
|   Blank " " is a gambit.              |   @ is a gambit.                      |
|                                       |                                       |
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|                                       |                                       |
|   If i is a gambit and                |   If i is a gambit and                |
|   if j is a gambit,                   |   if j is a gambit,                   |
|                                       |                                       |
|                                       |        i j                            |
|   then ij is one too.                 |   then  @  is one too.                |
|                                       |                                       |
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|                                       |                                       |
|   If i is a gambit and                |   If i is a gambit and                |
|   if j is a gambit,                   |   if j is a gambit,                   |
|                                       |                                       |
|                                       |         i   j                         |
|                                       |         o---o                         |
|                                       |         |                             |
|   then (i(j)) is one too.             |   then  @     is one too.             |
|                                       |                                       |
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In Step 1 of this recursive scheme, the
notation "ij" denotes the concatenation
of i and j, id est, just the writing of
string i and string j one after another.
The associated graphical notation means
that one joins tree i and tree j at the
root in order to construct the new tree.

In Step 2 of this scheme, i is called the "index"
and j is called the "exponent" of the gambit that
is constructed there.  The notation "(i(j))" says
to place the strings for i and j in the indicated
places of the frame "(_(_))".  The parallel graph
notation enjoins one to graft the tree i and then
to graft the tree j at the pair of sites that are
severally indicated, merging their roots with the
nodes that are currently sited in the underlying
gambit, in passing erasing their status as roots.

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