Laws of Triple Form: 7 of 12

          Chapter 6 : Three Marks

          Three Marks

          Trinary Algebra

          6A. Three Marks

          The bracket [x] swaps 0 and 1, and leaves 6 fixed. But there are other permutations of the three forms. Let <x> swap 6 and 0, leaving 1 fixed; and let {x} swap 1 and 6, leaving 0 fixed. Then for any form x we get these forms:

{x}      the brace

                    <x>    the wedge
                    [x]      the bracket

          In particular we have these forms:
                    {} = 0
                    <> = 6
                    []= 1

          There are also the “double-crosses”:

                    [[x]], <<x>>, {{x}}, {[x]}, [<x>], <{x}>, [{x}], <[x]>, {<x>}

They have these identities:

[[x]]  =   <<x>>  =  {{x}}  =   x

[<x>]  =  {[x]}  =  <{x}>

<[x]>  =  [{x}]  =  {<x>}

They have these tables:

                   x       {}  <>  []   _
         (x)      {}  <>  []
         {[x]}    <>  [] {}
         [{x}]    []  {}  <>
         {x}      {}  []  <>
         <x>      <>  {}  []
         [x]      []  <>  {}

                   x       0   6   1    _   
         (x)      0   6   1
         {[x]}    6   1   0
         [{x}]    1   0   6
         {x}      0   1   6
         <x>      6   0   1
         [x]      1   6   0

The crosses and double-crosses permute 0,6,1, and form the group S₃.
Here is S₃’s group table:

           

     f(g) | g = x    {[x]} [{x}] <x>  [x]  {x}

     ______|_________________________________________

f=   |

x    |    x    {[x]} [{x}] <x>  [x]  {x}

{[x]}|    {[x}] [{x}] x    [x]  {x}  <x>

[{x}]|    [{x}] x    {[x]} {x}  <x>  [x]

<x>  |    <x>  {x}  [x]  x    [{x}] {[x]}

[x]  |    [x]  <x>  {x}  {[x]} x    [{x}]

{x}  |    {x}  [x]  <x>  [{x}] {[x]} x   

6B. Trinary Algebra

These equations hold for all forms x:

{{x}} = <<x>> = [[x]] =  x        ‘Double-Cross’
{[x]} =  [<x>]  =  <{x}>            ‘Anti-commutativity’
[{x}] =  <[x]>  =  {<x>}

Since {}=void, it follows that
{{}}  = <<>>  =  [[]]  =  {}

{[]}    =  <{}> = [<>]  =  <>  
{<>}  =  <[]> =  [{}]   =   []

          Note that ()=void, and {}=void; but (x)=x for all x, and {x}=x only for void. The braces mark the unmarked without marking it; but it distinguishes between the other two marks. It’s a meta-distinction.

Here are some axioms for triple forms:

Order Laws:

xy = yx
xx = x
{}x = x
[]x = []

Group Laws:

x = {{x}} = [[x]] = <<x>>
{[x]} = [<x>] =<{x}> 
[{x}] =  <[x]>  =  {<x>}

Transpositions:

[[x][y]]z  =  [[xz][yz]]
<<x><y>>z  =  <<xz><yz>>
{<<{x}><{y}>>}z  =  {<<{xz}><{yz}>>}

Relocation:

[[x]x] [<>]   =  <>

Open questions for the ambitious student:

Do these triple-bracket axioms imply a normal form?

Are these axioms deductively complete?

Is there a more succinct, but complete set of axioms for triple forms?