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The extended operators[1] are an extension to the hyper operators by Jonathan Bowers. Their extensions eventually form BEAF.

### NotationEdit

Bowers uses a{b}c = a↑↑...↑↑c with b arrows in Arrow notation, so a{1}b is exponentiation, a{2}b is tetration, a{3}b is pentation, and so on. Originally a{1}b was addition, but it was changed to exponentiation on Bird's request.

Then Bowers defines a new operator, {{1}}, above all the hyper-operators. It is defined as follows:

$a\lbrace\lbrace1\rbrace\rbrace b = a\lbrace a\lbrace...\lbrace a\lbrace a\rbrace a\rbrace...\rbrace a\rbrace a$ with 2b-1 a's and b nested layers.

Then he defined $\lbrace\lbrace2\rbrace\rbrace$ to be the next operator after $\lbrace\lbrace1\rbrace\rbrace$. It behaves like {2}, but using {{1}} as a base. Then comes {{3}}, {{4}}, and so on.

Then he defines a structure after all that, {{{1}}}. It nests values in {{}} just like {{1}} nests values in {}. Then comes {{{2}}}, {{{3}}}, {{{{1}}}}, {{{{{1}}}}}, and so on.

Bowers eventually switched to {a,b,c,d}, meaning a{{...{{c}}...}}b with d pairs of braces.

Here is the definition:

{a,b,1,1} = a^b

{a,1,b,c} = a

{a,b,1,d} = {a,a,{a,b-1,1,d},d-1}

{a,b,c,d} = {a,{a,b-1,c,d},c-1,d}

### ReferencesEdit

1. http://www.polytope.net/hedrondude/array.htm