Extended operators

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

Notation
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,, 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 as a base. Then comes,  , and so on.

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

Bowers eventually switched to {a,b,c,d}, meaning ab 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}