Thanks to: http://www.polytope.net/hedrondude/array.htm
I created the "extended operator bisector notation" (xOBN) based on Jonathan Bowers' extended operator notation (BXO). These rule changes (in the primitive sequence level) are:
All variables must be positive integers (except numbers in the brackets, where 0 is allowed).
Rule 1: a{0}b = a*b
Rule 2: a{X + 1}b = a{X}(a{X + 1}(b - 1))
All the other rules work exactly the same.
You can notice that the expression a{c}b is exactly the same as in the Bowers' operator notation for any whole number c (c is any sequence smaller than {1}), as a{c}b is defined as a^^^^^...^^^^^b with c arrows in arrow notation.
The key difference is that {{1}} in the extended operator bisector notation (xOBN) is not the same as {{1}} in the Bowers' extended operator notation (BXO), in which the latter one is actually equal to {{1} + 1} in xOBN. Moreover, {{2}} in xOBN is significantly greater than {{2}} in BXO, as the former one eventually dominates the limit of BXO, and even more importantly, {{{1}}} in xOBN is way much greater than {{{1}}} in BXO, and even the limit of BXO as well.
So:
a{{1}}b in xOBN = a{b}a in BXO = {a, a, b} in BEAF
a{{1} + 1}b in xOBN = a{{1}}b in BXO = {a, b, 1, 2} in BEAF
a{{1} + 2}b in xOBN = a{{2}}b in BXO = {a, b, 2, 2} in BEAF
a{{1} + 3}b in xOBN = a{{3}}b in BXO = {a, b, 3, 2} in BEAF
a{{1} + 4}b in xOBN = a{{4}}b in BXO = {a, b, 4, 2} in BEAF
a{{1} + c}b in xOBN = a{{c}}b in BXO = {a, b, c, 2} in BEAF
a{{1}.2}b in xOBN = a{{b}}a in BXO = {a, a, b, 2} in BEAF
a{{1}.2 + 1}b in xOBN = a{{{1}}}b in BXO = {a, b, 1, 3} in BEAF
a{{1}.2 + 2}b in xOBN = a{{{2}}}b in BXO = {a, b, 2, 3} in BEAF
a{{1}.2 + c}b in xOBN = a{{{c}}}b in BXO = {a, b, c, 3} in BEAF
a{{1}.3}b in xOBN = a{{{b}}}a in BXO = {a, a, b, 3} in BEAF
a{{1}.3 + 1}b in xOBN = a{{{{1}}}}b in BXO = {a, b, 1, 4} in BEAF
a{{1}.4}b in xOBN = a{{{{b}}}}a in BXO = {a, a, b, 4} in BEAF
a{{1}.4 + 1}b in xOBN = a{{{{{1}}}}}b in BXO = {a, b, 1, 5} in BEAF
a{{1}.5}b in xOBN = a{{{{{b}}}}}a in BXO = {a, a, b, 5} in BEAF
a{{1}.5 + 1}b in xOBN = a{{{{{{1}}}}}}b in BXO = {a, b, 1, 6} in BEAF
a{{1}.6}b in xOBN = a{{{{{{b}}}}}}a in BXO = {a, a, b, 6} in BEAF
a{{1}.c}b in xOBN = a{{{...{{{b}}}...}}}a with c nests of brackets in BXO = {a, a, b, c} in BEAF
a{{1}.d + c}b in xOBN = a{{{...{{{c}}}...}}}b with d nests of brackets in BXO = {a, b, c, d} in BEAF
a{{2}}b in xOBN = a{{{...{{{a}}}...}}}a with b nests of brackets in BXO, which is the limit of BXO. It is also equal to {a, a, a, b} in BEAF.
a{{2} + 1}b in xOBN onwards also eventually dominates the limit of BXO as well.