It is not exactly the supefunction of zeration (max-kroenecker delta definition)...but it is the superfunction of Bennett's base 2 preaddition \( \odot_{-1} \) (-1th rank in base 2 commutative hos hierarchy).
\( A\odot_{i} B=\exp_2^{\circ i}[\log_2^{\circ i}(A) +\log_2^{\circ i}(B)] \)
\( A\odot_0 B=A+B \)
\( A\odot_{-1} B=\log_2(2^A +2^B) \)
In fact you are right: define plusation (set it as rank 1 of a new sequence)
\( b(+)_1x=b+\log_2(x) \)
\( b(-)_1 x=2^{x-b} \)
Let's take it's subfunction in the variable x (i.e.\( F\mapsto f \) where \( F(x+1)=f(F(x)) \) )
\( b(+)_0x=b(+)_1(1+b(-)_1 x)=\log_2(2^{x}+2^b)=b\odot_{-1}x \)
But the sequence \( (+)_t \) doesn't seem much interesting or natural imho... the only nice properties are that
1) it is based on the usual recursion/iteration law (ML \( b*_{i+1}x+1=b*_i(b*_{i+1}x) \) )
2) it intersects the 2-based commutative hos sequence at t=0 (\( b(+)_0x=b\odot_{-1}x \))
\( A\odot_{i} B=\exp_2^{\circ i}[\log_2^{\circ i}(A) +\log_2^{\circ i}(B)] \)
\( A\odot_0 B=A+B \)
\( A\odot_{-1} B=\log_2(2^A +2^B) \)
In fact you are right: define plusation (set it as rank 1 of a new sequence)
\( b(+)_1x=b+\log_2(x) \)
\( b(-)_1 x=2^{x-b} \)
Let's take it's subfunction in the variable x (i.e.\( F\mapsto f \) where \( F(x+1)=f(F(x)) \) )
\( b(+)_0x=b(+)_1(1+b(-)_1 x)=\log_2(2^{x}+2^b)=b\odot_{-1}x \)
But the sequence \( (+)_t \) doesn't seem much interesting or natural imho... the only nice properties are that
1) it is based on the usual recursion/iteration law (ML \( b*_{i+1}x+1=b*_i(b*_{i+1}x) \) )
2) it intersects the 2-based commutative hos sequence at t=0 (\( b(+)_0x=b\odot_{-1}x \))
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)