03/21/2015, 11:11 PM
Not that I have become a " believer " but apart from
a[0]b = max(a,b) + 1
reference : {max,+} algebra , linear algebra.
a[0]b = ln( exp(a) + exp(b) )
reference : Bennet Hyperoperations, homomorphic defined operations and Tommy's distributive property.
Another thing makes sense , and will probably lead to the above 2 ... but if not that would be intresting ( I doubt it ! ) :
Rethinking : what is required for an operator [q] such that
a[q]b = b[q]a
and this is somehow related to iterations ?
Probably this equation :
Let N be a " neutral element " as a function of q.
Call this N_q.
then
a[q]b = b[q]a
implies :
f and g are some function :
f^[a - N_q](N_q[q]b) = g^[b - N_q](N_q[q]a)
that makes sense !
example [q] = +
a+b = b+a
"add+1"^[a - 0](0+b) = "add+1"^[b - 0](0+a)
example [q] = *
a*b = b*a
"add+b"^[a-1](1*b) = "add+a"^[b-1](1*a)
However notice that an expression like
f^[a - oo](...) does not make sense as a nonconstant function.
Also combining " all nice properties " seems impossible for zeration.
For instance max(a,b)+1 is not an analytic function.
Or a^^b =/= b^^a ...
regards
tommy1729
a[0]b = max(a,b) + 1
reference : {max,+} algebra , linear algebra.
a[0]b = ln( exp(a) + exp(b) )
reference : Bennet Hyperoperations, homomorphic defined operations and Tommy's distributive property.
Another thing makes sense , and will probably lead to the above 2 ... but if not that would be intresting ( I doubt it ! ) :
Rethinking : what is required for an operator [q] such that
a[q]b = b[q]a
and this is somehow related to iterations ?
Probably this equation :
Let N be a " neutral element " as a function of q.
Call this N_q.
then
a[q]b = b[q]a
implies :
f and g are some function :
f^[a - N_q](N_q[q]b) = g^[b - N_q](N_q[q]a)
that makes sense !
example [q] = +
a+b = b+a
"add+1"^[a - 0](0+b) = "add+1"^[b - 0](0+a)
example [q] = *
a*b = b*a
"add+b"^[a-1](1*b) = "add+a"^[b-1](1*a)
However notice that an expression like
f^[a - oo](...) does not make sense as a nonconstant function.
Also combining " all nice properties " seems impossible for zeration.
For instance max(a,b)+1 is not an analytic function.
Or a^^b =/= b^^a ...
regards
tommy1729