(10/02/2014, 11:11 PM)tommy1729 Wrote: But what bothers me most is that zeration , unlike addition and multiplication does not have an inverse !!!

a - b , a / b

..............

Take the equations :

a + b = c

a = c - b

a * b = c

a = c/b

max(a,b) + 1 = c

a = max^-1(c - 1,b) ??

fail.

tommy1729

As a matter of fact, Max-Plus algebra, applied to the 'Reals', is not a 'Field' but an 'Idempotent Semi-Ring', where a[max]b has some 'peculiarities':

a[max](-oo) = (-oo)[max]a = a ; -oo is the 'unity element'

a[max]a = a ; [max] is idempotent

and there is NO max-inverse number.

For zeration, this is only partially true. In fact

a o (-oo) = (-oo) o a = a ; but: a o a = a + 2.

We should discuss here the fact that Rubtsov thinks (and I agree) that zeration HAS an inverse operation, that he calls 'Deltation', producing a new set of numbers, called the 'Delta Numbers', which can be put in correlation with the log of negative numbers (multi-valued Complex numbers, but also ... transfinite extension of the 'Reals'). This would need a separate detailed discussion.

A similar position is taken by Cesco Reale, with his definition of 'incrementation' and 'decrementation', giving an extension of the Reals that he calls 'Stigma-Reals'. Also to be seen.

Regards!