Zeration = inconsistant ?

[tommy1729]

Hi, Tommy 1729!

But with zeration that is A PROBLEM.
zeration tries to define itself in terms of HIGHER operators.
There are NO lower operators defined than zeration so its seems necc.
Any attempt at trying to define zeration with lower hyperoperations , requires those lowers to be defined as well by even lower hyperoperations ?
That would PROBABLY give an infinite descent problem.

However properties like distributive require use of a lower hyperoperator : a * (b+c) = a*b + a*c.
UNLESS , and now we are getting at the heart of the post ,
a \(0\) (b + c) = a \(0\) b + a \(0\) c.
However after many attempts that also seems to lead to paradoxes.

tommy1729

Dear Tommy1729,
Hi again!

Concerning the supposed inconsistency of zeration, due to the non-validity of the distributive property at the ZER-PLUS levels, because it is definable via higher rank hyperoperations, I should like to start by clarifying that there seems to be a peculiar double misunderstanding in your above-mentioned example.

In fact, I suppose by mis-attention, you have mentioned a correct relationship, together with a wrong one, such as:
a * (b + c) = a*b + a*c correct!
where the stronger operation ‘distributes’ over the weaker, as well as a wrong ‘equality’ such as:
a [0] (b + c) = a [0] b + a [0]c, impossible, (first misunderstanding) because [0] is weaker than [+].

Let us see how this works in the framework of the ‘Field’ of ‘Real Numbers’ and, therefore, at the 0 - 3 ranks of the HOH (Hyper-Operations Hierarchical family). Actually, let us start at the 1 / 2 ranks (the PLUS-TIME level), where we have:
a * (b + c) = a*b + a*c, correct, because TIMES (stronger) distributes over PLUS, but:
a + b*c >< (a + b) * (a +c) the equality would be wrong, because PLUS is weaker.
[It would also do it, but in the idempotent Boolean Algebra, not in the ‘Real Field’].

Considering, now, the 0 / 1 ranks (the ZER-PLUS level), we have (sorry for the brackets), with [0]=o:
a + (b o c) = (a + b) o (a + c) correct, PLUS ‘dominates’ ZER, but:
a o (b + c) >< (a o b) + (a o c) the equality would be wrong, because ZER is ‘dominated’.
Example - let’s put: a = 2, b = 4, c = 7, we get:
2 + (4 o 7) = 2 + 8 = 10; and: (2 + 4) o (2 + 7) = 6 o 9 = 10, etc., but:
2 o (4 + 7) = 2 o 11 = 12 and: (2 o 4) + (2 o 7) = 5 + 8 = 13, !!!!!!!!

It is also useful to notice that, at the 2 / 3 ranks (the TIMES-POW level), we have:
a ^ (b * c) >< (a ^ b) * (b ^ c) and:
a * (b ^ c) >< (a * b) ^ (a * c) both wrong equalities, because the ‘level’ is not distributive.
[In this case, for levels including ranks s > 2, the distributivity does not hold, even if the operations are defined, or described, by lower rank hyperoperations]. Second misunderstanding.

Having said so, we may see that the PLUS-ZER distributivity holds, even if this would not be absolutely necessary to consider zeration as consistent, because (anyway) it doesn’hold at higher ranks.

However, there is a problem about considering -oo (minus infinite) as the ‘unit element’ of zeration (like it also happens in the ‘tropical’ Max-Plus algebra), just because -oo is NaN (Not a Number) and it does not belong to the set of the ‘Reals’. We neeed to define the actual oo, different from the oo of the classical ‘Limit Theory’ of Mathematical Analysis.

By the way, nobody ever said that there is zeration 'inside' Max-Plus algebra, but it is easy to see that the 'maximation' Max-Plus operation is a fundamental component of the RR-zeration definition.

Thank you for your interest and your kind attention.
Best regards.

Hi.
Thanks for your swift reply.

There are indeed some misunderstandings , but I think most are on your side.

Since there are no lower operations than zeration defined , I could not write a distributive property where zeration was - what you call - STRONGER.

Im unaware of a nice consistant definition of zeration
such that

a + ( b [0] c ) = ( a + b ) [0] ( a + c ).

So I tried to reverse everything : define zeration in terms of higher operators , instead of lower , and define distributivity in the REVERSE way.

It was therefore done intentionally !
NOT to try to ridicule or dismiss zeration but rather as an attempt to " fix it ".

I understand it looks like a silly mistake but it was not.
Have a bit more confidence in me

Im not sure what you consider as the MUST HAVE PROPERTIES OF ZERATION.

That matters alot if we want to talk about the same thing.

It seems you too want distributivity ?

It also appears you have already picked a particular interpretation of zeration ?

So what definition of zeration do you use such that


a + ( b [0] c ) = ( a + b ) [0] ( a + c )

for all reals ?

That is the key question.
Otherwise we might be talking about different things due to lack of consensus and agreed definitions.

---

Secondly and I quote :

By the way, nobody ever said that there is zeration 'inside' Max-Plus algebra, but it is easy to see that the 'maximation' Max-Plus operation is a fundamental component of the RR-zeration definition

(end quote)

Then if zeration is not within Max-Plus , why does Max-Plus matters to zeration or vice versa ?
That is not clear to me.

Also I do not know what "RR-zeration definition" is.

The Max-Plus algebra is nice and I also like the amoebe.
I discussed variants of those 2 with Timothy Golden , mainly in the direction of (algebraicly closed) higher dimensional numbers and uncountable sets.
Amoebe set theory and Amoebe ring theory if you like.
That in the far past, although Im still considering them today.
" Amoebe numerical methods " is what I am currently considering.
( of course those terms are made up by me , Im unaware of official namings for these concepts or prior deep investigations )
My friend Mick had some ideas to apply Amoebe for physics.
But that is background and Im going a bit offtopic.

regards

tommy1729