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Initial values for hyper operations
#3
bo198214 Wrote:The problem can be equivalently (and even more canonically) posed with argument 0

a [0] 0 = 1, no right neutral element e, x[0]e=x => e=x-1
a [1] 0 = a, neutral element 0
a [2] 0 = 0, neutral element 1
a [3] 0 = 1, right neutral element 1, no left neutral element e[3]x=x => e=x^(1/x)
a [4] 0 = 1
a [3L] 0 = a
Thank you Henryk, for your comments. I know that KAR is carefully following this, but he is extremely busy with his job, sharing his time between Belgorod (Russia) and Martin (Slovakia). In the meantime, I shall try to keep contacts within this thread, awaiting that the "father" of zeration will react personally.

I agree that there are a lot of (infinite?) operations with a rank lower than (that of) addition. You proposed a variant of definition (I am still thinking of it) and Quickfur also defined one, by correctly adding that it was not zeration. The problem, as you also correctly said, is to define an operation (ONE hyperop) that would be the unique operation, exactly fitting in the hyperops hierarchy, at rank 0. Agreed. I am carefully studying all your comments and observations.

As a matter of documenting the KAR version of zeration, until now .... heavily supported also by me (no problem, I can easily change of mind, if necessary), I should like to supply the following tables, always supposing to take the following definition for zeration:

a[0]x = max[a,x] + 1 /; a >< x
a[0]x = a + 2 = x + 2 /; a ==x.


We should have:

a[0]0 = a + 1, for a >< 0
a[1]0 = a
a[2]0 = 0
a[3]0 = 1,
for a >< 0
a[4]0 = 1, for a >< 0
....

0[0]a = a + 1 for a >< 0
0[1]a = a
0[2]a = 0
0[3]a = 0
, for a < 0 ;
0[3]a = 1, for a > 0;
0[4]a = 1, for a > 0 (?)
....

a[0]1 = a + 1, for a >< 0
a[1]1 = a + 1
a[2]1 = a
a[3]1 = a

a[4]1 = a
....

1[0]a = a + 1 for a >< 0
1[1]a = a + 1
1[2]a = a
1[3]a = 1

1[4]a = 1, for a > 0 (!)

Concerning the uniformity of behaviour of the hyperops at what we might call their "initial values", the overall situation is rather messy, exactly for the classical ranks: 1, 2, 3.
Quote:.........
And now we also see the pattern. But this can be reformulated that there are no operations below [0] that obey our general rule (*), i.e. the hierarchy starts at 0.
So, no possibility to have: a[-1]a = a[0]2 ?
Schade ...! Aber, warum nicht. Smile It might even be justified by the Ackermann Function. Nevertheless, this is not the KAR's opinion, as far as I know.

GFR
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Messages In This Thread
Initial values for hyper operations - by bo198214 - 03/14/2008, 10:44 AM
RE: Initial values for hyper operations - by GFR - 03/19/2008, 11:03 AM

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