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i have been doing a lot of surreal analysis lately & it dawn'd on me that star in the surreal numbers has the same properties of a simple xor cipher.

star operations are given as:

x + * = { x | x }

0 + * = { 0 | 0 } 

* = { 0 | 0 }

0 + * = * 

* + * = 0

xor ciphers are given as:

⊕xor 

a ⊕ a = 0

a ⊕ 0 = a

surreal numbers in the normal form are constructed as follow:

x = { xl | xr } where xl < xr

the property of xl < xr in surreal numbers appears equivalent to xor linked lists where each address consists of the previous address & the next address:

...  A        B         C         D         E  ...

        <–>  A⊕C  <->  B⊕D  <->  C⊕E  <->

update

after playing around w/ tiny & miny games, i have found a simple connection between xor, xnor & *.

tiny & miny games are of the form

tiny(x) = {0|{0|-x}}
-tiny(x) = miny(x) = {{x|0}|0}

xnor

Y = ((A XOR B) XOR C)

xor/xnor in tiny/miny terms

a xor b = {tiny(a)|miny(b)} = {{0|{0|-a}}|{{b|0}|0}
a xnor b = {miny(a)|tiny(b)} = {{{a|0}|0}|{0|{0|-b}}}

x xor x = {tiny(x)|mini(x)} = {{x|{x}}|{{x}|x} 
x xnor x = {miny(x)|tiny(x)} = {{{x}|x}|{{x|{x}}}

zero

0 xor 0 = {{0|*}|{*|0}} = {UP|DOWN} = *
0 xnor 0 = {{*|0}|{0|*}} = {DOWN|UP} = 0
0 xor 0 xor 0 = 0 xnor 0 = 0
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closed as off-topic by YCor, Mark Sapir, Gerald Edgar, Gro-Tsen, Mikhail Katz May 23 '18 at 9:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Gerald Edgar, Mikhail Katz
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Not sure that the question makes much sense, but at least for context: en.wikipedia.org/wiki/XOR_cipher $\endgroup$ – YCor May 22 '18 at 22:19
  • $\begingroup$ just found this question (re: xor & star) here : mathoverflow.net/questions/211745/nimbers-and-surreal-numbers $\endgroup$ – meowzz May 22 '18 at 22:22
  • $\begingroup$ please edit your question properly, giving the relevant definitions explicitly, so it stands on its own, make an effort; what's nimber addition? etc. etc. Otherwise it will sink without a trace, especially since it doesnt seem to be research level. $\endgroup$ – kodlu May 22 '18 at 22:47
  • $\begingroup$ @kodlu hopefully that is better! $\endgroup$ – meowzz May 22 '18 at 23:05
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    $\begingroup$ Probably you should explain that $\ast$ is a particular game, and that it makes sense to add it with surreal numbers, which can also be viewed as particular kinds of games. This will help with the reception of your question, among those who are not necessarily aware that there is a rich algebra here. I think the question is interesting. $\endgroup$ – Joel David Hamkins May 22 '18 at 23:15