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I assume the infinitesimal gives a new sort algebra similar to for instance binary algebra.
For me to understand this matter better I would like to know how to work with infinitesimal theory.
Therefore it would like to have the algebraic rules:
^+^=^^ ^+DOWN=0 ^+*=^*
^=-DOWN
Also, at some point it is mentioned that infinitesimal algebra transforms to numbers. But also for the numbers the algebra is missing.
Am I approaching this from the wrong perspective, or can you give this information.
Thanks in advance
Combinatorial games, including infinitesimals, form a group under addition. As for infinitesimals, * + * = 0, and otherwise, an infinitesimal plus its negative equals 0. Otherwise, infinitesimal expressions do not reduce. Irreducible combinations of infinitesimals are often written without the plus sign. Thus ^* is the same as ^+*.
If you want to delve deeper into CGT, see On Numbers and Games or Winning Ways.
Thx Bill,
One remark about this, it seems to me that you reduced the expression ^+* to ^*, since i can imagine that you have a group that is ^* and that inherently this group equals two different groups, a * and a ^.
So to conclude we can have
^*+^+*+^^-^-^=^
or is this calculation wrong?
and then, when we have a surplus of ^^ we always have the final play, I read that somewhere, is it true?
One remark about this, it seems to me that you reduced the expression ^+* to ^*, since i can imagine that you have a group that is ^* and that inherently this group equals two different groups, a * and a ^.
Yes, you can have a group that is ^* in chilled go. And it is equal to ^ + *.
So to conclude we can have
^*+^+*+^^-^-^=^
or is this calculation wrong?
As I count the ^s, it reduces to ^^.
and then, when we have a surplus of ^^ we always have the final play, I read that somewhere, is it true?
I think that has to do with atomic weight. Atomic weight is approximately equal to a number of ^s. An atomic weight of 2 is enough to get the final play.
^ itself is positive, which means that Left can get the last play. ^* also has an atomic weight of 1, but Right can win by playing first. However, ^^* has an atomic weight of 2, and Left wins even if Right plays first.