# Conway Chained Arrow Notation - Properties

Properties

1. A chain of length 3 corresponds to Knuth's up-arrow notation and hyper operators:
$begin{matrix} p to q to r = text{hyper}(p,r+2,q) = p !!! & underbrace{ uparrow dots uparrow } & !!! q = puparrow^r q.\ & !!! r text{ arrows} !!! end{matrix}$
2. a chain XY is of the form Xp; hence:
3. a chain starting with a is a power of a
4. a chain 1 → Y is equal to 1
5. a chain X → 1 → Y is equal to X
6. a chain 2 → 2 → Y is equal to 4
7. a chain X → 2 → 2 is equal to X → (X) (chain X with its value concatenated to it)

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The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.
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