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By Brian Osserman

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**Extra info for Axioms for the integers**

**Example text**

In our case, we have (r;,r;) = 2 and otherwise each (r;,rj) is 0 or 1. Supposing this, we select a Weyl vector w with (^,rj) < 0 for each r;. Then the test is 0. if (r, w) > 0, replace r by r 1. if r G {±rx, ± r 2 , . . , ±r n } then r is a root vector 2. otherwise replace r by its reflection in any r* for which (r,ri) and (r,w) have opposite signs, and repeat. ) This is easily proved. The reflecting hyperplanes of a discrete reflection group in hyperbolic space divide that space into copies of what is called the fundamental region, which may be taken to be the intersection of certain half-spaces, say (rt-,x) < 0.

Where L= 000111 000111 , R= 000012 111000 111111 000111 , sothat £ + # = 000222 = N, say. 000021 000033 This is equivalent to any equation of the form u = v, where u and v are any two Monster roots whose sum or difference is N. Geometrically, the fundamental root vectors of Y$2i define a semidefinite space in which the unique null vector (up to scalar factors) is N. The corresponding abstract Coxeter group is an infinite group of structure Z 8 : W, where W, the Weyl group of jBg, is the finite group defined by the Coxeter relations of Y^2\.

For example we shall show that we have (omitting the redundant coordinate s) 000111 111000 000111 = 111000 00001* 00001* (in System 2) which in view of our Si 2 x S$ of coordinate permutations implies that the reflection in any vector of shape 0 6 l 6 | 04l represents the same element of G as that in the corresponding vector 16O6 | 04l obtained by interchanging digits "0" and "1" before the |, but fixing those after it. We shall say that 16O6 | 04l is an alias for 0 6 l 6 | 0 4 l, and that the digit permutation (0 1) |~ belongs to the alias group for this vector.

### Axioms for the integers by Brian Osserman

by Steven

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