Axioms for the integers by Brian Osserman

By Brian Osserman

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We make P into a monoid with binary operation (aI/3). For a E P we set AaE = k if a = 0 and AaE = AalE ®. ®A'-E if a = (a,, ... ). In the same way, we setAaV=kifa=0andAaV=Aa1V® ®AO-Vif a = (a1, ... , a,,,). We set P(n, 0) = {0}. For r > 0 we write P(n, r) for the set of a = (a1, ... , a,,,) E P(n) such that El ai = r. We write P+(n, r) for the set of partitions in P(n, r) (for r > 0). e. the partition obtained by arranging the terms in a in descending order. We shall use the natural (dominance) partial order on partitions.

9) Let 1 < a < r and o E Sym(r). We have _ C'U3a,160 - if o(a) < o(a + 1); if o(a) > o(a + 1) gcu,uasa , cu,uasa + (q - 1)cu,uo and cvo,vs, _ if o(a) < o(a + 1); if o(a) > o(a + 1). gcvosa,v , vvasa,v - (q - 1)wo,v, From (9) we obtain the following. (10) Let 1 < a < r and o E Sym(r). We have if l(osa) = 1(0) + 1; if 1(o8a) = 1(o) - 1 b3a ba - bas, b'a 3a b' = r bO3a, if l(osa) = 1(o) + 1; l gbO3a - (q - 1)bo, if 1(0-sa) = 1(o) - 1. gba5a + (q - 1)ba, and We write Ta for ba-,, o E Sym(r). The first part of (10) may thus be reformulated as follows.

14, that I(n, r) denotes the set of functions i : [1, r] [1, n]. We shall often write i E I(n, r) as the sequence (i1, ... , ir) (where is = i(a), 1 < a < r). Let i = (i1 i ... , ir) E I(n, r). We define ei = ei, ® ® eir E E®'* and vi = vi, ® ® vi,. E V®'. We define ei = ei, A A eir E AE, the image of ei under the natural map E®r -> ArE and define vi = vi, A A vi, E A' V, the image of vi under the natural map V®r->ArV. We write P = P(n) for a certain set of sequences of non-negative integers. We define P to be 0 together with sequences a = (a1, ...

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