... be l-isomorphic. Theorem D. If A is a root system satisfying the descending chain condition and .... On 77, define (qx, m)(q2, n) to equal .... -g + M + g c -2g + M + 2g. Since M ...... there exists an r £ 32 such that g, = rx + t for some / £ 5ZA 32x .
transactions of the american mathematical society Volume 330, Number 2, April 1992
LATTICE-ORDEREDGROUPS WHOSE LATTICES DETERMINE THEIR ADDITIONS PAUL F. CONRAD AND MICHAELR. DARNEL Abstract. In this paper it is shown that several large and important classes of lattice-ordered groups, including the free abelian lattice-ordered groups, have their group operations completely determined by the underlying lattices, or determined up to /-isomorphism.
1. Introduction In the group of integers Z with the usual order 0, (c) For all g£G,
g/\x(g) >0. r(g) e g".
(d) For every polar P, x(P) —P. (e) For every minimal prime M, x(M) = M.
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P. F. CONRAD AND M. R. DARNEL
Proof, (a) =>(b) If £ a x(g) = 0, then 0 = x(g) A x(g) = x(g Ag) = x(g) and so g = 0. (b) => (c) Assume g > 0 but x(g) £ g" . Then there exists 0 < h £ g' such that x(g)Ah > 0 and so gAx~x(h) > 0. But then [^At_1(«)]At[^At_1(«)] > 0, and so 0 < g A x~x(h) A x(g) Ah•(d) =>■(e) Is easily adapted from similar results about p-endomorphisms. (e) =» (a) Suppose \g\ A |«| = 0 but \g\ A \x(h)\ > 0. There exists a minimal prime M not containing \g\ and \x(h)\. Since \g\ A \h\ = 0, h £ M, contradicting x(h) £ M. D
Proposition 2.5. The p-permutations of an I-group G form a normal subgroup of the group of all lattice automorphisms of G that preserve the identity. Proposition 2.6. If x is a p-permutation -[t(-x)] is also a p-permutation of G.
of an l-group G, then a : x —>
Proposition 2.7. [4] If (G, -g + (x ® g) is a + g = M © g, and thus
=> is pG
We now look at the class %?* of /-groups that satisfy (*) of Theorem 3.4.
Proposition 3.10. If (G, we can assume that each g¡ is the characteristic function of a subset A, of A. Then
G= l^
+^ts,]) œ-œ í$^
+^te"]) .
So it suffices to show that each £A &+&[gi] has essentially only one addition. So let © be another addition on 77 = £A 32 + 32[g¡] so that we get a real vector lattice. We can assume that © is connected to + by some lattice automorphism x = (..., xx, ...), that (t(77), (0,0) if a > 0 and bai > 0. Then with componentwise addition + , (G,
