A lattice homomorphism of a lattice ordered group

Abstract

The homomorphism we have in mind has been introduced by Jaffard, [1],2 for lattice ordered groups, and has been discussed by Pierce, [2], for any distributive lattice L with a minimal element. It may be defined as the homomorphism a of L for which a(x) =a(y) if and only if the set of elements in L disjoint with x is the same as the set disjoint with y. Pierce has shown that this is the maximal lattice homomorphism of L whose kernel is [0], the set whose only element is 0, and that it is the only lattice homomorphism of L, with kernel [0], whose image is disjunctive. Jaffard called the elements of the image lattice (i.e., the equivalence sets of the homomorphism) filets in L, but we prefer the term carriers in L because of their role in function spaces. Our purpose is to obtain a characterization of a, which involves the group operation, for the case where L is the positive cone of an archimedean lattice ordered group. Jaffard has shown, in this case, that a has the following properties: (a) a: O implies a(a) >a(O). (b) a is a lattice homomorphism; i.e., a(aUb) =a(a)Ua(b), a(anb) a (xa) rxab) . (c) a(a+b) =a(aUb). We show that a has the additional property: (d) If x=sup [xi|iEI], then sup [a(xi)IiEI] exists and a(x) = sup [a(xifl|iEII]. PROOF. First, a(x) _a(xi) for all iEI. Suppose there is z>O such that a(z) >a(x,), for all iEI, but a(z) 0 such that a(w) ? a(x) and a(w)ra(z) =a(0). Then a(w)Oa(xi) =a(0), for all iEI. Let v=xCxw. Then a(v) =a(x) a(w) =a(w), and v>O. It follows that vqxi=O, for all iEI. Now, v<x, xs<x, and vflxi=O implies that v+xi=vUxi<x for all iEI. If we let y=x-v, then y<x and y2xi for all iEI. But this contradicts the definition of x.