J. Austral. Math. Soc. 19 (Series A) (1975), 337-342. FREE I-GROUPS AND VECTOR LATTICES

The purpose of this paper is to present three somewhat disparate results on free objects in three different classes of /-groups. The first is that no proper ideal of a finitely generated free vector lattice can itself be a free vector lattice. Second, each free abelian /-group is characteristically simple. The third result is that each disjoint subset of a free (non-abelian) /-group is countable. The reader is referred to Conrad (1970) for the general algebraic theory of /-groups and (real) vector lattices. We review here only the standard definitions of freedom. A vector lattice Fis free if V ^ 0 and V possesses a generating subset S such that each function a: S-> W, where Wis a vector lattice, extends to a vec-tor lattice homomorphism a: V-> W. We say V is free on S. Free /-groups and free abelian /-groups are similarly defined. Given any non-empty set S there exists a (unique up to isomorphism) free vector lattice on S. A very useful model for this free vector lattice was given in Baker (1968), and earlier in a generalized form in Henriksen and Isbell (1962). We confine our attention here to the case that S is finite of cardinality n. The set W of all functions/: R"->-R with pointwise operations is a vector lattice. The sublattice of W generated by the linear functions is (isomorphically) the free vector lattice on S. It will be denoted by FVLn. S is to be identified with the set of coordinate projection maps nt: R "-> R (i = 1, ••-,«). We have FVLn = {/: R "-> R | / = V/Aj/y where / and J are finite and the/y are linear}. In particular, each feFVLn is continuous and positively homogeneous (f(rx) = rf(x) for all xeW and O g r e R). The set T(f) = {XGR" | / (X) = £ 0} is open since / is continuous. Let S""1 denote the unit sphere in R". The map restricting feFVLn to S""1 is an isomorphism (by positive homogeneity). It is often convenient to identify FVLn with this isomorphic copy. In this case T(f) = {xeS"'1 \f(x) # 0} is an open subset of the sphere. K. Baker (1968; Lemma 3.2) showed that the sets T(f), feFVLn, form a base for the topology of S"-i