Full convexl-subgroups and the existence ofa∗-closures of lattice ordered groups
Richard N. Ball · Pacific Journal of Mathematics · 1975
An affirmative answer to the question of whether an arbitrary lattice-ordered group has an α*-closure is the main result of this paper.This result is obtained by first introducing the notion of a full convex Z-subgroup which is closely analogous to the notion of a closed convex Z-subgroup.The first two sections of this paper are a development of the basic properties of full convex Z-subgroups.In §3 we define an /extension of an Z-group, the direct analogue of the definition of an α*-extension.It is the existence of /-closures which we prove in §4; the existence of enclosures is a corollary to the proof.We believe the study of full convex Z-subgroups will continue to enrich the theory of lattice-ordered groups.G and H will denote lattice-ordered groups throughout.G ^ H will mean that G is an Z-subgroup of H.For a subset X of G, Cn (G, X) and Cl (G, X) will denote the smallest convex Z-subgroup of G containing X and the smallest closed convex Z-subgroup of G containing X, respectively.This notation will be shortened whenever the result is unambiguous; for example, Cn(G, {x}) and Cn(G, X[j{x}) may be written Cn (x) and Cn (X (J {x}) l Full convex ^-subgroups* A set X of positive elements of G is full if, for positive y, Cl (x) = Cl(y) and xeX imply yeX.A convex Z-subgroup will be said to be full if the set of its positive elements is full.THEOREM 1.1.For a convex l-subgroup C of the l-group G the following are equivalent:(i ) C is full.(ii) C is a union of closed convex l-subgroups.