Composition-closed l-groups of almost-piecewise-linear functions

We construct some non-essential H-closed epireflections of W, the category of archimedean l-groups with weak order unit, that are not comparable with any known H-closed epireflections of W other than the divisible hull and the epicompletion. In preparation, we show that the free objects in any Hclosed epireflective subcategory of W must be closed under composition (see section 2 for a precise definition), and that any epic extension of a finiteor countably-generated free W-object that is closed under composition is actually a free object in some H-closed epireflective subcategory of W. We apply these results to certain l-groups of almost-piecewise-linear Baire functions on R to obtain the examples. By definition, a function f : R → R is almost-piecewiselinear if there is a finite point set S ⊂ R such that f is piecewise-linear on the complement of any neighborhood of S. MSC classification: primary 06F20, secondary 06D22, 18B30, 26A30