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

In the present work, we shall construct some non-essential H-closed epireflections of W that are not comparable with any other known H-closed epireflections of W other than the divisible hull and the epicompletion. We show first that the free objects in any H-closed epireflective subcategory must be closed under composition (see Section 2 for a precise definition), and that any epic extension of a free W-object on n generators that is closed under composition is actually the free object on n generators in some H-closed epireflective subcategory of W. We then apply these results to certain ℓ-groups of almost-piecewise-linear Baire functions on R. By definition, a function f:R→R is almost-piecewise-linear if there is a finite point set S⊂R such that f is piecewise-linear on the complement of any neighborhood of S.