Hereditarily projectable Archimedean lattice-ordered groups with unit

Let W be the category of Archimedean lattice-ordered groups with distinguished weak unit and unit-preserving l-group homomorphisms. We denote by P, wP, Loc, HA, respectively, the collections of projectable, weakly projectable, local, and hyperarchimedean W-objects. For any C subset of W, we say that a W-object H is hereditarily C, denoted H is an element of hC, if G is an element of C whenever G is a W-subobject of H. Our focus is hP. Taking off from the known items P = wP boolean AND Loc, h HA = HA subset of P, and C (X) is an element of P iff X is basically disconnected (BD), we show For C (X), equivalent are: hP; hLoc and X BD; X finite.If the Yosida space Y G is BD, then (1) If G is hLoc, then G is bounded; (2) If G is hP, then G is HA.hP = hwP, and every G is an element of hP with G is not an element of HA has Yosida space Y G = alpha N, the one-point compactification of N.There are various G is an element of hP with Y G = alpha N, including: G bounded not HA; G not bounded with bounded part HA; G not bounded with bounded part not HA.