Hereditarily projectable Archimedean lattice-ordered groups with unit

Abstract

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 ⊆ W, we say that a W-object H is hereditarily C, denoted H ∈ hC, if G ∈ C whenever G is a W-subobject of H. Our focus is hP. Taking off from the known items P = wP ∩ Loc, h HA = HA ⊆ P, and C (X) ∈ 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 ∈ hP with G ∉ HA has Yosida space Y G = αN, the one-point compactification of N. There are various G ∈ hP with Y G = αN, including: G bounded not HA; G not bounded with bounded part HA; G not bounded with bounded part not HA.