Weakly least integer closed groups

Feebly projectable ℓ-groups

“…For more information on the full subcategory of W consisting of singular W-objects, the reader is referred to [10]. Surprisingly, our next proposition is not stated anywhere, even in [9]. Proposition 4.10.…”

“…Let (G, e G ) ∈ W. We call (G, e G ) weakly projectable if for each a ∈ G, cl Y G coz(a) is a clopen subset of Y G. (Note: this definition has nothing to do with the stranded primes property. See [9] for more information on weakly projectable -groups.) It follows that an archimedean -group G containing any weak order unit is projectable if and only if G has a weak order unit e G such that (G, e G ) is local and weakly projectable.…”

“…If (G, e G ) is a singular W-object, then loc(G) is a projectable -group by Proposition 5.5 of [10]. But then Proposition 4.1 of [9] states that loc(G) is projectable if and only if G is weakly projectable. Therefore, a singular W-object is weakly projectable.…”

From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra

“…The next lemma is a consequence of more general results, cf. [14,Proposition 2.4]. We provide a proof here, for the reader's convenience.…”

“…(1)⇒(2) Recall that P (G) is the ℓ-subgroup of C (w G Z G ) generated by the set (14). Hence by the elementary theory of lattice-groups we can write e as i∈I j∈J…”