[23] On coproducts in varieties, quasivarieties and prevarieties,
Algebra and Number Theory
**3** (2009)
847-879,
MR 2587406.
abstract.
preprint

As noted there, under slight additional conditions, the group case of Question 4.4 remains open. A variant of Question 4.5 is shown there to have a negative answer, while other possible variants are noted which remain open.

Reference [20] has appeared:

[20] Wehrung, F. -- Embedding coproducts of partition lattices,
Acta Sci. Math. (Szeged) **73** (2007), 429--443,
MR **2008m**:06011.

However, a result that I described as occurring
in [20], that Self ( Ω )^{op} cannot be embedded
in Self ( Ω ), is actually proved in

[22] Araújo J. and Wehrung F. --
Embedding properties of endomorphism semigroups.
Fund. Math. **202** (2009), 125--146,
MR **2010c**:20077.