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  has appeared:
 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 , that Self ( Ω )op cannot be embedded in Self ( Ω ), is actually proved in
 Araújo J. and Wehrung F. -- Embedding properties of endomorphism semigroups. Fund. Math. 202 (2009), 125--146, MR 2010c:20077.
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