Update on "Some results on embeddings of algebras, after de Bruijn and McKenzie"

UPDATE ON Some results on embeddings of algebras, after de Bruijn and McKenzie, Indagationes Mathematicae 17(2007) 349-403.

Question 4.4, for the case of groups, has a negative answer, and Question 4.5 has a positive answer. See:

[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.

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