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

In 1957, N. G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ω contains a free subgroup on 2 generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality ≤ card(Ω), the group Sym(Ω) contains a coproduct of 2 copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras V, and formulated as a statement about functors Set→ V. From this one easily obtains analogs of the results stated above with “group” and Sym(Ω) replaced by “monoid” and the monoid Self(Ω) of endomaps of Ω, by “associative K-algebra” and the K-algebra EndK(V ) of endomorphisms of a K-vector-space V with basis Ω, and by “lattice” and the lattice Equiv(Ω) of equivalence relations on Ω . It is also shown, extending another result from de Bruijn’s 1957 paper, that each of Sym(Ω), Self(Ω) and EndK(V ) contains a coproduct of 2 copies of itself. That paper also gave an example of a group of cardinality 2 that was not embeddable in Sym(Ω), and R. McKenzie subsequently established a large class of such examples. Those results are shown here to be instances of a general property of the lattice of solution sets in Sym(Ω) of sets of equations with constants in Sym(Ω). Again, similar results – this time of varying strengths – are obtained for Self(Ω), EndK(V ), and Equiv(Ω), and also for the monoid Rel(Ω) of binary relations on Ω . Many open questions and areas for further investigation are noted. 1 Conventions, and outline. Throughout this note, Ω will be an infinite set. Each ordinal (in particular, each natural number) is understood to be the set of all smaller ordinals; the set of all natural numbers is denoted ω. Functions, including elements of permutation groups, will be written to the left of their arguments and composed accordingly. The word “algebra” will be used in the sense of general algebra (universal algebra), except in the combination “K-algebra”, which will always mean an associative unital algebra in the sense of ring theory, over a field K assumed fixed throughout this note. In those contexts, V will denote a vector space with basis Ω over that field K. In §§2-3 we develop results to the effect that algebras arising as values of certain sorts of functors can be embedded in certain infinite direct product algebras, and obtain, as immediate corollaries, results on embeddability of groups, monoids, K-algebras, and lattices in the group Sym(Ω), the monoid Self(Ω), the K-algebra EndK(V ), and the lattice Equiv(Ω) respectively (all defined as in the abstract). The remaining sections obtain results specific to embeddings in one or another of those four structures, and in the monoid Rel(Ω). In §4 (and two appendices, §§10-11) it is shown that one can embed into each of the first three of these algebras a coproduct of 2 copies of that same algebra, while §§5-8 obtain restrictions on algebras A embeddable in these five algebras, in terms of order-properties of chains of solution sets of systems of equations in A. §9 suggests some ways in which the results of this note might be extended. ∗2000 Mathematics Subject Classifications. Primary: 08B25. Secondary: 06Bxx, 08B20, 16S50, 18A99, 20B07, 20M20, 54Hxx. This preprint is readable online at http://math.berkeley.edu/∼ gbergman/papers/ , and arXiv:math/0606407 . The former version is likely to be updated more frequently than the latter.