Some results on embeddings of algebras, after de Bruijn and McKenzie
Abstract
In 1957, N. G. de Bruijn showed that the symmetric group Sym() on an infinite set contains a free subgroup on 2card() generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality ≤ card(), Sym() contains a coproduct of 2card() 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() replaced by "monoid" and the monoid Self() of endomaps of , by "associative K-algebra" and the K-algebra EndK(V) of endomorphisms of a K-vector-space V with basis , and by "lattice" and the lattice Equiv() of equivalence relations on . It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(), Self() and EndK (V) contains a coproduct of 2card() copies of itself. That paper also gave an example of a group of cardinality 2card() that was not embeddable in Sym(), and R. McKenzie subsequently established a large class of such examples. Those results are shown to be instances of a general property of the lattice of solution sets in Sym() of sets of equations with constants in Sym(). Again, similar results -- this time of varying strengths -- are obtained for Self(), EndK (V), and Equiv(), and also for the monoid of binary relations on . Many open questions and areas for further investigation are noted.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.