Some results on embeddings of algebras, after de Bruijn and McKenzie
- Autores: George M. Bergman
- Localización: Indagationes mathematicae, ISSN 0019-3577, Vol. 18, Nº 3, 2007, págs. 349-403
- Idioma: inglés
- DOI: 10.1016/s0019-3577(07)00018-3
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Resumen
In 1957, N.G. de Bruijn showed that the symmetric group Sym(O) on an infinite set O contains a free subgroup on 2card(O) generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality card(O), the group Sym(O) contains a coproduct of 2card(O) 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(O) replaced by �monoid� and the monoid Self(O) of endomaps of O, by �associative K-algebra� and the K-algebra EndK (V) of endomorphisms of a K-vector-space V with basis O, and by �lattice� and the lattice Equiv(O) of equivalence relations on O. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(O), Self(O) and EndK(V) contains a coproduct of 2card(O) copies of itself.
That paper also gave an example of a group of cardinality 2card(O) that was not embeddable in Sym(O), 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(O) of sets of equations with constants in Sym(O). Again, similar results - this time of varying strengths - are obtained for Self(O), EndK(V), and Equiv(O), and also for the monoid Rel(O) of binary relations on O.
