George M. Bergman - abstracts of preprints and recent reprints

Homomorphic images of pro-nilpotent algebras

It is shown that any finite-dimensional homomorphic image of an inverse limit of nilpotent not-necessarily-associative algebras over a field is nilpotent.  More generally, this is true of algebras over a general commutative ring  k,  with "finite-dimensional" replaced by "of finite length as a  k-module". 

These results are obtained by considering the multiplication algebra  M(A)  of an algebra  A  (the associative algebra of  k-linear maps  A --> A  generated by left and right multiplications by elements of  A),  and its behavior with respect to nilpotence, inverse limits, and homomorphic images. 

As a corollary, it is shown that a finite-dimensional homomorphic image of an inverse limit of finite-dimensional solvable Lie algebras over a field of characteristic 0 is solvable. 

Examples are given showing that infinite-dimensional homomorphic images of inverse limits of nilpotent algebras can have properties far from those of nilpotent algebras; in particular, properties that imply that they are not residually nilpotent. 

Several open questions are noted. 

17pp., 13 Oct., 2009 (slightly revised 14 Oct.).  tex dvi pdf ps.  Back to publications-list. 

Homomorphisms on infinite direct product algebras, especially Lie algebras  (with Nazih Nahlus)

Our results have the following consequences when  k  is an infinite field, the algebras are Lie algebras, and  B  is finite-dimensional:

If all the Lie algebras  A_i  are solvable, then so is  B. 

If all the Lie algebras  A_i  are nilpotent, then so is  B. 

If  k  is not of characteristic 2 or 3, and all the Lie algebras  A_i  are finite-dimensional and are direct products of simple algebras, then, (i) so is  B,  (ii)  f  splits, and (iii) under a weak cardinality bound on  I,  f  is continuous in the pro-discrete topology.  A key fact used in getting (i)-(iii) is that over any such field, every finite-dimensional simple Lie algebra  L  can be written  L = [x₁, L ] + [x₂, L ].  for some  x₁, x₂ \in L , which we prove from a recent result of J. M. Bois. 

The general technique of the paper involves studying conditions under which a homomorphism on  \prod_I A_i  must factor through the direct product of finitely many ultraproducts of the  A_i. 

Several open questions are noted. 

32pp., last revised 28 Oct., 2009.  tex dvi pdf ps.  arXiv:0910.5181  Back to publications-list. 

Linear maps on  k^I,  and homomorphic images of infinite direct product algebras  (with Nazih Nahlus)

These results are used to prove that any homomorphism from a direct product  \prod_I A_i  of not-necessarily-associative algebras  A_i onto an algebra  B,  where  dim_k(B)  is "not too large" (in the same senses) must factor through the projection of  \prod_I A_i  onto the product of finitely many of the  A_i,  modulo a map into the subalgebra  {b \in B | bB = Bb = {0}} \subseteq B. 

Detailed consequences are noted in the case where the  A_i  are Lie algebras. 

A partial generalization of the above results is proved with the field  k  replaced by a commutative valuation ring. 

13pp., last revised 28 Oct., 2009.  tex dvi pdf ps.  arXiv:0910.5183  Back to publications-list. 

An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

The inner automorphisms of a group  G  can be characterized in terms of the category of groups, without reference to group elements: they are precisely those automorphisms of  G  that can be extended, in a functorial manner, to all groups  H  given with homomorphisms  G --> H .  (Precise statement in sec.1.)  However, unlike the group of inner automorphisms, the group of such extended systems of automorphisms is always isomorphic to  G .  A similar characterization holds for inner automorphisms of an associative algebra  R  over a field  K ; here the group of functorial systems of automorphisms is isomorphic to the group of units of  R  modulo units of  K . 

If one substitutes "endomorphism" for "automorphism" in these considerations, then in the group case, the only additional example is the trivial endomorphism; but in the K-algebra case, a construction unfamiliar to ring theorists (but known to functional analysts) also arises. 

Systems of endomorphisms with the above functoriality property are examined in some other categories; other uses of the phrase "inner endomorphism" in the literature, some of which overlap the one introduced here, are noted; the concept of an inner derivation of an associative algebra or Lie algebra is looked at from the same point of view, and a dual concept of "co-inner" endomorphism is briefly examined.  Several questions are posed. 

17pp., 31 Dec., 2008.  tex dvi pdf ps.  Back to publications-list. 

On common divisors of multinomial coefficients

Erdös and Szekeres showed in 1978 that for any four positive integers satisfying  m₁+ m₂ = n₁+ n₂,  the two binomial coefficients  (m₁+ m₂)! / m₁! m₂!  and  (n₁+ n₂)! / n₁! n₂!  have a common divisor  >1.  The analogous statement for families of  k   k-nomial coefficients  (k > 1)  was conjectured in 1997 by David Wasserman. 

Erdös and Szekeres remark that if  m₁, m₂, n₁, n₂  as above are all  > 1,  there is probably a lower bound on the common divisor in question which goes to infinity as a function of  m₁+ m₂.  Such a bound is obtained. 

Criteria are developed for narrowing the class of possible counterexamples to Wasserman's conjecture.  On the other hand, several plausible generalizations of that conjecture are shown to be false. 

18pp., last revised 3 June, 2008.  tex dvi pdf ps.  arXiv:0806.0607  Errata to earlier versions.  Back to publications-list. 

On coproducts in varieties, quasivarieties and prevarieties

If the free algebra  F  on one generator in a variety  V  of algebras (in the sense of universal algebra) has a subalgebra free on two generators, must it also have a subalgebra free on three generators?  In general, no; but yes if  F  generates the variety  V. 

Generalizing the argument, it is shown that if we are given an algebra and subalgebras,  A₀ \supseteq ...\supseteq A_n,  in a prevariety (SP-closed class of algebras)  P  such that  A_n  generates  P,  and also subalgebras  B_i \subseteq A_i-1  (0 < i ≤ n)  such that for each  i > 0  the subalgebra of  A_i-1  generated by  A_i  and  B_i  is their coproduct in  P,  then the subalgebra of  A  generated by  B₁, ..., B_n  is the coproduct in  P  of these algebras. 

Some further results on coproducts are noted:

If  P  satisfies the amalgamation property, then one has the stronger "transitivity" statement, that if  A  has a finite family of subalgebras  (B_i)_{i \in I}  such that the subalgebra of  A  generated by the  B_i  is their coproduct, and each  B_i  has a finite family of subalgebras  (C_ij)_j\in Ji  with the same property, then the subalgebra of  A  generated by all the  C_ij  is their coproduct. 

For  P  a residually small prevariety or an arbitrary quasivariety, relationships are proved between the least number of algebras needed to generate  P  as a prevariety or quasivariety, and behavior of the coproduct operation in  P. 

It is shown by example that for  G  a subgroup of the group  S = Sym(\Omega)  of all permutations of an infinite set  \Omega,  the group  S  need not have a subgroup isomorphic over  G  to the coproduct with amalgamation  S \coprod_G S.  But under weak additional hypotheses, that question remains open. 

Following a request by the editors of Algebra and Number Theory, I have included a "Prologue" for the reader who is not an expert in universal algebra, and (going further than what they asked) a short glossary at the end for the same set of readers.

20pp., last revised 22 November, 2009.  tex dvi pdf ps.  arXiv.  Back to publications-list. 

On lattices and their ideal lattices, and posets and their ideal posets

For  P  a poset or lattice, let  Id(P)  denote the poset, respectively, lattice, of upward directed downsets in  P,  including the empty set, and let  id(P) = Id(P)-\{\emptyset\}.  This note obtains various results to the effect that  Id(P)  is always, and  id(P)  often, "essentially larger" than  P.  In the first vein, we find that a poset  P  admits no < -respecting map (and so in particular, no one-to-one isotone map) from  Id(P)  into  P,  and, going the other way, that an upper semilattice  S  admits no semilattice homomorphism from any subsemilattice of itself onto  Id(S). 

The slightly smaller object  id(P)  is known to be isomorphic to  P  if and only if  P  has ascending chain condition.  This result is strengthened to say that the only posets  P₀  such that for every natural number  n  there exists a poset  P_n  with  idⁿ(P_n) =~ P₀  are those having ascending chain condition.  On the other hand, a wide class of cases is noted here where  id(P)  is embeddable in  P. 

Counterexamples are given to many variants of the results proved. 

Preprint.  tex dvi pdf ps.  Back to publications-list. 

Colimits of representable algebra-valued functors

If  C  and  D  are varieties of algebras in the sense of general algebra, then by a representable functor  C --> D  we understand a functor which, when composed with the forgetful functor  D --> Set,  gives a representable functor in the classical sense; Freyd showed that these functors are determined by D-coalgebra objects of  C.  Let  Rep(C, D)  denote the category of all such functors, a full subcategory of  Cat(C, D),  opposite to the category of D-coalgebras in  C.  It is proved that  Rep(C, D)  has small colimits, and in certain situations, explicit constructions for the representing coalgebras are obtained. 

In particular,  Rep(C, D)  always has an initial object.  This is shown to be "trivial" unless  C  and  D  either both have no zeroary operations, or both have more than one derived zeroary operation.  In those two cases, the functors in question may have surprisingly opulent structures. 

It is also shown that every set-valued representable functor on  C  admits a universal morphism to a D-valued representable functor. 

Several examples are worked out in detail, and areas for further investigation noted. 

MR 2009c:18012.  Preprint.  tex dvi pdf ps.  Back to publications-list. 

Mapping radii of metric spaces

It is known that every closed curve of length  ≤ 4 in  Rⁿ   (n>0)  can be surrounded by a sphere of radius  1,  and that this is the best bound.  Letting  S  denote the circle of circumference  4,  with the arc-length metric, we here express this fact by saying that the mapping radius of  S  in  Rⁿ  is  1. 

Tools are developed for estimating the mapping radius of a metric space  X  in a metric space  Y.  In particular, it is shown that for  X  a bounded metric space, the supremum of the mapping radii of  X  in all convex subsets of normed metric spaces is equal to the infimum of the  sup  norms of all convex linear combinations of the functions  d(x,-): X --> R  (x\in X). 

Several explicit mapping radii are calculated, and open questions noted. 

Article.   Addenda.  MR 2009g:54064.  Preprint: tex dvi pdf ps.  Errata to July 2007 and March 2008 preprints.  Back to publications-list. 

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

In 1957, N. G. de Bruijn showed that the symmetric group  Sym(\Omega)  on an infinite set  \Omega  contains a free subgroup on  2^card(\Omega)  generators, and proved a more general statement, a sample consequence of which is that for any group  A  of cardinality  ≤ card(\Omega),  the group  Sym(\Omega)  contains a coproduct of  2^card(\Omega)  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(\Omega)  replaced by "monoid" and the monoid  Self(\Omega)  of endomaps of  \Omega,  by "associative K-algebra" and the K-algebra  End_K(V)  of endomorphisms of a K-vector-space  V  with basis  \Omega,  and by "lattice" and the lattice  Equiv(\Omega)  of equivalence relations on  \Omega.  It is also shown, extending another result from de Bruijn's 1957 paper, that each of  Sym(\Omega),  Self(\Omega)  and  End_K(V)  contains a coproduct of  2^card(\Omega)  copies of itself. 

That paper also gave an example of a group of cardinality  2^card(\Omega)  that was not embeddable in  Sym(\Omega),  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(\Omega)  of sets of equations with constants in  Sym(\Omega).  Again, similar results - this time of varying strengths - are obtained for  Self(\Omega),  End_K(V)  and  Equiv(\Omega),  and also for the monoid  Rel(\Omega),  of binary relations on  \Omega. 

Many open questions and areas for further investigation are noted.  A couple of these have been solved:  see "update" link below. 

MR 2008m:08016.  Preprint version: tex dvi pdf ps.  DOI.  Update.  Back to publications-list. 

Two statements about infinite products that are not quite true

This one is hard to summarize concisely; below are some high points.  In statements (i) and (ii) below,  R^\omega  and  (+)_\omega R  are the direct product, respectively, the direct sum, of countably many copies of the ring  R,  as a left  R-module. 

(i) There exist nontrivial rings  R  for which one has surjective module  homomorphisms  (+)_\omega R --> R^\omega ;   but in such cases, the module  R^\omega  must in fact be finitely generated. 
(ii) There exist nontrivial rings  R  for which one has surjective module  homomorphisms  R^\omega --> (+)_\omega R ;   but in such cases,  R  must have DCC on finitely generated right ideals. 
(iii) The full permutation group  S  on an infinite set  \Omega  has the property that the card(\Omega)-fold direct product of copies of  S  is generated over its diagonal subgroup by a single element. 
(iv) Whenever an algebra  S  in the sense of universal algebra has the property that the countable direct product  S^\omega  is finitely generated over its diagonal subalgebra (or even when the corresponding property holds with a nonprincipal countable ultrapower in place of this direct product),  S  has some of the other strange properties known to hold for infinite symmetric groups.  (The consequences are slightly weaker in the ultrapower case than in the direct product case.) 

MR 2007k:16008.  Preprint version: tex dvi pdf ps.  update.  Back to publications-list.