Eggert's Conjecture says that if R is a finite-dimensional nilpotent commutative algebra over a perfect field F of characteristic p, and R(p) is the image of the p-th power map on R, then dimF R ≥ p dimF R(p). Whether this very elementary statement is true is not known.
We examine heuristic evidence for this conjecture, versions of the conjecture that are not limited to positive characteristic and/or to commutative R, consequences the conjecture would have for finite abelian semigroups, and examples that give equality in the conjectured inequality.
We pose several related questions, and briefly survey the literature on the subject.
10 pp., 29 July, 2011.
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Let G be an abelian group, and F a downward directed family of subsets of G. The finest topology T on G under which F converges to 0 has been described by I. Protasov and E. Zelenyuk. In particular, their description yields a criterion for T to be Hausdorff. They then show that if F is the Fréchet filter on a countable subset A ⊆ G, there is a simpler criterion: T is Hausdorff if and only if for every g ∈ G−{0} and positive integer n, there is an S ∈ F such that g ∉ n (S ∪ {0} ∪ −S).
In this note, their proof is adapted to a larger class of families F. In particular, if A is an arbitrary subset of G, κ a regular cardinal ≤ card(A), and F the set of complements in A of subsets of cardinality < κ, then the above criterion holds.
On the other hand, we give an example of a countable downward directed set F of subsets of Z not of the above sort which satisfies the "g ∉ n (S ∪ {0} ∪ −S)" condition, but does not induce a Hausdorff topology.
We end with a version of our result for noncommutative G (probably far from best possible).
10 pp., 25 July 2011.
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It is shown that if a bilinear map f : A × B → C of modules over a commutative ring k is nondegenerate (i.e., if no nonzero element of A annihilates all of B, and vice versa), and if A and B are Artinian, then A and B are of finite length.
Some immediate consequences are noted. Counterexamples are given to certain generalizations of this statement to balanced bilinear maps of bimodules over noncommutative rings, while the question is raised whether other such generalizations may be true.
6 pp., last revised 13 Nov., 2011.
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In answer to a question of A. Blass, J. Irwin and G. Schlitt, a subgroup G of the additive group Zω is constructed whose dual, Hom(G, Z ), is free abelian of rank 2ℵ0. The question of whether Zω has subgroups whose duals are free of still higher rank is discussed, and some further classes of subgroups of Zω are noted.
9 pp., last revised 24 Sept., 2011.
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To appear, Algebra Universalis.
Let ( Li | i ∈ I ) be a family of lattices in a nontrivial lattice variety V, and let φi : Li → M, for i ∈ I, be isotone maps (not assumed to be lattice homomorphisms) to a common lattice M (not assumed to lie in V). We show that the maps φi can be extended to an isotone map φ: FreeV( Li ) → M, where FreeV( Li ) is the free product of the Li in V. This was known for L = V, the variety of all lattices.
The above free product L can be viewed as the free lattice in V on the partial lattice P formed by the disjoint union of the Li. The analog of the above result does not, however, hold for free lattices L on arbitrary partial lattices P. We show that the only codomain lattices M for which that more general statement holds are the complete lattices. On the other hand, we prove the analog of our main result for a class of partial lattices P that are not-quite-disjoint unions of lattices.
We also obtain some results similar to our main one, but with the relationship lattices : orders replaced either by semilattices : orders or by lattices : semilattices.
Some open questions are noted.
21 pp., last revised 1 Sept., 2011.
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To appear, Proc. A.M.S.
It is shown that any left module A over a ring R can be written as the intersection (and hence the inverse limit) of a downward directed system of injective submodules of an injective R-module. If R is left Noetherian, A can also be written as the inverse limit of a system of surjective homomorphisms of injectives.
Some questions are raised, and an example is noted.
7 pp., last updated 22 Sept., 2011.
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To appear, Illinois J. Math..
Let V be a variety of not necessarily associative algebras, and A an inverse limit of nilpotent algebras Ai ∈ V, such that some finitely generated subalgebra S ⊆ A is dense in A under the inverse limit of the discrete topologies on the Ai .
A sufficient condition on V is obtained for all algebra homomorphisms from A to finite-dimensional algebras B to be continuous; in other words, for the kernels of all such homomorphisms to be open ideals. This condition is satisfied, in particular, if V is the variety of associative, Lie, or Jordan algebras.
Examples are given showing the need for our hypotheses, and some questions are raised.
22 pp., last revised 15 Feb., 2011.
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To appear, Illinois J. Math..
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 and directions for further investigation are noted.
19 pp., last revised 24 Sept., 2010.
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We study surjective homomorphisms f : ∏I Ai → B of not-necessarily-associative algebras over a commutative ring k, for I a generally infinite set; especially when k is a field and B is countable-dimensional over k.
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 Ai are solvable, then so is B.
If all the Lie algebras Ai are nilpotent, then so is B.
If k is not of characteristic 2 or 3, and all the Lie algebras Ai 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 ∈ 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 ∏I Ai must factor through the direct product of finitely many ultraproducts of the Ai.
Several open questions are noted.
MR2785938.
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To appear, J. Alg..
Let k be an infinite field, I an infinite set, V a k-vector-space, and g : kI → V a k-linear map. It is shown that if dimk(V) is not too large (under various hypotheses on card(k) and card(I), if it is finite, respectively < card(k), respectively less than continuum), then ker(g) must contain an element (ui)i∈ I with all but finitely many components ui nonzero.
These results are used to prove that any homomorphism from a direct product ∏I Ai of not-necessarily-associative algebras Ai onto an algebra B, where dimk(B) is "not too large" (in the same senses) must factor through the projection of ∏I Ai onto the product of finitely many of the Ai, modulo a map into the subalgebra {b ∈ B | bB = Bb = {0}} ⊆ B.
Detailed consequences are noted in the case where the Ai are Lie algebras.
A partial generalization of the above results is proved with the field k replaced by a commutative valuation ring.
14 pp., last revised 10 Jan., 2012. tex. pdf. arXiv. DOI. Back to publications-list.
Diagram-chasing arguments frequently lead to "magical" relations between distant points of diagrams: exactness implications, connecting morphisms, etc.. These long connections are usually composites of short "unmagical" connections, but the latter, and the objects they join, are not visible in the proofs. I try to remedy this situation.
Given a double complex in an abelian category, we consider, for each object A of the complex, the familiar horizontal and vertical homology objects at A, and two other objects, which we name the "donor" A□ and the "receptor" □A at A. For each arrow of the double complex, we prove the exactness of a 6-term sequence of these objects (the "Salamander Lemma"). Standard results such as the 3×3-Lemma, the Snake Lemma, and the long exact sequence of homology associated with a short exact sequence of complexes, are obtained as easy applications of that lemma.
We then obtain some further generalizations of the long exact sequence of homology, getting various exact diagrams from double complexes with all but a few rows and columns exact.
The total homology of a double complex is also examined in terms of the constructions we have introduced. We end with a brief look at the world of triple complexes, and a couple of exercises.
TeX source file of preprint (requires
this
style-class file).
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Publicacions Matemàtiques
, 56 (2012) 91-126.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 section 1.) The group of such extended systems of automorphisms, unlike the group of inner 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 looks at the above functorial extendibility property for endomorphisms, rather than just automorphisms, 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.
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Erdős 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.
Erdős 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.
(Early preprints of this note contained many digressions, which I moved into 8 pages of "Addenda"; I don't plan on publishing the Addenda.)
MR2765421. TeX. PDF. arXiv. DOI. Back to publications-list.
Algebra and Number Theory 3 (2009) 847-879.
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 ⊇ ... ⊇ An, in a prevariety (SP-closed class of algebras) P such that An generates P, and also subalgebras Bi ⊆ Ai-1 (0 < i ≤ n) such that for each i > 0 the subalgebra of Ai-1 generated by Ai and Bi is their coproduct in P, then the subalgebra of A generated by B1, ..., Bn 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 (Bi)i ∈ I such that the subalgebra of A generated by the Bi is their coproduct, and each Bi has a finite family of subalgebras (Cij)j∈ Ji with the same property, then the subalgebra of A generated by all the Cij 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(Ω) of all permutations of an infinite set Ω, the group S need not have a subgroup isomorphic over G to the coproduct with amalgamation S \coprodG S. But under weak additional hypotheses, that question remains open.
For the reader who is not an expert in universal algebra, a "Prologue" introducing the concepts, and a short glossary at the end, are provided.
MR2011e:08014. Preprint version: tex. dvi. pdf. ps. arXiv. Back to publications-list.
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)- {∅}. 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 Pn with idn(Pn) ≅ 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.
MR2010k:06001. Preprint: tex. dvi. pdf. ps. Back to publications-list.
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.
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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 vector spaces is equal to the infimum of the sup norms of all convex linear combinations of the functions d(x,-): X → R (x∈ X).
Several explicit mapping radii are calculated, and open questions noted.
Article.
Addenda and erratum.
MR
2009g:54064.
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In 1957, N. G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ω contains a free subgroup on 2card(Ω) 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 2card(Ω) 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 2card(Ω) copies of itself.
That paper also gave an example of a group of cardinality 2card(Ω) 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. A couple of these have since been solved: see "update" link below.
MR
2008m:08016.
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This one is hard to summarize concisely; below are some high points. In statements (i) and (ii) below, Rω and ⊕ω 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
⊕ω R → Rω ;
but in such cases, the module Rω must
in fact be finitely generated.
(ii) There exist nontrivial rings R for which one has
surjective module homomorphisms
Rω →
⊕ω R ; but in such
cases, R must have DCC on finitely generated
right ideals.
(iii) The full permutation group S on an infinite
set Ω has the property that the card(Ω)-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 Sω 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.
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Let S = Sym(Ω) be the group of all permutations of a countably infinite set Ω, and for subgroups G1, G2 ≤ S let us write G1 ~ G2 if there exists a finite set U ⊆ S such that <G1 ∪ U> = <G2 ∪ U>. It is shown that the subgroups closed in the function topology on S lie in precisely four equivalence classes under this relation. Which of these classes a closed subgroup G belongs to depends on which of the following statements about pointwise stabilizer subgroups G(Γ) of finite subsets Γ ⊆ Ω holds:
(i) For every finite set Γ, the
subgroup G(Γ) has
at least one infinite orbit in Ω.
(ii) There exist finite
sets Γ such that all orbits
of G(Γ)
are finite, but none for which the cardinalities
of such orbits have a common finite bound.
(iii) There exist finite sets Γ
such that the cardinalities
of the orbits of G(Γ)
have a common finite bound,
but none such that G(Γ) = {1}.
(iv) There exist finite sets Γ such
that G(Γ) = {1}.
Some related results and topics for further investigation are noted.
Afterthoughts.
MR 2008a:20005.
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Let S = Sym(Ω) be the group of all permutations of an infinite set Ω. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, respectively as a monoid, then there exists a positive integer n such that every element of S may be written as a group word, respectively a monoid word, of length ≤ n in the elements of U.
Some related questions and recent results by others are noted, and a brief proof is given of a result of Ore's on commutators that is used in the proof of the above result.
MR 2007e:20004.
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Let k be an integral domain, n a positive integer, X a generic n×n matrix over k (i.e., the matrix (xij) over a polynomial ring k[xij] in n2 indeterminates xij), and adj(X) its classical adjoint. For char k = 0 it is shown that if n is odd, adj(X) is not the product of two noninvertible n×n matrices over k[xij], while for n even, only one special sort of factorization occurs. Whether the corresponding results hold in positive characteristic is not known.
The operation adj on matrices arises from the (n − 1)-st exterior power functor on modules; the analogous factorization question for matrix constructions arising from other functors is raised, as are several other questions.
MR 2006k:15037. Preprint: tex. dvi. pdf. ps. galley proofs: pdf. Back to publications-list.