arXiv subjects

George M. Bergman

Publications and source records attributed to George M. Bergman.

51 records · Page 3Linked to original sources

arXiv2011

Homomorphisms on infinite direct product algebras, especially Lie algebras

We study surjective homomorphisms f:\prod_I A_i\to 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 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=[x_1,L]+[x_2,L] for some x_1, x_2\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 examples are given, and open questions noted.

math.RA↗

arXiv2010

Continuity of homomorphisms on pro-nilpotent algebras

Let V be a variety of not necessarily associative algebras, and A an inverse limit of nilpotent algebras A_i\in V, such that some finitely generated subalgebra S \subseteq A is dense in A under the inverse limit of the discrete topologies on the A_i. 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 open questions are noted.

math.RA↗

arXiv2010

On common divisors of multinomial coefficients

Erdős and Szekeres showed in 1978 that for any four positive integers satisfying m_1+m_2 = n_1+n_2, the two binomial coefficients (m_1+m_2)!/m_1! m_2! and (n_1+n_2)!/n_1! n_2! 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 m_1, m_2, n_1, n_2 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 m_1+m_2. Such a bound is here obtained. Results are proved that narrow the class of possible counterexamples to Wasserman's conjecture. On the other hand, several plausible generalizations of that conjecture are shown to be false.

math.NT↗

arXiv2009

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 and directions for further investigation are noted.

math.RA↗

arXiv2009

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, A_0\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 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 B_1, ..., B_n is the coproduct in P of these algebras. Some further results on coproducts are noted: If P satisfies the amalgamation property, one has the stronger "transitivity" statement: 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 J_i} 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 B 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 B to the coproduct with amalgamation S \cP_B S. But under weak additional hypotheses on B, the question remains open.

math.RA↗

arXiv2008

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.

math.CT↗

arXiv2008

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 P_0 such that for every natural number n there exists a poset P_n with id^n(P_n)\cong P_0 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.

math.RA↗

arXiv2008

Mapping radii of metric spaces

It is known that every closed curve of length \leq 4 in R^n (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 R^n 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.

math.MG↗

arXiv2007

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 Ωcontains a free subgroup on 2^{card(Ω)} generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality \leq card(Ω), Sym(Ω) contains a coproduct of 2^{card(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 \bf{V}, and formulated as a statement about functors Set --> \bf{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 End_K(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 End_K (V) contains a coproduct of 2^{card(Ω)} copies of itself. That paper also gave an example of a group of cardinality 2^{card(Ω)} that was {\em not} embeddable in Sym(Ω), 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(Ω) of sets of equations with constants in Sym(Ω). Again, similar results -- this time of varying strengths -- are obtained for Self(Ω), End_K (V), and Equiv(Ω), and also for the monoid \Rel of binary relations on Ω. Many open questions and areas for further investigation are noted.

math.RA↗

arXiv2005

Two statements about infinite products that are not quite true

Hard to summarize concisely; here are the high points. The first two statements below are ring-theoretic; in these R is a nontrivial ring, R^ω, and \bigoplus_ωR are the direct product, respectively direct sum, of countably many copies of R; the remaining two statements are in the context of general algebra (a.k.a. universal algebra): (i) There exist nontrivial rings R for which one has surjective homomorphisms \bigoplus_ωR -> R^ω-- but in such cases, R^ωis in fact finitely generated as a left R-module. (ii) There exist nontrivial rings R for which one has surjective homomorphisms R^ω-> \bigoplus_ωR -- but in such cases, R must have DCC on finitely generated right ideals. (iii) The full permutation group S on an infinite set Ωhas the property that the |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 S^ωis finitely generated over its diagonal subalgebra (or even when the corresponding property holds with an ultrapower in place of this direct product), S has some of the other strange properties known to hold for infinite symmetric groups (cf. math.GR/0401304).

math.RA↗

arXiv2005

Direct limits and fixed point sets

For which groups G is it true that whenever we form a direct limit of G-sets, dirlim_{i\in I} X_i, the set of its fixed points, (dirlim_I X_i)^G, can be obtained as the direct limit dirlim_I(X_i^G) of the fixed point sets of the given G-sets? An easy argument shows that this holds if and only if G is finitely generated. If we replace ``group G'' by ``monoid M'', the answer is the less familiar condition that the improper left congruence on M be finitely generated. Replacing our group or monoid with a small category E, the concept of set on which G or M acts with that of a functor E --> Set, and the concept of fixed point set with that of the limit of a functor, a criterion of a similar nature is obtained. The case where E is a partially ordered set leads to a condition on partially ordered sets which I have not seen before (pp.23-24, Def. 12 and Lemma 13). If one allows the {\em codomain} category Set to be replaced with other categories, and/or allows direct limits to be replaced with other kinds of colimits, one gets a vast area for further investigation.

math.CT↗

arXiv2005

Closed subgroups of the infinite symmetric group

Let S=Sym(Ω) be the group of all permutations of a countably infinite set Ω, and for subgroups G_1, G_2\leq S let us write G_1\approx G_2 if there exists a finite set U\subseteq S such that < G_1\cup U > = < G_2\cup 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\subseteqΩholds: (i) For every finite set Γ, the subgroup G_{(Γ)} has at least one infinite orbit in Ω. (ii) There exist finite sets Γsuch that all orbits of G_{(Γ)} are finite, but none such that the cardinalities of these orbits have a common finite bound. (iii) There exist finite sets Γsuch 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 Γsuch that G_{(Γ)}=\{1\}. Some questions for further investigation are discussed.

math.GR↗

arXiv2005

Generating infinite symmetric groups

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 \leq n in the elements of U. Several related questions 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.

math.GR↗

arXiv2004

On lattices of convex sets in R^n

Properties of several sorts of lattices of convex subsets of R^n are examined. The lattice of convex sets containing the origin turns out, for n>1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of R^n and the lattice of all convex subsets of R^{n-1}. The lattices of arbitrary, of open bounded, and of compact convex sets in R^n all satisfy the same identities, but the last of these is join-semidistributive, while for n>1 the first two are not. The lattice of relatively convex subsets of a fixed set S \subseteq R^n satisfies some, but in general not all of the identities of the lattice of ``genuine'' convex subsets of R^n.

math.MG↗

arXiv2004

Can one factor the classical adjoint of a generic matrix?

Let k be a field, n a positive integer, X a generic nxn matrix over k (i.e., a matrix (x_{ij}) of n^2 independent indeterminates over the polynomial ring k[x_{ij}]), and adj(X) its classical adjoint. It is shown that if char k=0 and n is odd, then adj(X) is not the product of two noninvertible nxn matrices over k[x_{ij}]. If n is even and >2, a restricted class of nontrivial factorizations occur. The nonzero-characteristic case remains open. The operation adj on matrices arises from the (n-1)st exterior power functor on modules; the same question can be posed for matrix operations arising from other functors.

math.AC↗