For *P* a poset, the *dimension* of *P* is
defined to be the least cardinal κ such that *P* is
embeddable in a direct product of κ totally ordered sets.
We study the behavior of this function on finite-dimensional
(not necessarily finite) posets.
In general, the dimension dim(*P* × *Q*) of a
product of two posets can be smaller
than dim(*P*) + dim(*Q*), though no examples are known where
the discrepancy is greater than 2.
We obtain a result that gives upper bounds on the
dimensions of certain products of posets, including cases
where the discrepancy 2 is achieved.
But the paper is mainly devoted to stating questions, old
and new, about dimensions of product posets, noting implications
among their possible answers, and introducing
concepts that might be helpful in tackling these questions.

We end by noting an ultraproduct argument showing that an
infinite poset *P* has finite
dimension *n* if and only if *n* is the
supremum of the dimensions of its finite subposets.

12pp., last revised 11 Feb., 2024. tex. pdf. arXiv:2208.06511. Back to publications-list.

Let *R* be a ring, and consider a left *R*-module given with
two (generally infinite) direct sum decompositions
*A* ⊕ (⊕_{i∈I}*
C _{i}*) =

We then note some ways that this result can and cannot be generalized, and pose some related questions.

Preprint version:. tex. pdf. arXiv:2208.06511. Back to publications-list.

F. Wehrung has asked: Given a family * C* of subsets
of a set Ω, under what conditions will there exist
a total ordering on Ω under which every member of

Note that if *A* and *B* are nondisjoint convex subsets of
a totally ordered set, neither of which contains the other,
then *A*∪*B*, *A*∩*B*, and
*A*∖*B* are also convex.
So let * C* be an arbitrary set of subsets of a set
Ω, and form its closure

We establish bounds on the cardinality of the subset
* P* generated as above by an

We note connections with the theory of *interval graphs*
and *hypergraphs*, which lead to
other ways of answering Wehrung's question.

Preprint version: tex. pdf. arXiv. Back to publications-list.

We review the definition of a *quandle,* and in particular
of the *core quandle* Core(*G*) of a group *G,* which
consists of the underlying set of *G,* with the binary operation
*x* ◁ *y* = *x y*^{-1} *x*.
This is an *involutory quandle*, i.e., satisfies the identity
*x* ◁ (*x* ◁ *y*) = *y*
in addition to the identities in the definition of
a quandle (which we recall).

*Trajectories*
(*x _{i}*)

**Addenda**: An example answering Questions 5.3 and 5.5 of
this paper is given here.

Georgii Kadantsev has pointed me to [11] below, which discusses
earlier articles which studied involutory quandles under
other names, in particular [12] and [13].

[11] David Stanovský, *The origins of involutory quandles*, 8pp.,
https://arxiv.org/abs/1506.02389.

[12] R. S. Pierce, *Symmetric groupoids,*
Osaka J. Math. 15 (1978), 51–76. MR0539589.

[13] R. S. Pierce, *Symmetric groupoids II,*
Osaka J. Math. 16 (1979), 317–348. MR0480780.

Preprint version: tex. pdf. arXiv:2006.00641. MR3787805. Back to publications-list.

Let us call a group *G* *resistant* if
for all fields *k*,
every non-monomial element of the group algebra *k G*
generates a proper 2-sided ideal.
We would like to know which groups are resistant;
the only groups for which we know this to be true are the
torsion-free abelian groups.
We would in particular like to know
whether all free groups are resistant.

We show that wide classes of
groups are non-resistant: every group *G* that contains an
element *g* ≠ 1 whose image
in *G* / [*g,G*] has finite order (in particular, every
group containing a *g* ≠ 1 that itself has finite order, or
that satisfies *g* ∈ [*g,G*]); and every group
containing an element *g* which commutes with a distinct
conjugate *hgh*^{-1} ≠ *g* (in particular,
every nonabelian solvable group).

Closure properties of the class of resistant groups are noted. Further questions are raised. In particular, a plausible Freiheitssatz for group algebras of free groups is proposed, which would imply that free groups are resistant.

Preprint version: tex. pdf. arXiv:1905.12704. DOI. Back to publications-list.

In Question 19.35 of the
Kourovka Notebook, M. H. Hooshmand
asks whether, given a finite group *G* and a factorization
card(*G*) = *n*_{1} ... *n _{k}*, one
can always find subsets

We show that for *G* the alternating group on 4 elements,
*k* = 3, and
(*n*_{1}, *n*_{2}, *n*_{3})
= (2, 3, 2), the answer is negative.
We then generalize some of the tools used in our proof,
and note an open question.

tex. journal pdf. arXiv:2003.12866. Back to publications-list.

P. M. Cohn showed in 1971 that given a ring *R,* to describe,
up to isomorphism, a division ring *D* generated by a homomorphic
image of *R* is equivalent to specifying the set of square
matrices over *R* which map to singular matrices
over *D,* and he determined
the conditions that such a set of matrices must satisfy.
I later developed another version of this data,
in terms of closure operators on free *R*-modules.

This note examines the latter concept further, and shows how
an *R*-module *M* satisfying certain conditions
can be made to induce such data.
In an appendix we make some observations on Cohn's
original construction,
and note how the data it uses can similarly be induced by
appropriate sorts of *R*-modules.

Our motivation is the longstanding question of whether,
for *G* a right-orderable group and *k* a field, the group
algebra *kG* must be embeddable in a division ring.
Our hope is that the right *kG*-module
*M* = *k*((*G*)) might induce
a closure operator of the required sort.
We review a partial result in this direction due to N. I. Dubrovin,
note a plausible generalization thereof which would give the
desired embedding, and briefly sketch some other
thoughts on ways of approaching the problem.

Addendum to the bibliography of this paper:
N. I. Dubrovin,
*Skew field of fractions of the group algebra of the universal
covering of* SL(2, **R**),
Trans. Moscow Math. Soc., 32 pp., to appear 2020.

Preprint version: tex. dvi. pdf. ps. arXiv:1812.06123. Back to publications-list.

The Boolean ring *B* of measurable subsets of the unit interval,
modulo sets of measure zero, has proper
*primary* ideals that are closed
under the natural metric (e.g., {0}), but has no *prime*
ideals closed
under that metric; hence closed primary ideals are not, in general,
intersections of closed prime ideals. Moreover, *B* is known
to be complete in its metric.
Together, these facts answer a question
posed by J. Gleason. From this example, rings of arbitrary
characteristic with the corresponding properties are obtained.

The result that *B* is complete is generalized to
show that if *L* is a lattice given with a metric *d*
satisfying identically either the relation
*d*(*x*∨*y*, *x*∨*z*) ≤
*d*(*y*, *z*) or the relation
*d*(*x*∧*y*, *x*∧*z*) ≤
*d*(*y*, *z*), and if in *L* if every increasing
Cauchy sequence converges and every decreasing
Cauchy sequence converges, then every Cauchy sequence converges;
i.e., *L* is complete as a metric space.

We end with an example showing that if the above inequalities
are replaced by the weaker conditions
*d*(*x*, *x*∨*y*) ≤
*d*(*x*, *y*), respectively
*d*(*x*, *x*∧*y*) ≤
*d*(*x*, *y*), then
this completeness statement can fail.

arXiv:1808.04455. Back to publications-list.

M. Kuczma asked in 1980 whether for every positive integer *n*,
there exists a subsemigroup *M* of a group *G*, such
that *G* is equal to the *n*-fold product
*M M*^{−1} *M M*^{−1} ...
*M*^{(−1)n-1}, but not to any
proper initial subproduct of this product.
We answer his question affirmatively,
and prove a more general result on representing
a certain sort of relation algebra by a family of subsets of a group.
We also sketch several variants of the latter result.

MR3787805. DOI. Preprint version: tex. dvi. pdf. ps. arXiv:1702.06088. Back to publications-list.

For *V* a vector space over a field, or more generally, over a
division ring, it is well-known that
every *x* ∈ End(*V*)
has an *inner inverse*, i.e., an element
*y* ∈ End(*V*) satisfying *xyx=x*.
We show here that a large class of
such *x* have inner inverses *y* that satisfy an infinite
family of additional monoid relations, making the monoid generated
by *x* and *y* an *inverse monoid*
(definition recalled).
We obtain consequences of these relations, and related results.

P. Nielsen and J. Šter (TAMS **370** (2018)
1759-1782)
show that a much
larger class of elements *x* of rings *R*, including
all elements
of von Neumann regular rings, have inner inverses satisfying any
*finite* subset of the abovementioned system of relations.
But we show by example that the
endomorphism ring of an infinite-dimensional vector space
contains elements having no inner inverse that
simultaneously satisfies all those relations.

A tangential result proved gives a necessary and sufficient condition
on an endomap *x* of
a set *S*, for *x* to have a strong inner inverse in
the monoid of all endomaps of *S*.

MR3738191. DOI. Preprint version: tex. dvi. pdf. ps. arXiv:1611.00972. Back to publications-list.

A standard construction associates to every finite partially
ordered set *P* a finite simplicial complex Δ(*P*),
called the "order complex" of *P*.
Simplicial complexes come in two versions, the "abstract" complex
(a set of points called "vertices", with distinguished subsets
called "simplices") and its "geometric realization" (a topological
space in which the 2-element simplices
are replaced by line-segments, the 3-element simplices
by triangles, etc.); we shall
understand Δ(*P*) to denote the geometric realization.
I give a variant way of describing this object,
which leads to a natural partial ordering of Δ(*P*),
such that as a poset, Δ(*P*) is a subdirect product
of copies of *P*, and for *L* a lattice,
the poset Δ(*L*) is likewise a lattice.

The lattice Δ(*M*_{3}) answers
W. Taylor's question "Is there a
simplicial complex which admits a continuous structure
of lattice, but not a continuous structure of distributive lattice?"
Properties of the construction Δ(*L*)
suggest some further questions, such as whether in
a topological lattice whose underlying space is a finite
simplicial complex, "almost every" point must have a neighborhood
that is a distributive sublattice.

Variants of the construction Δ(*L*) are sketched,
and some of their properties noted.

I also describe a construction of "stitching" together a
family of lattices along a common chain, and note that
Δ(*M*_{3})
can be obtained by stitching together three
copies of Δ(*C*), where *C* is a 3-element chain.

DOI. MR3604382. Preprint version: tex. dvi. pdf. ps. arXiv:1602.00034. Back to publications-list.

Vaughan Pratt has
introduced objects consisting of pairs (*A, W*) where *A*
is a set and *W* a set of subsets of *A*, such that

(i) *W* contains ∅ and *A*,

(ii) if *C* is a subset of *A* × *A* such
that for every *a ∈ A*,
both {*b* | (*a,b*) ∈ *C*}
and {*b* | (*b,a*)∈ *C*} are
members of *W* (a "crossword" with
all "rows" and "columns" in *W*),
then {*a* | (*a,a*)∈ *C*} (the "diagonal word")
also belongs to *W*, and

(iii) for all *a, b ∈ A, W* has an element which
contains *a* but not *b*.

He has asked whether for every *A*, the only such *W* is
the set of *all* subsets of *A*.

We answer that question in the negative.
We also
obtain several positive results, in particular, a positive answer
to the above question if *W* is closed under complementation.
We obtain partial results
on the problem of whether Pratt's question has a
positive answer if *W* is required to be countable.

MR3509966. DOI Preprint version: tex. dvi. pdf. ps. arXiv:1504.07310. Back to publications-list.

Let *R* be an algebra over a field *k*, let *p* be
an element of *R*, and let
*R*' = *R*< *q* | *pqp* = *p*>.
We obtain normal forms for elements of *R*', and for elements
of *R*'-modules arising by extension of scalars
from *R*-modules.
The details depend on where in the
chain *pR* ∩ *Rp* ⊆
*pR* ∪ *Rp* ⊆
*pR* + *Rp* ⊆ *R*
the unit 1 of *R* first appears.

This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring; but that goal remains distant.

We end with a normal form result for the algebra obtained by
tying together
a *k*-algebra *R* given with a nonzero element *p*
satisfying 1∉*pR*+*Rp* and
a *k*-algebra *S* given with a nonzero element *q*
satisfying 1∉*qS*+*Sq*
via the pair of relations *p* = *pqp*,
*q* = *qpq*.

MR3448178. DOI. Preprint version: tex. dvi. pdf. ps. arXiv:1505.02312. Back to publications-list.

Let *G* be an abelian group,
and *F* a downward directed family of subsets of *G*.
The finest topology ** T** on

In this note, their proof is adapted to a larger
class of families *F*.
In particular, if *X* is any infinite
subset of *G*, *κ* any regular
infinite cardinal ≤ card(*X*), and *F* the set of
complements in *X* of subsets of
cardinality < *κ*, then the above criterion holds.

We then give some negative examples, including 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 main result for noncommutative *G*.

(The printed version has some errors -- in particular, an added "the" after the first word of the title, and some spelling errors in the added Russian translation of the abstract -- but these have been corrected in the online version.)

MR3444020. DOI. Preprint version: tex. dvi. pdf. ps. arXiv:1311.2648. Back to publications-list.

Let *R* be a left
Artinian ring, and *M* a faithful left *R*-module which
is minimal, in the sense that no proper submodule *or*
proper homomorphic image of *M* is faithful.

If *R* is local, and socle(*R*) is central
in *R*, we show that length(*M/J*(*R*)*M*)
+ length(socle(*M*)) ≤
length(socle(*R*)) + 1, strengthening a result of T. Gulliksen.

Most of the rest of the paper is devoted to the situation
where the Artinian ring *R* is not necessarily local, and
does not necessarily have central socle.
In the case where *R* is a finite-dimensional algebra over an
algebraically closed field, we get an inequality similar to the
preceding,
with the length of socle(*R*) interpreted as
its length as a bimodule,
and with the final summand +1 replaced
by the Euler characteristic of a bipartite graph
determined by the bimodule structure of socle(*R*).
More generally, that inequality holds if, rather than
assuming *k* algebraically closed, we assume
that *R*/*J*(*R*) is a direct
product of full matrix algebras
over *k*, and exclude the case where *k* has small
finite cardinality.
Examples show that the restriction on the cardinality
of *k* is needed; we do not know whether versions of our
result are true with the hypotheses
on *R*/*J*(*R*) significantly weakened.

The situation for faithful modules with only one minimality property,
i.e., having no faithful proper submodules *or* having
no faithful proper homomorphic images, is more straightforward:
The length of *M*/*J*(*R*)*M* in the former case,
and of socle(*M*) in the latter,
is ≤ length(socle(*R*)) (where this
again means length as a bimodule).
We end with a section, essentially independent of the rest of
the note, which obtains these bounds,
and shows that every faithful module over a left Artinian ring
has a faithful submodule with the former minimality condition,
and a faithful factor module with the latter.
The proofs involve some nice
general results on decompositions of modules.

MR3374608. DOI. Preprint version: tex. dvi. pdf. ps. arXiv:1310.5365. Back to publications-list.

We study properties of a group, abelian group, ring, or
monoid *B* which (a) guarantee that every homomorphism
from an infinite
direct product Π_{I}* A _{i}* of
objects of the same sort onto

We note a number of open questions, and topics for further investigation.

MR3332912. DOI. Preprint version: tex. dvi. pdf. ps. arXiv:1406.1932. Back to publications-list.

Sections 1-4, by Trevor Stuart, sketch Cohn's personal and professional life: his childhood in Hamburg, his move to England on the Kindertransport at age 15, his years directing Ph.D. students, and his service to the London Mathematical Society. Of the sections I wrote on his research and publications, §5 tells the non-mathematician what noncommutative rings are, §§6-11 sketch his research for the specialist, and §12 surveys his publications.

The memoir is supplemented by two online files, a 209-item bibliography of his works, and an English translation of an account he wrote of his early childhood.

DOI. pdf. Back to publications-list.

Criteria are obtained for a filter ** F** of subsets of a
set

We briefly examine the question of how the
common technique used in applying the general results of this note
to *k*-algebras on the one
hand, and to nonabelian groups on the other,
might be extended to more general varieties of algebras in the
sense of universal algebra.

In a final section, the Erdős-Kaplansky Theorem on dimensions
of vector spaces *D ^{I}* (

MR3226022. DOI. Preprint version: tex. dvi. ps. pdf. arXiv:1301.6383. Back to publications-list.

Several authors have studied the question of when the
monoid ring *DM* of a monoid *M* over a
ring *D* is a right and/or
left fir (free ideal ring), a semifir, or a 2-fir (definitions
recalled in section 1).
It is known that for *M* nontrivial, a necessary condition for
any of these properties to hold is that *D* be a division ring.
Under that assumption, necessary and sufficient conditions
on *M* are known for *DM* to be a right or left fir,
and various
conditions on *M* have been proved necessary or sufficient
for *DM* to be a 2-fir or semifir.

A sufficient condition for *DM* to be a semifir is
that *M* be a direct limit of monoids which are free products
of free monoids and free groups.
Warren Dicks has conjectured that this is also necessary.
However, F. Cedó has given an example of a
monoid *M* which
is not such a direct limit, but satisfies all
the known necessary conditions for *DM* to be a semifir.
It is an open question whether for this *M,* the
rings *DM* are semifirs.

We note here some reformulations of the known necessary conditions
for a monoid ring *DM* to be a 2-fir or a semifir,
motivate Cedó's construction and a variant of that
construction, and recover Cedó's results for both constructions.

Any homomorphism from a monoid *M* into **Z** induces a
**Z**-grading on *DM,* and we show that for the two
monoids in question,
the rings *DM* are "homogeneous semifirs" with respect to
all such nontrivial gradings; i.e., have (roughly) the property that
every finitely generated homogeneous one-sided ideal is free.

If *M* is a monoid such that *DM* is an *n*-fir,
and *N* a "well-behaved" submonoid *N* of *M,* we
prove some properties of *DN*.
Using these, we show that if *M* is any monoid having a monoid
ring *DM* which is a 2-fir, then mutual commutativity
is an equivalence
relation on nonidentity elements of *M,* and each equivalence
class, together with the identity element, is a directed union
of infinite cyclic groups or infinite cyclic monoids.

Several open questions are noted.

MR3258484. DOI. Preprint version: tex. dvi. pdf. ps. arXiv:1309.0564. Back to publications-list.

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
dim* _{F} R* ≥

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.

MR3204348. DOI. Preprint version: tex. dvi. pdf. arXiv:1206.0326. Back to publications-list.

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.

(** Erratum:**
In the second sentence of the Introduction, which recalls the
definitions of right,
left, and 2-sided ideals using a "respectively" construction,
the conditions for "right" and "left" are reversed.)

MR3069281. tex file of preprint. pdf file of reprint. arXiv. Back to publications-list.

Illinois J. Math.,
**55** (2011) 749–748.
(Actually appeared 2013.)

Let **V** be a variety of not
necessarily associative algebras, and *A* an inverse
limit of nilpotent algebras
*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 questions are raised.

MR3069282. tex file of preprint. pdf file of reprint. arxiv. Back to publications-list.

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.

MR2983182. DOI. Preprint: tex. dvi. pdf. ps. arXiv. Back to publications-list.

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.

(The MR review points out that, calling a
module *E finitely injective* if every homomorphism from
a finitely generated module *A* into *E* extends to all
overmodules of *A*, we can, in the second result above,
delete "Noetherian" if we weaken "injective" to
"finitely injective".)

MR3008865. DOI. preprint: tex. dvi. pdf . ps. arXiv. Back to publications-list.

Let ( *L _{i}* |

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 *L _{i}*.
The analog of the above result does not, however, hold for
free 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.

MR3008736. DOI. preprint: tex. pdf. arXiv. Back to publications-list.

In answer to a question of A. Blass, J. Irwin and G. Schlitt,
a subgroup *G* of the additive
group **Z**^{ω} is constructed whose
dual, Hom(G, **Z** ), is free abelian of
rank 2^{ℵ0}.
The question of whether **Z**^{ω} has subgroups
whose duals are free of still higher rank is discussed, and some
further classes of subgroups of **Z**^{ω} are noted.

MR2900652. DOI. preprint: tex. dvi. ps. pdf. arXiv. Back to publications-list.

Let *k* be an infinite field,
and *g* : *k ^{I} → V*
a

These results are used to prove that any homomorphism from a direct
product ∏* _{I} A_{i}*
of not-necessarily-associative
algebras

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.

It is shown in a subsequent note that cardinality assumptions of this note can be weakened.

MR2891132. DOI. preprint: tex. pdf. arXiv. Back to publications-list.

We study surjective homomorphisms
*f* : ∏_{I}* A _{i} →
B* of
not-necessarily-associative algebras over a commutative ring

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

If all the Lie algebras *A _{i}* are nilpotent,
then so is

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

The general technique of the paper
involves studying conditions under which
a homomorphism on ∏_{I}* A _{i}*
must factor through the direct product of finitely many
ultraproducts of the

Several open questions are noted.

MR2785938. DOI. preprint: tex. dvi. pdf. ps. arXiv. 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 here 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.

MR2909639.
Published version:
dvi.
pdf.
ps.
TeX source file of preprint (requires
this
style-class file).
arXiv:1108.0958.
Back to publications-list.

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.

MR2918185. DOI. Preprint: tex. dvi. pdf. ps. arXiv. Back to publications-list.

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 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. DOI preprint: TeX. PDF. arXiv. Back to publications-list.

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} ⊇ ... ⊇
*A _{n}*,
in a prevariety (

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}*)

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* \coprod_{G }*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.

MR2587406. DOI. preprint: 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 *P*_{0} such that
for every natural number *n* there exists a
poset *P _{n}* with
id

Counterexamples are given to many variants of the results proved.

MR2563808. 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.

MR2425552. Preprint: tex. dvi. pdf. ps. Back to publications-list.

It is known that every closed curve of length ≤ 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
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.

MR2407106. Article. DOI. preprint: tex. dvi. pdf. ps. Back to publications-list.

In 1957, N. G. de Bruijn showed that the symmetric
group Sym(Ω) on an infinite set Ω contains
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 ≤ card(Ω),
the group 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 **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 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 *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), End_{K}(*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.

MR2373687. DOI. preprint: tex. dvi. pdf. ps. Back to publications-list.

This one is hard to summarize concisely;
below are some high points.
In statements (i) and (ii) below,
*R*^{ω} 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 → R*^{ω} ;
but in such cases, the module *R*^{ω} must
in fact be finitely generated.

(ii) There exist nontrivial rings *R* for which one has
surjective module homomorphisms
*R*^{ω} →
⊕_{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 *S*^{ω} 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.)

MR2279231. DOI. preprint: tex. dvi. pdf. ps. Back to publications-list.

Let *S* = Sym(Ω)
be the group of all permutations of a countably
infinite set Ω, and for
subgroups *G*_{1},
*G*_{2} ≤ *S* let us
write *G*_{1} ~ *G*_{2}
if there exists a finite set *U* ⊆ *S*
such that <*G*_{1} ∪ *U*> =
<*G*_{2} ∪ *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.

MR2280223. DOI. preprint: tex. dvi. ps. Back to publications-list.

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.

MR2239037. DOI. galley proofs: pdf. preprint: tex. ps. dvi. Back to publications-list.

Let *k* be an integral domain, *n* a positive
integer, *X* a generic *n*×*n* matrix
over *k* (i.e., the
matrix (*x _{ij}*) over a polynomial
ring

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.

MR2205070. DOI. preprint: tex. dvi. ps. galley proofs: pdf. Back to publications-list.