If C is a category of algebras closed under finite direct products, and MC the commutative monoid of isomorphism classes of members of C, with operation induced by direct product, A.Tarski defines a nonidentity element p of MC to be prime if, whenever it divides a product of two elements in that monoid, it divides one of them, and calls an object of C prime if its isomorphism class has this property.
McKenzie, McNulty and Taylor have asked whether the category of nonempty semigroups has any prime objects. We show in section 2 of this note that it does not. However, for the category of monoids, and some other subcategories of semigroups, we obtain examples of prime objects in sections 3-4. In section 5 two related questions, open as far as I know, are recalled.
In section 6, which can be read independently of the rest of this note, we recall two other conditions that are called primeness by semigroup theorists, and obtain results and examples on the relationships among those two conditions and Tarski's in categories of groups. Section 7 notes an interesting characterization of one of those conditions on finite algebras in an arbitrary variety.
Several questions are raised.
17 pp., last revised Oct. 17, 2024.
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arXiv:2409.15541.
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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.
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arXiv:2208.06511.
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Let R be a ring, and consider a left R-module given with two (generally infinite) direct sum decompositions A ⊕ (⊕i∈I Ci) = M = B ⊕ (⊕j ∈J Dj), such that the submodules A and B and the Dj are each finitely generated. We show that there then exist finite subsets I0 ⊆ I, J0 ⊆ J, and a direct summand Y ⊆ ⊕i∈I0 Ci, such that A ⊕ Y = B ⊕ (⊕j ∈J0 Dj).
We then note some ways that this result can and cannot be generalized, and pose some related questions.
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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 C is convex?
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 P under forming, whenever A and B are nondisjoint and neither contains the other, the sets A∪B, A∩B, and A∖B. We determine the form P can take when C, and hence P, is finite, and for this case get necessary and sufficient conditions for there to exist an ordering of Ω of the desired sort. From this we obtain a condition which works without the finiteness hypothesis.
We establish bounds on the cardinality of the subset P generated as above by an n-element set C.
We note connections with the theory of interval graphs and hypergraphs, which lead to other ways of answering Wehrung's question.
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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 (xi)i∈Z in groups and in involutory quandles (in the former context, sequences of the form xi = x zi where x, z ∈ G, among other characterizations; in the latter, sequences satisfying xi+1 = xi ◁ xi-1) are examined. A necessary condition is noted for an involutory quandle to be embeddable in the core quandle of a group. Some implications are established between identities holding in a group and in its core quandle. Upper and lower bounds are obtained on the number of elements needed to generate the quandle Core(G) for a finitely generated group G. Several questions are posed.
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.
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arXiv:2006.00641.
MR3787805.
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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.
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arXiv:1905.12704.
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In Question 19.35 of the Kourovka Notebook, M. H. Hooshmand asks whether, given a finite group G and a factorization card(G) = n1 ... nk, one can always find subsets A1, ..., Ak of G with card(Ai) = ni such that G = A1 ... Ak; equivalently, such that the group multiplication map A1 × ... × Ak → G is a bijection.
We show that for G the alternating group on 4 elements, k = 3, and (n1, n2, n3) = (2, 3, 2), the answer is negative. We then generalize some of the tools used in our proof, and note an open question.
tex.
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arXiv:2003.12866.
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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.
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arXiv:1812.06123.
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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.
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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.
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arXiv:1702.06088.
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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.
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arXiv:1611.00972.
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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(M3) 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(M3) can be obtained by stitching together three copies of Δ(C), where C is a 3-element chain.
Erratum: In section 3.4, the first sentence of the paragraph before Corollary 14 refers to connected components "S1 and S2". For consistency with the notation in the rest of the paper, that should be "S0 and S1".
DOI.
MR3604382.
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arXiv:1602.00034.
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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.
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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.
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arXiv:1505.02312.
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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 filter of cofinite subsets of a countable subset X ⊆ G (the Fréchet filter on X), then 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 does not lie in the n-fold sum n (S ∪ {0} ∪ −S).
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.
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arXiv:1311.2648.
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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.
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arXiv:1310.5365.
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We study properties of a group, abelian group, ring, or monoid B which (a) guarantee that every homomorphism from an infinite direct product ΠI Ai of objects of the same sort onto B factors through the direct product of finitely many ultraproducts of the Ai (possibly after composition with the natural map B → B/Z(B) or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters in question must be principal.
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 I to be an intersection of finitely many ultrafilters, respectively, finitely many κ-complete ultrafilters for a given uncountable cardinal κ. From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the Łoś-Eda theorem (characterizing homomorphisms from a not-necessarily-countable direct product of modules to a slender module), and some results of N. Nahlus and the author on homomorphisms on infinite direct products of not-necessarily-associative k-algebras. The same tools allow other results of N. Nahlus and the author on that topic to be nontrivially strengthened, and yield an analog to one of their results, with nonabelian groups taking the place of k-algebras.
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 DI (D a division ring) is extended to reduced products DI/F and used to generalize to division rings another result of N. Nahlus and the author.
MR3226022.
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arXiv:1301.6383.
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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.
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arXiv:1309.0564.
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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.
MR3204348.
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arXiv:1206.0326.
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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.
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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 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.
MR3069282.
tex file of preprint.
pdf file of reprint.
arxiv.
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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.
MR2983182.
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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.
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preprint:
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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.
MR3008736.
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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.
MR2900652.
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Let k be an infinite field, and g : kI → V a k-linear map, where I is an infinite set and V a k-vector-space. It is shown that if dimk(V) is not too large (where depending on the hypotheses on card(k) and card(I), this can mean that dimk(V) 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.
It is shown in a subsequent note that cardinality assumptions of this note can be weakened.
MR2891132.
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arXiv.
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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. 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.
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dvi.
pdf.
ps.
TeX source file of preprint (requires
this
style-class file).
arXiv:1108.0958.
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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 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. 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, 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.
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 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.
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.
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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.
MR2407106.
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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.
MR2373687.
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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.)
MR2279231.
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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.
MR2280223.
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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.
MR2239037.
DOI.
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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.
MR2205070. DOI. preprint: tex. dvi. ps. galley proofs: pdf. Back to publications-list.