32 research outputs found
Groups where free subgroups are abundant
Given an infinite topological group G and a cardinal k>0, we say that G is
almost k-free if the set of k-tuples in G^k which freely generate free
subgroups of G is dense in G^k. In this note we examine groups having this
property and construct examples. For instance, we show that if G is a
non-discrete Hausdorff topological group that contains a dense free subgroup of
rank k>0, then G is almost k-free. A consequence of this is that for any
infinite set X, the group of all permutations of X is almost 2^|X|-free. We
also show that an infinite topological group is almost aleph_0-free if and only
if it is almost n-free for each positive integer n. This generalizes the work
of Dixon and Gartside-Knight.Comment: 13 page
Generating self-map monoids of infinite sets
Let I be a countably infinite set, S = Sym(I) the group of permutations of I,
and E = End(I) the monoid of self-maps of I. Given two subgroups G, G' of S,
let us write G \approx_S G' if there exists a finite subset U of S such that
the groups generated by G \cup U and G' \cup U are equal. Bergman and Shelah
showed that the subgroups which are closed in the function topology on S fall
into exactly four equivalence classes with respect to \approx_S. Letting
\approx denote the obvious analog of \approx_S for submonoids of E, we prove an
analogous result for a certain class of submonoids of E, from which the theorem
for groups can be recovered. Along the way, we show that given two subgroups G,
G' of S which are closed in the function topology on S, we have G \approx_S G'
if and only if G \approx G' (as submonoids of E), and that cl_S (G) \approx
cl_E (G) for every subgroup G of S (where cl_S (G) denotes the closure of G in
the function topology in S and cl_E (G) its closure in the function topology in
E).Comment: 26 pages. In the second version several of the arguments have been
simplified, references to related literature have been added, and a few minor
errors have been correcte
Polynomials of small degree evaluated on matrices
A celebrated theorem of Shoda states that over any field K (of characteristic
0), every matrix with trace 0 can be expressed as a commutator AB-BA, or,
equivalently, that the set of values of the polynomial f(x,y)=xy-yx on the
nxn-matrix K-algebra contains all matrices with trace 0. We generalize Shoda's
theorem by showing that every nonzero multilinear polynomial of degree at most
3, with coefficients in K, has this property. We further conjecture that this
holds for every nonzero multilinear polynomial with coefficients in K of degree
m, provided that m is at most n+1.Comment: 9 page
Traces on Semigroup Rings and Leavitt Path Algebras
The trace on matrix rings, along with the augmentation map and Kaplansky
trace on group rings, are some of the many examples of linear functions on
algebras that vanish on all commutators. We generalize and unify these examples
by studying traces on (contracted) semigroup rings over commutative rings. We
show that every such ring admits a minimal trace (i.e., one that vanishes only
on sums of commutators), classify all minimal traces on these rings, and give
applications to various classes of semigroup rings and quotients thereof. We
then study traces on Leavitt path algebras (which are quotients of contracted
semigroup rings), where we describe all linear traces in terms of central maps
on graph inverse semigroups and, under mild assumptions, those Leavitt path
algebras that admit faithful traces.Comment: 21 page
Generating Endomorphism Rings of Infinite Direct Sums and Products of Modules
Let R be a ring, M a left R-module, I an infinite set, N either the direct
sum or product of |I| copies of M, and E the endomorphism ring of N as a left
R-module. In this note it is shown that E is not the union of a chain of |I| or
fewer proper subrings, and also that given a generating set U for E as a ring,
there exists a positive integer n such that every element of E is represented
by a ring word of length at most n in elements of U.Comment: 3 page
Endomorphism rings generated using small numbers of elements
Let R be a ring, M a nonzero left R-module, X an infinite set, and E the
endomorphism ring of the direct sum of copies of M indexed by X. Given two
subrings S and S' of E, we will say that S is equivalent to S' if there exists
a finite subset U of E such that the subring generated by S and U is equal to
the subring generated by S' and U. We show that if M is simple and X is
countable, then the subrings of E that are closed in the function topology and
contain the diagonal subring of E (consisting of elements that take each copy
of M to itself) fall into exactly two equivalence classes, with respect to the
equivalence relation above. We also show that every countable subset of E is
contained in a 2-generator subsemigroup of E.Comment: 12 pages. In the new version the main result has been slightly
generalized, references have been added (particularly in connection with
Corollary 2, which had been known before), and several of the proofs have
been rewritten to improve clarit
The Structure of a Graph Inverse Semigroup
Given any directed graph E one can construct a graph inverse semigroup G(E),
where, roughly speaking, elements correspond to paths in the graph. In this
paper we study the semigroup-theoretic structure of G(E). Specifically, we
describe the non-Rees congruences on G(E), show that the quotient of G(E) by
any Rees congruence is another graph inverse semigroup, and classify the G(E)
that have only Rees congruences. We also find the minimum possible degree of a
faithful representation by partial transformations of any countable G(E), and
we show that a homomorphism of directed graphs can be extended to a
homomorphism (that preserves zero) of the corresponding graph inverse
semigroups if and only if it is injective.Comment: 19 pages; corrected errors, improved organization, strengthened a
result (Theorem 20), added reference
Graded Semigroups
We systematically develop a theory of graded semigroups, that is semigroups S
partitioned by groups G, in a manner compatible with the multiplication on S.
We define a smash product S#G, and show that when S has local units, the
category S#G-Mod of sets admitting an S#G-action is isomorphic to the category
S-Gr of graded sets admitting an appropriate S-action. We also show that when S
is an inverse semigroup, it is strongly graded if and only if S-Gr is naturally
equivalent to S_1-Mod, where S_1 is the partition of S corresponding to the
identity element 1 of G. These results are analogous to well-known theorems of
Cohen/Montgomery and Dade for graded rings. Moreover, we show that graded
Morita equivalence implies Morita equivalence for semigroups with local units,
evincing the wealth of information encoded by the grading of a semigroup. We
also give a graded Vagner-Preston theorem, provide numerous examples of
naturally-occurring graded semigroups, and explore connections between graded
semigroups, graded rings, and graded groupoids. In particular, we introduce
graded Rees matrix semigroups, and relate them to smash product semigroups. We
pay special attention to graded graph inverse semigroups, and characterise
those that produce strongly graded Leavitt path algebras.Comment: 45 pages. The second version has minor error and typo fixes,
additional references, and improvements in the expositio