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