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 nonRees 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.

To every directed graph E one can associate a graph inverse semigroup G(E), where elements roughly correspond to possible paths in E. These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C * -algebras, and Toeplitz C * -algebras. We investigate topologies that turn G(E) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G(E) \ {0} must be discrete for any directed graph E. On the other hand, G(E) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E, G(E) admits a T 1 semigroup topology in which G(E) \ {0} is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G(E) in larger topological semigroups.

A ring is called a commutator ring if every element is a sum of additive commutators. In this note we give examples of such rings. In particular, we show that given any ring R, a right ii-module N, and a nonempty set il, Endft(® n N) and End^^n N) a r e commutator rings if and only if either fl is infinite or Endii(N) is itself a commutator ring. We also prove that over any ring, a matrix having trace zero can be expressed as a sum of two commutators.

For a field K and directed graph E, we analyze those elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E), L_K(E)]. This analysis allows us to give easily computable necessary and sufficient conditions to determine which Lie algebras of the form [L_K(E), L_K(E)] are simple, when E is row-finite (i.e., has finite out-degree) and L_K(E) is simple.Comment: 18 pages. In the second version the exposition has been improved, and various typos and minor errors have been correcte

Abstract. 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.