We define a functor which gives the “global rank of a quiver representation” and prove that it has nice properties which make it a generalization of the rank of a linear map. We demonstrate how to construct other “rank functors” for a quiver Q, which induce ring homomorphisms (called “rank functions”) from the representation ring of Q to Z. These rank functions give discrete numerical invariant...

We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries include efficient algorithms for computing weighted sums approximating the number of k-colorings and the number of independent sets in a graph, as well as a...

List homomorphisms are functions which can be e ciently computed in parallel since they ideally suit the divide-and-conquer paradigm. However, some interesting functions, e.g., the maximum segment sum problem, are not list homomorphisms. In this paper, we propose a systematic way of embedding them into list homomorphisms so that parallel programs are derived. We show, with an example, how a sim...

In a previous paper, the author has introduced a number of homomorphisms of an arbitrary semigroup into the translational hull of certain Rees matrix semigroups or orthogonal sums thereof. For regular semigroups, it is proved here that all of these homomorphisms have the property that the image is a densely embedded subsemigroup, i.e., is a densely embedded ideal of its idealizer, and that the ...

In this note we study umkehr maps in generalized (co)homology theories arising from the Pontrjagin-Thom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which any umkehr homomorphism must satisfy. We use a version of Brown representability to show that these axioms ...

Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number. We investigate here homomorphisms from orienta-tions of undirected planar graphs to graphs (not necessarily planar) containing as few digons as possible. We relate the existence of such homomorphisms to girth and it appears that these questions remain interesting even under large girth assumption ...