The structure of a graph inverse semigroup

Abstract

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.

Embedding of graph inverse semigroups into CLP-compact topological semigroups

“…By [32,Corollary 2], two non-zero elements ab −1 and cd −1 of a GIS G(E) are D-equivalent iff r(a) = r(b) = r(c) = r(d). Observe that each non-zero D-class contains exactly one vertex.…”

An alternative look at the structure of graph inverse semigroups

“…Graph inverse semigroups play an important role in the study of rings and C * -algebras (see [1,3,12,18,25]). Algebraic theory of graph inverse semigroups is well developed (see [2,5,16,17,20,22]). Topological properties of graph inverse semigroups were investigated in [6,7,8,23].…”

“…Topological properties of graph inverse semigroups were investigated in [6,7,8,23]. This paper is inspired by the paper of Mesyan and Mitchell [22] and can be regarded as an alternative look at the structure of graph inverse semigroups.…”

“…By Theorem 4, the semigroup D 0 e is isomorphic to the semigroup B 0 Ie (P 0 ). By [22,Corollary 2], two non-zero elements ab −1 and cd −1 of a GIS G(E) are J -equivalent iff there exist elements u, v ∈ Path(E) such that s(u) = r(a) = r(v) and r(u) = r(c) = s(v). There exists a one to one correspondence between the set of strongly connected components of a graph E and non-zero J -classes of a GIS G(E).…”

FP-injective semirings, semigroup rings and Leavitt path algebras

“…We now consider a special class of inverse semigroup rings which are constructed from directed graphs E = (E 0 , E 1 , s, r) with vertex set E 0 and edge set E 1 , where the maps s, r : E 1 −→ E 0 describe the source and the range of each edge. The graph inverse semigroup G(E) (see for example [15] for further reference) is the semigroup with zero generated by the sets E 0 and E 1 , together with a set of variables {e * |e ∈ E 1 }, satisfying the following relations for all v, w ∈ E 0 and e, f ∈ E 1 :…”