Ends of semigroups

Abstract

We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf's Theorem, stating that an infinite group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.

Ends for subsemigroups of finite index

“…Also, it was proven in [5] that the number of directed ends of a f.g. left cancellative infinite semigroup is 1, 2 or ∞.…”

“…The semigroups S and T from Example 4.2 are inverse, so the inverse semigroups are not a good candidate. As with directed ends [5], we find that the class of cancellative semigroups is the right one:…”

“…5, we present several semigroups that serve as counterexamples to our attempts to formulate a semigroup analogue of Stallings Theorem. For more details on terminology and results from semigroup theory, we refer the reader to [4,11,13] In a recent preprint [5] the notion of directed ends of semigroups is introduced. In it, we prove that the number of directed ends is preserved by subsemigroups of finite Rees index, and that the same holds for Green index when working with the class of finitely generated left cancellative semigroups.…”

Hyperbolicity in semimetric spaces, digraphs and semigroups

“…The results of Section 15 together with results on the number of ends of semigroups by Craik et al [11] enable us to obtain some results on the size of the quasi-geodesic boundary of hyperbolic semigroups. First, we immediately have the following corollary of Corollary 15.7.…”

“…Craik et al [11,Corollary 2.3] proved that Zuther's definition of ends of digraphs extends to a notion of ends of finitely generated semigroups that is preserved under changing the finite generating set. More precisely, for a finitely generated semigroup every right Cayley digraph has the same isomorphism type (as a partially ordered set) of their ends.…”