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.
Highlights
The study of ends in group theory has been extensive and has had widespread influence
In the case of a finitely generated group, the ends of the group are defined to be the ends of the Cayley graph of G with respect to the generating set A
Jackson and Kilibarda prove that the number of ends of a semigroup is invariant under change of finite generating set and provide examples of semigroups with n ends in the left Cayley graph and m ends in the right Cayley graph for any prescribed positive integers n and m
Summary
The study of ends in group theory has been extensive and has had widespread influence. The number of ends was shown to be invariant under taking subsemigroups of finite Green index when restricted to the class of cancellative semigroups These notions of index are defined later in the paper. When restricted to the case of locally finite graphs there is a natural bijection between the set of ends considered by Hopf and the equivalence classes of rays under Halin’s definition. . ., the set B of vertices that can be reached from v by a path of length at most K contains the final vertex of wi for infinitely many. Since m ≥ 2, α1 = α0 and, by construction, there is a path from α1 to every element β of the infinite set 2 (consisting of the vertices between α1 and β in qβ ∈ P1) such that the only vertex in p1 and this path is α1. By construction the only vertex on both pi and pi+1 is αi+1 and so the paths from αi+1 ∈ r to βi+1 ∈ are disjoint
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