Ends of semigroups

Semigroups;Directed graphs

Craik, Simon ; Mitchell, James David ; Mitchell, James David

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Semigroups;    Directed graphs;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/3590/SimonCraikPhDThesis.pdf?sequence=3&isAllowed=y

来源: DR-NTU

【 摘 要 】

The aim of this thesis is to understand the algebraic structure of a semigroupby studying the geometric properties of its Cayley graph. 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 subsemigroupsand extensions of finite Rees index. We prove an analogue of Hopfs Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellativesemigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.We classify all semigroups with one end and make use of this classificationto prove various finiteness properties for semigroups with one end.We also consider the ends of digraphs with certain algebraic properties.We prove that two quasi-isometric digraphs have isomorphic end sets. We also prove that vertex transitive digraphs have 1, 2 or infinitely many ends and construct a topology that reflects the properties of the ends of a digraph.