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Summary: arXiv:1104.2301v1[math.GR]12Apr2011
GEOMETRIC SEMIGROUP THEORY
JON MCCAMMOND, JOHN RHODES, AND BENJAMIN STEINBERG
Abstract. Geometric semigroup theory is the systematic investigation
of finitely-generated semigroups using the topology and geometry of
their associated automata. In this article we show how a number of
easily-defined expansions on finite semigroups and automata lead to sim-
plifications of the graphs on which the corresponding finite semigroups
act. We show in particular that every finite semigroup can be finitely
expanded so that the expansion acts on a labeled directed graph which
resembles the right Cayley graph of a free Burnside semigroup in many
respects.
Contents
1. Introduction 2
2. The topology of directed graphs 2
2.1. Directed graphs 3
2.2. Morphisms of directed graphs 6
2.3. Semigroups and automata 7
2.4. Rooted graphs 9
2.5. The unique simple path property 14
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