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PARTIAL ACTIONS OF GROUPS

J. KELLENDONK

Institut Girard Desargues, Université Claude Bernard Lyon 1, France

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and

MARK V. LAWSON

School of Informatics, University of Wales, Bangor, Dean Street, Bangor, Gwynedd LL57 1UT, United Kingdom

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https://doi.org/10.1142/S0218196704001657Cited by:91 (Source: Crossref)

Abstract

A partial action of a group G on a set X is a weakening of the usual notion of a group action: the function G×X→X that defines a group action is replaced by a partial function; in addition, the existence of g·(h·x) implies the existence of (gh)·x, but not necessarily conversely. Such partial actions are extremely widespread in mathematics, and the main aim of this paper is to prove two basic results concerning them. First, we obtain an explicit description of Exel's universal inverse semigroup , which has the property that partial actions of the group G give rise to actions of the inverse semigroup . We apply this result to the theory of graph immersions. Second, we prove that each partial group action is the restriction of a universal global group action. We describe some applications of this result to group theory and the theory of E-unitary inverse semigroups.

Keywords:

AMSC: 20M18

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