Globalization of Confluent Partial Actions on Topological and Metric Spaces

Abstract

We generalize Exel's notion of partial group action to monoids. For partial monoid actions that can be defined by means of suitably well-behaved systems of generators and relations, we employ classical rewriting theory in order to describe the universal induced global action on an extended set. This universal action can be lifted to the setting of topological spaces and continuous maps, as well as to that of metric spaces and non-expansive maps. Well-known constructions such as Shimrat's homogeneous extension are special cases of this construction. We investigate various properties of the arising spaces in relation to the original space; in particular, we prove embedding theorems and preservation properties concerning separation axioms and dimension. These results imply that every normal (metric) space can be embedded into a normal (metrically) ultrahomogeneous space of the same dimension and cardinality.

New presentation of material on rewriting

References in corpus (3)

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• Geometric partial comodules over flat coalgebras in Abelian categories are globalizable
• Borel Globalizations of Partial Actions of Polish Groups
• Geometric presentations for Thompson's groups
• Partial monoid actions on objects in categories with pullbacks and their globalizations
• Partial actions and proper extensions of two-sided restriction semigroups
• $F$-birestriction monoids in enriched signature
• Partial Semigroupoid Actions on Sets