Globalization of Confluent Partial Actions on Topological and Metric Spaces (bibtex)

by Michael Megrelishvili and Lutz Schröder

Abstract:

Given a partial action of a monoid on a set, equipped with a suitable system 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.

Reference:

Michael Megrelishvili and Lutz Schröder: Globalization of Confluent Partial Actions on Topological and Metric Spaces, In Topology and Applications, 145, pp. 119–145, 2004. [preprint]

Bibtex Entry:

@Article{MegrelishviliSchroder04,
  author = {Michael Megrelishvili and Lutz Schr{\"o}der},
  title = {Globalization of Confluent Partial Actions on Topological and Metric Spaces},
  year = {2004},
  journal = {Topology and Applications},
  volume = {145},
  pages = {119-145},
  keywords = {Partial actions ultrahomogeneuos spaces rewriting globalizations},
  comment = { <a href = "http://www8.informatik.uni-erlangen.de/~schroeder/papers/partact.pdf"> [preprint] </a>},
  abstract = {Given a partial action of a monoid on a set, equipped with a suitable system 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.
},
}