Globalization of confluent partial actions on topological and metric spaces

By Michael Megrelishvili, Lutz Schroeder. 2002

51 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bff74392fc24f58c8237ce7815097ba4a1cc2bce

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Mikhailo Dokuchaev
2. Patrick Dehornoy
3. Andrés Navas
4. Matthew G. Brin
5. Eric Jespers
6. Jan Okninski

In Mathematical Physics

1. Mikhailo Dokuchaev
2. Patrick Dehornoy
3. Andrés Navas
4. Matthew G. Brin
5. Antonio Paques

Citations in the corpus

• Recent developments around partial actions (2018, background): “Subsequent C-algebraic and topological developments on partial actions were made in [5], [159], [233], [6], [141], [222], [60], [112], [93], [11] and [163].”
• Partial actions: a survey (2011, background): “A very close concept of “homogeneity” (in its various forms) is especially important in model theory [69] and topology (see, for example [76], [94] and the bibliography therein).”
• Geometric presentations for Thompson's groups (2004, result): “It is essentially equivalent to the one investigated in [18] (in the case of groups)—see also [19]— and in [ 22 ] (in the case of monoids).”