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.

Partial Actions of Monoids

“…The notion of partial monoid action previously adopted by Megrelishvili and Schröder [12] is precisely that of Definition 2.4. Whenever we need to emphasise the distinction between the partial actions of Definitions 2.2 and 2.4, we will refer to the former as a weak partial action.…”

“…This was one of the approaches taken for partial group actions by Kellendonk and Lawson and was subsequently generalised to partial monoid actions by Megrelishvili and Schröder [12]. An important difference between the methods of [10] and [12] is the fact that Kellendonk and Lawson define their 'globalisation' in terms of generators, whilst Megrelishvili and Schröder do not. We will compromise between these two methods by describing the 'globalisation' of a partial monoid action in terms of generators.…”

“…For example, Megrelishvili and Schröder [12] consider so-called confluent partial monoid actions on topological and metric spaces and show that these can be 'globalised' in a manner analogous to that of Kellendonk and Lawson. Megrelishvili and Schröder then go on to study the connection between their starting space (with partial action) and their larger space (with action), proving embedding theorems and describing certain preservation properties.…”

“…These are equivalent to a class of premorphisms, which we call strong premorphisms. We describe two distinct methods for constructing a monoid action from a partial monoid action: the expansion method provides a generalisation of a result of Kellendonk and Lawson [10] in the group case, whilst the approach via globalisation extends results of both [12] and [10]. …”

“…We investigate partial monoid actions, in the sense of Megrelishvili and Schröder [12]. These are equivalent to a class of premorphisms, which we call strong premorphisms.…”