This term is related to: Extensible automorphisms problem
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The extensibility operator is a map from the function property space to itself that takes as input a function property and outputs the property of being a function that can, for any embedding in a bigger group, be extended to a function satisfying property in that bigger group.
Here by function we mean a function from any group to itself, and by function property we mean a property over the collection of all functions.
Definition with symbols
The extensibility operator is a map from the function property space to itself that takes as input a function property and outputs a function property defined as follows:
satisfies property if and only if for every group containing , there is a function satisfying such that the restriction of to is .
In terms of the qualified extensibility operator
For the qualified extensibility operator, we replace the any embedding with any embedding satisfying a given subgroup property. Thus, the extensibility operator is a special case of the qualified extensibility operator where the qualifying subgroup property is the tautology.
Relation with other operators
If are function properties, then .
Any function satisfying property also satisfies property . That is, .
A function property that is invariant under the extensibility operator is termed an extensibility-stable function property. Thus, for instance, the property of being an inner automorphism is an extensibility-stable function property.
Effect of metaoperators
The extensibility operator is not in general an idempotent operator, and we can thus apply the iteration metaoperator to it. For any ordinal , the -extensibility operator is defined inductively as follows:
- When , then the property operator -extensibility is obtained by composing the xtensibility operator wit the -extensibility operator.
- When is a limit ordinal, then the -extensibility operator is the logical conjunction of the -extensibility operators for all .