# Extensibility operator

Template:Function property modifier

This term is related to: Extensible automorphisms problem

View other terms related to Extensible automorphisms problem | View facts related to Extensible automorphisms problem

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## Definition

### Symbol-free definition

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, .

## Fixed points

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

### Iteration

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 .