Extensibility operator

Symbol-free definition
The extensibility operator is a map from the function property space to itself that takes as input a function property $$p$$ and outputs the property of being a function that can, for any embedding in a bigger group, be extended to a function satisfying property $$p$$ 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 $$E$$ is a map from the function property space to itself that takes as input a function property $$p$$ and outputs a function property $$q = E(p)$$ defined as follows:

$$f: G \to G$$ satisfies property $$q$$ if and only if for every group $$H$$ containing $$G$$, there is a function $$f': H \to H$$ satisfying $$p$$ such that the restriction of $$f'$$ to $$G$$ is $$f$$.

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 $$p_1 \le p_2$$ are function properties, then $$E(p_1) \le E(p_2)$$.

Any function satisfying property $$E(p)$$ also satisfies property $$p$$. That is, $$E(p) \le p$$.

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.

Iteration
The extensibility operator is not in general an idempotent operator, and we can thus apply the iteration metaoperator to it. For any ordinal $$\alpha$$, the $$\alpha$$-extensibility operator is defined inductively as follows:


 * When $$\alpha = \beta + 1$$, then the property operator $$\alpha$$-extensibility is obtained by composing the xtensibility operator wit the $$\beta$$-extensibility operator.
 * When $$\alpha$$ is a limit ordinal, then the $$\alpha$$-extensibility operator is the logical conjunction of the $$\gamma$$-extensibility operators for all $$\gamma < \alpha$$.