Extensibility operator

This term is 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 $p$ 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 $p_{1}\leq p_{2}$ are function properties, then $E(p_{1})\leq E(p_{2})$.

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

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.

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 $\alpha$, 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 $\gamma <\alpha$.