Extensibility operator

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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 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

Template:Monotone fpm

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

Template:Descendant fpm

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.

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.