# Composition rule for function restriction

## Contents

## Statement

### Symbolic statement

If are function properties such that , then:

where is for the function restriction formalism and denotes the composition operator.

### Verbal statement

Let be groups, such that:

- Every function on satisfying property in restricts to a function on satisfying property in
- Every function on satisfying property in restricts to a function on satisfying property in .

Further suppose .

Then every function on satisfying property in restricts to a function on satisfying property in .

## Definitions

### Subgroup properties and function properties

A subgroup property is a property that, given any group and any subgroup of it, is either true for the subgroup in the group or is not true for the subgroup in the group.

A function property is a property that, given any group and any function from the group to itself, is either true for the function on the group, or is not true for the function on the group.

if and are function properties (respectively subgroup properties) then means that every function (respectively subgroup) satisfying must also satisfy .

### Function restriction formalism

We are using the function restriction formalism for expressing subgroup properties. A subgroup property has a function restriction formal expression if the following holds:

A subgroup has property in if and only if every function on satisfying property restricts to a function on satisfying property .

### Composition operator

The composition operator on subgroup properties is defined as follows: given two subgroup properties and , the composition is defined as the following subgroup property: satisfies in if and only if there exists an intermediate subgroup such that satisfies property in and satisfies property in .

The result we have relates the function restriction formalism with the composition operator.

## Proof

The proof is direct from the verbal statement. To prove the symbolic statement, we simply *unravel* the definition and obtain the verbal statement.