Composition rule for function restriction

Statement

Symbolic statement

If $a,b,c,d$ are function properties such that $b\leq c$, then:

$(c\to d)*(a\to b)\leq a\to d$

where $\to$ is for the function restriction formalism and $*$ denotes the composition operator.

Verbal statement

Let $H\leq K\leq G$ be groups, such that:

• Every function on $G$ satisfying property $a$ in $G$ restricts to a function on $K$ satisfying property $b$ in $K$
• Every function on $K$ satisfying property $c$ in $K$ restricts to a function on $H$ satisfying property $d$ in $H$.

Further suppose $b\leq c$.

Then every function on $G$ satisfying property $a$ in $G$ restricts to a function on $H$ satisfying property $b$ in $H$.

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 $a$ and $b$ are function properties (respectively subgroup properties) then $a\leq b$ means that every function (respectively subgroup) satisfying $a$ must also satisfy $b$.

Function restriction formalism

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

A subgroup $H$ has property $p$ in $G$ if and only if every function on $G$ satisfying property $a$ restricts to a function on $H$ satisfying property $b$.

Composition operator

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

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.