Right transiter

Symbol-free definition
The right transiter is an operator from the subgroup property space to itself, defined as follows. The right transiter of a subgroup property $$p$$ is the maximum among all subgroup properties $$q$$ for which the following holds:

$$p * q \le p$$

where $$*$$ denotes the composition operator on subgroup properties.

Definition with symbols
The right transiter of a subgroup property $$p$$ is the property $$q$$ defined by the following criterion:

A subgroup $$H$$ in a group $$G$$ is said to have $$q$$ if and only if whenever $$K$$ has property $$p$$ as a subgroup of $$H$$, $$K$$ also has property $$p$$ in $$G$$.

We denote the right transiter of a property $$p$$ as $$R(p)$$.

In terms of the residual operator
The right transiter of a property is its right residual by itself, with respect to the composition operator.

A general notion of right transiter can be given for any associative quantalic binary operator on a property space. The generic notion inherits many of the nice behaviours from this special case.

Effect on subgroup metaproperties
If $$p$$ satisfies the intermediate subgroup condition, so does the right transiter of $$p$$.

In the function restriction formalism
If a subgroup property can be expressed by a function restriction formal expression $$a \to b$$, thne we can do the following:


 * Use the left tightening operator to obtain a left tight restriction formal expression for the subgroup property
 * Then the right transiter is the balanced subgroup property with respect to the left side in the left tight restriction formal expression.