Right transiter

This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup property

View a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

This property modifier is idempotent and a property is a fixed-point, or equivalently, an image of this if and only if it is a:t.i. subgroup property

Definition

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:

$p*q\leq p$

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

Template:Intsubcondn-preserving

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

Computing the right transiter

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.