# 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 propertyView 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

## Contents

## 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 is the maximum among all subgroup properties for which the following holds:

where denotes the composition operator on subgroup properties.

### Definition with symbols

The right transiter of a subgroup property is the property defined by the following criterion:

A subgroup in a group is said to have if and only if whenever has property as a subgroup of , also has property in .

We denote the right transiter of a property as .

### 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 satisfies the intermediate subgroup condition, so does the right transiter of .

## Computing the right transiter

### In the function restriction formalism

If a subgroup property can be expressed by a function restriction formal expression , 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.