Left 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)

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

$q * p \le p$

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

Definition with symbols

The left 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 $G$ has property $p$ as a subgroup of a group $K$, $H$ also has property $p$ in $K$.

We denote the left transiter of a property $p$ as $L(p)$.

In terms of the residual operator

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

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

Facts

Transiter master theorem

The transiter master theorem for composition operator on subgroup properties, which is a corollary of the residuation master theorem, states the following:

1. If $p$ is transitive, $p$ is stronger than its left transiter. That is, $p \le L(p)$.
2. If $p$ is identity-true, the left transiter of $p$ is stronger than $p$. That is, $L(p) \le p$.
3. If $p$ is a t,i. subgroup property, i.e., $p$ is both transitive and identity-true, then $p = L(p)$.
4. The left transiter of any property is a t.i. subgroup property, that is, it is both transitive and identity-true.

Combining parts (3) and (4) of the transiter master theorem, we see that the left transiter operator is an idempotent operator and the fixed point space is precisely the collection of t.i. subgroup properties (that is, subgroup properties that are transitive and identity-true).

The fixed point space of the left transiter operator is the same as the fixed point space of the subordination operator and the right transiter operator.

Relation with metaproperties

A subgroup metaproperty that is preserved on taking left transiters is termed a left transiter-preserved subgroup metaproperty. Some subgroup metaproperties that we encounter that are left transiter-preserved may satisfy a stronger condition: they are left residual-preserved. For instance, the metaproperties of being intersection-closed, join-closed and upward-closed are all left residual-preserved.

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