Left transiter: Difference between revisions

Latest revision as of 09:30, 8 January 2026

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

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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:

$q * p \leq p$

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:

If $p$ is transitive, $p$ is stronger than its left transiter. That is, $p \leq L (p)$ .
If $p$ is identity-true, the left transiter of $p$ is stronger than $p$ . That is, $L (p) \leq p$ .
If $p$ is a t.i. subgroup property, i.e., $p$ is both transitive and identity-true, then $p = L (p)$ .
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.

@@ Line 1: / Line 1: @@
 {{subgroup property modifier}}
-{{fpsace|[[t.i. subgroup metaproperty]]}}
+{{wikilocal}}
+{{fpspace|t.i. subgroup property}}
 ==Definition==
@@ Line 24: / Line 26: @@
 ===In terms of the residual operator===
-The left transiter of a property is its left residual by itself, with respect to the composition operator.
+The left transiter of a property is its [[left residual operator for composition|left residual]] by itself, with respect to the [[composition operator]].
-A [[left transiter (generic)|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 nice behaviours from this special case.
+A [[left transiter (generic)|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==
@@ Line 32: / Line 34: @@
 ===Transiter master theorem===
-The [[transiter master theorem]], which is a corollary of the [[residuation master theorem]], states the following:
+The [[transiter master theorem for composition operator on subgroup properties]], which is a corollary of the [[residuation master theorem]], states the following:
-* If <math>p</math> is [[transitive subgroup property|transitive]], <math>p</math> is stronger than its left transiter. That is, <math>p</math> &le; <math>L(p)</math>.
+# If <math>p</math> is [[transitive subgroup property|transitive]], <math>p</math> is stronger than its left transiter. That is, <math>p \le L(p)</math>.
-* If <math>p</math> is [[identity-true subgroup property|identity-true]], the left transiter of <math>p</math> is stronger than <math>p</math>. That is, <math>L(p)</math> &le; <math>p</math>.
+# If <math>p</math> is [[identity-true subgroup property|identity-true]], the left transiter of <math>p</math> is stronger than <math>p</math>. That is, <math>L(p) \le p</math>.
-* The left transiter of any property is a [[t.i. subgroup property]], that is, it is both transitive and identity-true.
+# If <math>p</math> is a [[t.i. subgroup property]], i.e., <math>p</math> is both transitive and identity-true, then <math>p = L(p)</math>.
+# The left transiter of any property is a [[t.i. subgroup property]], that is, it is both transitive and identity-true.
-Combining the above three parts 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. subgorup properties (that is, subgroup properties that are 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 transiter master theorem holds more generally for any associative quantalic binary operator, and hence t.i. properties with respect to that operator are the fixed point space of the left transiter operator with respect to that operator.
 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.
-===Interplay with the intersection operator===
+===Relation with metaproperties===
-The interplay stems from the subdistributivity relation between the [[intersection operator]] and the [[composition operator]].
-<math>(q_1 \cap q_2) * p</math> &le; <math>q_1 * p \cap q_2 * p</math>
-Some implications:
-* The intersection operator applied to the left transiter of a property is stronger than the left transiter applied to its intersection. That is:
-<math>L(p)</math> &cap; <math>L(p)</math> &le; <math>L(p</math> &cap; <math>p)</math>
-* If <math>p</math> is a [[finite-intersection-closed subgroup property]], so is <math>L(p)</math>.
-The corresponding results for arbitrary intersections are also true.
-===Interplay with restriction formalisms===
-The [[right tightness theorem]] for the [[function restriction formalism]], or more generally, for any [[injective restriction formalism]], gives a method to compute the left transiter for a subgroup property that has been expressed using such a restriction formalism. The left transiter of any property expressed using the formalims is simply the [[balanced subgroup property]] corresponding to the ''tightest possible'' right side.
-==Effect on subgroup metaproperties==
-{{intersection-closedness-preserving}}
-From the above observations on the interplay between left transiter and the intersection operator, we conclude that the left transiter of any [[intersection-closed subgroup property]] is again intersection-closed.
-{{join-closedness-preserving}}
-For reasons very similar to that for intersections, the left transiter of any [[join-closed subgroup property]] is again join-closed.
-{{upward-closedness-preserving}}
-The left transiter of an [[upward-closed subgroup property]] is also upward-closed.
+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 subgroup metaproperty|left residual-preserved]]. For instance, the metaproperties of being [[intersection-closed subgroup property|intersection-closed]], [[join-closed subgroup property|join-closed]] and [[upward-closed subgroup property|upward-closed]] are all left residual-preserved.
 ==Computing the left transiter==