Direct factor of a loop: Difference between revisions

Revision as of 06:27, 20 August 2011

This article defines a property that can be evaluated for a subloop of a loop| View other such properties

ANALOGY: This is an analogue in algebra loop of a property encountered in group. Specifically, it is a subloop property analogous to the subgroup property: direct factor
View other analogues of direct factor | View other analogues in algebra loops of subgroup properties (OR, View as a tabulated list)

Definition

A direct factor of an algebra loop $L$ is a subloop $S$ such that there exists a subloop $T$ of $S$ satisfying:

Every element of $L$ can be written uniquely in the form $s * t, s \in S, t \in T$ .
For any $s_{1}, s_{2} \in S, t_{1}, t_{2} \in T$ , we have $(s_{1} * t_{1}) * (s_{2} * t_{2}) = (s_{1} * s_{2}) * (t_{1} * t_{2})$ . In particular, by an idea analogous to the Eckmann-Hilton principle, every element of $S$ commutes with every element of $T$ .

Relation with other properties

Weaker properties

Property	Meaning	Proof of strictness (reverse implication failure)	Intermediate notions
central factor of a loop	Product with another subloop is whole loop, they commute in the strong sense indicated here	(proof for groups suffices)	\|FULL LIST, MORE INFO
normal subloop			\|FULL LIST, MORE INFO
Lagrange-like subloop			\|FULL LIST, MORE INFO
retract of a loop	image of the whole loop under a retraction, i.e., an endomorphism whose fixed point set equals its image		\|FULL LIST, MORE INFO

@@ Line 21: / Line 21: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Stronger than::Central factor of a loop]] || Product with another subloop is whole loop, they commute in the strong sense indicated here || || ([[central factor not implies direct factor|proof for groups suffices]]) || {{intermediate notions short|central factor of a loop|direct factor of a loop}}
+| [[Stronger than::central factor of a loop]] || Product with another subloop is whole loop, they commute in the strong sense indicated here || || ([[central factor not implies direct factor|proof for groups suffices]]) || {{intermediate notions short|central factor of a loop|direct factor of a loop}}
 |-
-| [[Stronger than::Normal subloop]] || || || || {{intermediate notions short|normal subloop|direct factor of a loop}}
+| [[Stronger than::normal subloop]] || || || || {{intermediate notions short|normal subloop|direct factor of a loop}}
 |-
 | [[Stronger than::Lagrange-like subloop]] || || || || {{intermediate notions short|normal subloop|Lagrange-like subloop}}
 |-
-| [[Stronger than::Retract of a loop]] || image of the whole loop under a retraction, i.e., an endomorphism whose fixed point set equals its image || || || {{intermediate notions short|retract of a loop|direct factor of a loop}}
+| [[Stronger than::retract of a loop]] || image of the whole loop under a retraction, i.e., an endomorphism whose fixed point set equals its image || || || {{intermediate notions short|retract of a loop|direct factor of a loop}}
 |}