Power-associative loop: Difference between revisions

Revision as of 19:16, 5 March 2010

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Definition

A power-associative loop is an algebra loop satisfying the following equivalent conditions:

The subloop generated by any element is a cyclic subgroup, i.e., it is associative.
Every element is contained in a subgroup with the same identity element.

Note that this is somewhat stronger than simply being a power-associative magma because we care here about positive and negative powers.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group				\|FULL LIST, MORE INFO
Moufang loop				\|FULL LIST, MORE INFO
Left Bol loop				\|FULL LIST, MORE INFO
Left Bruck loop				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Algebra loop				\|FULL LIST, MORE INFO
Quasigroup				\|FULL LIST, MORE INFO
Power-associative magma				\|FULL LIST, MORE INFO

@@ Line 8: / Line 8: @@
 * Every element is contained in a subgroup with the same identity element.
-==Relation with other structures==
+Note that this is somewhat ''stronger'' than simply being a [[power-associative magma]] because we care here about positive ''and'' negative powers.
-===Stronger structures===
+==Relation with other properties==
-* [[Weaker than::Group]]
+===Stronger properties===
-* [[Weaker than::Moufang loop]]
-* [[Weaker than::Alternative loop]]
-===Weaker structures===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::Group]] || || || || {{intermediate notions short|power-associative loop|group}}
+|-
+| [[Weaker than::Moufang loop]] || || || || {{intermediate notions short|power-associative loop|group}}
+|-
+| [[Weaker than::Left Bol loop]] || || || || {{intermediate notions short|power-associative loop|left Bol loop}}
+|-
+| [[Weaker than::Left Bruck loop]] || || || || {{intermediate notions short|power-associative loop|left Bruck loop}}
+|}
+===Weaker properties===
-* [[Stronger than::Algebra loop]]
+{| class="sortable" border="1"
-* [[Stronger than::Quasigroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::Algebra loop]] || || || || {{intermediate notions short|algebra loop|power-associative loop}}
+|-
+| [[Stronger than::Quasigroup]] || || || || {{intermediate notions short|quasigroup|power-associative loop}}
+|-
+| [[Stronger than::Power-associative magma]] || || || || {{intermediate notions short|power-associative magma|power-associative loop}}
+|}