Alternative loop: Difference between revisions

Latest revision as of 03:37, 4 March 2010

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This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.
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An alternative loop is an algebra loop that is also an alternative magma.

An alternative loop is an algebra loop $(L, *)$ satisfying the following two conditions:

@@ Line 1: / Line 1: @@
 {{variation of|group}}
+{{magma property}}
+==Definition==
+===Symbol-free definition===
+An '''alternative loop''' is an [[algebra loop]] that is also an [[alternative magma]].
-==Definition==
+===Definition with symbols===
-An '''alternative loop''' or '''di-associative loop''' is an [[algebra loop]] satisfying the following equivalent conditions:
+An '''alternative loop''' is an [[algebra loop]] <math>(L,*)</math> satisfying the following two conditions:
-* The subloop generated by any two elements is a subgroup
+* <math>\! x * (x * y) = (x * x) * y \ \forall \ x,y \in L</math>.
-* For any two elements, there is a subgroup of the algebra loop (with the same identity element) containing them
+* <math>\! x * (y * y) = (x * y) * y \ \forall \ x,y \in L</math>.
-==Relation with other structures==
+==Relation with other properties==
-===Stronger structures===
+===Stronger properties===
+* [[Weaker than::Di-associative loop]]
+* [[Weaker than::Moufang loop]]
 * [[Weaker than::Group]]
-* [[Weaker than::Moufang loop]]
 ===Weaker structures===
-* [[Stronger than::Power-associative loop]]
 * [[Stronger than::Algebra loop]]
+* [[Stronger than::Left-alternative loop]]
+* [[Stronger than::Right-alternative loop]]
+* [[Stronger than::Alternative magma]]
 * [[Stronger than::Quasigroup]]