Reducible multiary group
From Groupprops
Definition
Recall that a multiary group is a -ary group for some . A -ary group with multiplication is termed reducible if it satisfies the following equivalent conditions:
No. | Shorthand | Statement with symbols | Note |
---|---|---|---|
1 | arises from a group | there exists a group structure on such that for all (possibly repeated) , with the multiplication on the right being as per the group structure. For more on this, see group is n-ary group for all n | If we're assuming this, we don't need to check the -ary associativity and unique solution conditions; they follow automatically. |
2 | existence of neutral element | there exists a neutral element for : evaluated at any tuple where of the entries are equal to and the remaining entry is , gives output . This is true regardless of where we place and is also true if . | Note that we do need to separately check the associativity and unique solution conditions. |
Equivalence of definitions
Further information: equivalence of definitions of reducible multiary group