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Reducible multiary group

Definition

Recall that a multiary group is a n-ary group for some n \ge 2. A n-ary group with multiplication f:G^n \to G 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 G such that f(a_1,a_2,\dots,a_n) = a_1a_2\dots a_n for all (possibly repeated) a_1,a_2,\dot,a_n \in G, 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 n-ary associativity and unique solution conditions; they follow automatically.
2 existence of neutral element there exists a neutral element e \in G for f: f evaluated at any tuple where n - 1 of the entries are equal to e and the remaining entry is a \in G, gives output a. This is true regardless of where we place a and is also true if a = e. 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

Facts