Equivalence of definitions of reducible multiary group

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This article gives a proof/explanation of the equivalence of multiple definitions for the term reducible multiary group
View a complete list of pages giving proofs of equivalence of definitions

Statement

Recall that a multiary group is a n-ary group for some n \ge 2. The following are equivalent for a n-ary group with multiplication f:G^n \to G:

  1. 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
  2. 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.

Related facts

Proof

(1) implies (2)

This is immediate: we can take e to be the neutral element (i.e., the identity element) of G as a group.

(2) implies (1)

Given: A set G, function f:G^n \to G making it a n-ary group, an element e that is neutral for f.

To prove: There exists a group structure on G with respect to which f(a_1,a_2,\dots,a_n) = a_1a_2 \dots a_n for all a_1,a_2,\dots,a_n \in G.

Proof: We use the notation e_i to denote e written in succession i times.

Step no. Assertion/construction Given data used Previous steps used Explanation
1 f(x,y,e_{n-2}) = f(e_i,x,e_j,y,e_k) for all x,y \in G and all i,j,k \ge 0 with i + j + k = n - 2. e is neutral for f, f satisfies associativity becaus it gives a n-ary group operation By asociativity, we have f(e_{k+1},f(e_i,x,e_j,y,e_k),e_{i+j}) = f(f(e_{k+1+i},x,e_j),y,e_{n-2}). Using that e is neutral, the outer f of the left side vanishes and the left side becomes f(e_i,x,e_j,y,e_k). On the right side, using that f is neutral, the inside f vanishes and we are left with f(x,y,e_{n-2}), completing the proof.
2 We can define an operation xy := f(x,y,e_{n-2}), and this equals f evaluated on any tuple with (n-2) es, one x and one y, with the x appearing to the left of the y. Step (1) Step-direct
3 The operation defined in Step (2) is associative, i.e., (ab)c = a(bc) for all a,b,c \in G f satisfies n-ary associativity Step (2) We write the operation as xy := f(x,e_{n-2},y). We thus get(ab)c = f(f(a,e_{n-2},b),e_{n-2},c) = f(a,e_{n-2},f(b,e_{n-2},c)) = a(bc)
4 The operation defined in Step (2) has identity element e e is neutral for f Step (2) e works as an identity element: for all a \in G, ae = f(a,e_{n-1}) = a. Similarly, ea = f(e_{n-1},a) = a.
5 The operation defined in Step (2) admits two-sided inverses Steps (2), (3), (4) We show one-sided inverses from the unique solution condition: the right inverse of a is the unique solution b to f(a,e,\dots,e,b) = e. Similarly, the left inverse of a is the unique c such that f(c,e,\dots,e,a) = a. Now, we use the equality of left and right inverses in monoid to conclude that two-sided inverses exist.
6 The operation defined in Step (2) gives a group structure on G Steps (2), (3), (4), (5) Step-combination direct
7 For any elements a_1,a_2,\dots,a_i \in G with i \le n, f(a_1,a_2,\dots,a_i,e_{n-i}) = a_1a_2\dots a_i. f is associative Steps (2), (6) We prove this by induction on i. Assuming true for i - 1, write f(f(a_1,a_2,\dots,a_i,e_{n-i}),e_{n-1}) = f(a_1,f(a_2,\dots,a_i,e_{n-i+1})e_{n-2}). The left side simplifies to the desired left side. On the right side, the second input simplifies to a_2\dots a_i by inductive hypothesis, so now using the definition of group product, the product simplifies to a_1a_2 \dots a_i as desired.
8 We are done Steps (2), (6), (7) By Steps (2) and (6), we have a group structure on G, which setting i = n in Step (7), gives the desired n-ary group, completing the proof.

Caution about non-uniqueness of neutral elements

For a reducible multiary group, there may be many different choices of neutral element. Each choice yields a potentially different binary operation, though the group structures we obtain are isomorphic. For instance, for the n-ary group obtained from the cyclic group of order n - 1, every element is a neutral element. The different group structures correspond to affine shifts in the original group, i.e., relabeling all elements by adding a group element to them.