Although not a TQG, the simplest and most familiar nondecomposable ternary function is the majority operation M over values a and b, defined by:

1. M (a, a, a) = a
2. M (a, a, b) = a
3. M (a, b, a) = a
4. M (a, b, b) = b
5. M (b, a, a) = a
6. M (b, a, b) = b
7. M (b, b, a) = b
8. M (b, b, b) = b

Suppose that there were a decomposition M (p, q, r) = F (p, G (q, r)). Then we would have:

1. F (a, G (a, a)) = a
2. F (a, G (a, b)) = a
3. F (a, G (b, a)) = a
4. F (a, G (b, b)) = b
5. F (b, G (a, a)) = a
6. F (b, G (a, b)) = b
7. F (b, G (b, a)) = b
8. F (b, G (b, b)) = b

Observe:

• From 5 and 6, G (a, a) ≠ G (a, b).
• From 2 and 4, G (a, b) ≠ G (b, b).
• From 1 and 4, G (b, b) ≠ G (a, a).