Definition of sets that are cyclic under ternary XOR

Let us define a set to be cyclic under ternary XOR when XOR'ing any three numbers from the set results in a new number that belongs to the set itself.

def ternary_xor(𝑎, 𝑏, 𝑐):
    return 𝑎 ^ 𝑏 ^ 𝑐

def is_cyclic_under_ternary_xor(𝛹):
    for 𝑎 in 𝛹:
        for 𝑏 in 𝛹:
            for 𝑐 in 𝛹:
                𝑥 = ternary_xor(𝑎, 𝑏, 𝑐)
                if 𝑥 not in 𝛹:
                    return False
    return True

If there exists a non-empty set 𝛹 that satisfies the is_cyclic_under_ternary_xor definition given above, what happens if we XOR more than three numbers from 𝛹?

Let us XOR 𝑎, 𝑏, 𝑐, 𝑑, 𝑒 from 𝛹. We do this by first XOR'ing 𝑎, 𝑏, 𝑐 to get 𝑥, and then XOR 𝑥 with the remaining 𝑑, 𝑒. We know by definition that 𝑎 ⊕ 𝑏 ⊕ 𝑐 = 𝑥 and that 𝑥 belongs to 𝛹. Since 𝑥, 𝑑, 𝑒 all belong to 𝛹, the result of 𝑥 ⊕ 𝑑 ⊕ 𝑒 also belongs to 𝛹. So it turns out that when 𝑎, 𝑏, 𝑐, 𝑑, 𝑒 all belong to 𝛹 then the result of 𝑎 ⊕ 𝑏 ⊕ 𝑐 ⊕ 𝑑 ⊕ 𝑒 belongs to 𝛹.

This process can be repeated for 𝑛 numbers from 𝛹 where 𝑛 ≥ 3 and 𝑛 is odd.

It follows by induction that: if 𝑥 is the result of XOR'ing 𝑛 numbers from 𝛹 where 𝑛 ≥ 3 and 𝑛 is odd, then 𝑥 itself is a member of 𝛹.

Visualization of a set that is cyclic under ternary XOR

The following tables illustrate all 16³ combinations of 𝑎 ⊕ 𝑏 ⊕ 𝑐 for 𝑎, 𝑏, 𝑐 in 𝛹' where 𝛹' is a set of 16 carefully selected numbers.

Each number in 𝛹' has been assigned a unique color. The set is also illustrated by a colored graph under the tables.