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§5B1 Rationale. As mentioned on the main page, the definition of a loop under ternary operation O calls for an ordered pair of constants (p, q) satisfying all three of the following formulas for all x:

Instead of the q_p identity, some researchers might have preferred the p_q identity, namely O (p, x, q) = x. That is certainly a valid option.

A five-input operation H is helpful in clarifying the rationale for the choice of q_p in this report:

Under the straight loop criteria, input p might appear as the first or second operand, but never third, fourth, or fifth. Similarly, q is limited to the second and third operand positions, r to the third and fourth, and s to the fourth and fifth. By contrast, under the cyclic loop criteria, each input appears in each position exactly once. The present author preferred the symmetry of the cyclic arrangement, hence q_p. (Although cycle and loop happen to be near-synonyms, that vocabularial fact did not influence this decision.)


§5B2 Some loop patterns. The first table below lists all the possible combinations of 3SIs when C ≤ 3. The "nature" column indicates whether the two elements of each ordered pair are equal or not.

3SIs operationnature
(a, a) 1:0 =
(a, a) (b, b) 2:0 = =
(a, b) (b, a) 2:1 ≠ ≠
(a, a), (b, c), (c, b) 3:0 = ≠ ≠
(b, b), (a, c), (c, a) 3:15
(c, c), (a, b), (b, a) 3:16

The next table shows all the possibilities when C = 4; as many as 18 operations might share one combination.

3SIs quantity of
operations
examplenature
(a, a) 184:148 =
(b, b) 184:797
(c, c) 184:235
(d, d) 184:233
(a, a), (b, b) 144:1 = =
(a, a), (c, c) 144:18
(a, a), (d, d) 144:16
(b, b), (c, c) 144:44
(b, b), (d, d) 144:89
(c, c), (d, d) 14 4:8
(a, b), (b, a) 144:16353 ≠ ≠
(a, c), (c, a) 144:38458
(a, d), (d, a) 144:54671
(b, c), (c, b) 144:650
(b, d), (d, b) 144:515
(c, d), (d, c) 144:147
(a, b), (b, a), (c, d), (d, c) 74:16443 ≠ ≠ ≠ ≠
(a, c), (c, a), (b, d), (d, b) 74:25256
(a, d), (d, a), (b, c), (c, b) 74:22241
(a, a), (b, b), (c, d), (d, c) 14:139 = = ≠ ≠
(a, a), (c, c), (b, d), (d, b) 14:452
(a, a), (d, d), (b, c), (c, b) 14:626
(b, b), (c, c), (a, d), (d, a) 14:54669
(b, b), (d, d), (a, c), (c, a) 14:38456
(c, c), (d, d), (a, b), (b, a) 14:16352
(a, a), (b, b), (c, c), (d, d) 14:0 = = = =
total268