§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 | operation | nature |
---|---|---|
(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 | example | nature |
---|---|---|---|
(a, a) | 18 | 4:148 | = |
(b, b) | 18 | 4:797 | |
(c, c) | 18 | 4:235 | |
(d, d) | 18 | 4:233 | |
(a, a), (b, b) | 14 | 4:1 | = = |
(a, a), (c, c) | 14 | 4:18 | |
(a, a), (d, d) | 14 | 4:16 | |
(b, b), (c, c) | 14 | 4:44 | |
(b, b), (d, d) | 14 | 4:89 | |
(c, c), (d, d) | 14 | 4:8 | |
(a, b), (b, a) | 14 | 4:16353 | ≠ ≠ |
(a, c), (c, a) | 14 | 4:38458 | |
(a, d), (d, a) | 14 | 4:54671 | |
(b, c), (c, b) | 14 | 4:650 | |
(b, d), (d, b) | 14 | 4:515 | |
(c, d), (d, c) | 14 | 4:147 | |
(a, b), (b, a), (c, d), (d, c) | 7 | 4:16443 | ≠ ≠ ≠ ≠ |
(a, c), (c, a), (b, d), (d, b) | 7 | 4:25256 | |
(a, d), (d, a), (b, c), (c, b) | 7 | 4:22241 | |
(a, a), (b, b), (c, d), (d, c) | 1 | 4:139 | = = ≠ ≠ |
(a, a), (c, c), (b, d), (d, b) | 1 | 4:452 | |
(a, a), (d, d), (b, c), (c, b) | 1 | 4:626 | |
(b, b), (c, c), (a, d), (d, a) | 1 | 4:54669 | |
(b, b), (d, d), (a, c), (c, a) | 1 | 4:38456 | |
(c, c), (d, d), (a, b), (b, a) | 1 | 4:16352 | |
(a, a), (b, b), (c, c), (d, d) | 1 | 4:0 | = = = = |
total | 268 |