Some ternary quasigroups over small sets.   Page one contains sections 1 through 4. Page two contains sections 5 through 10. This is page three, with sections 11 and 12. Page four contains sections 13 through 16.

§11 Operation numbers. To each ternary operation has been assigned a reference number in the form C:n, while binary operations have two colons. As always, C stands for the cardinality. The specification of n requires a little more work, but two things can be said now:

• n is a nonnegative integer
• numbers are never skipped; in other words, whenever C:(n + 1) denotes an operation, so does C:n
Because the TQGs studied here are over small sets, only finitely many operations are possible, and large values of n will not be needed. Some maxima are:

 C maximum n 1 2 3 4 5 0 1 23 55,295 2,781,803,519

Step 1. Create any aribitrary ordering for the TQG's set members; alphabetical is as good as any other, so by definition a < b, b < c et cetera.

Step 2. Observe that the inputs to an invocation of an operation form an ordered triplet. For instance, in the invocation O (p, q, r), the triplet is (p, q, r). Now create a lexicographic ordering on those triplets. Given triplets T1 = (p1, q1, r1) and T2 = (p2, q2, r2), specify:

• if p1 < p2, then T1 < T2
• if p1 > p2, then T1 > T2
• if p1 = p2:
• if q1 < q2, then T1 < T2
• if q1 > q2, then T1 > T2
• if q1 = q2:
• if r1 < r2, then T1 < T2
• if r1 > r2, then T1 > T2
• if r1 = r2, then T1 = T2
For example, (b, a, e) < (b, c, d) because b = b and then a < c. In this case, the e and d make no difference. As before, parentheses and commas can be omitted when no confusion will result, resulting in the briefer notation bae < bcd.

Now an ordering of input triplets can be constructed. As a case in point, here are all 27 possible input triplets for C = 3, in lexicographic order from least (left) to greatest (right):

Step 3. Consider two ternary operations, C:k and C:l over the same set, and ordered triplets T1 and T2. If these two conditions are satisfied:

• C:k (T2) < C:l (T2)
• C:k (T1) = C:l (T1) for every T1 that is less than T2
then require the number k to be less than l. With the nonskipping rule from above, this is sufficient to define n for each operation, and it generates a lexicographic ordering in its own right.

Remark. This numbering scheme arose naturally from the author's C++ program, which used a simple recursive scheme to find all the operations for each cardinality.

In an extended example, each row in the table below is one of the 24 operations of C = 3, while each column represents an input triplet. The table defines the same operations as found in another chart, but in a different format.

 aaa aab aac aba abb abc aca acb acc baa bab bac bba bbb bbc bca bcb bcc caa cab cac cba cbb cbc cca ccb a b c b c a c a b b c a c a b a b c c a b a b c b c a a b c b c a c a b c a b a b c b c a b c a c a b a b c a b c c a b b c a b c a a b c c a b c a b b c a a b c a b c c a b b c a c a b b c a a b c b c a a b c c a b a c b b a c c b a b a c c b a a c b c b a a c b b a c a c b b a c c b a c b a a c b b a c b a c c b a a c b a c b c b a b a c b a c a c b c b a c b a b a c a c b a c b c b a b a c c b a b a c a c b b a c a c b c b a b a c a c b c b a a c b c b a b a c c b a b a c a c b b a c a c b c b a c b a b a c a c b a c b c b a b a c b a c c b a a c b a c b b a c c b a c b a a c b b a c b a c c b a a c b c b a a c b b a c a c b b a c c b a b c a a b c c a b a b c c a b b c a c a b b c a a b c b c a a b c c a b c a b b c a a b c a b c c a b b c a b c a c a b a b c a b c b c a c a b c a b a b c b c a b c a c a b a b c c a b a b c b c a a b c b c a c a b c a b a b c b c a a b c b c a c a b b c a c a b a b c c a b a b c b c a b c a c a b a b c a b c b c a c a b c a b b c a a b c a b c c a b b c a b c a a b c c a b c a b b c a a b c b c a a b c c a b a b c c a b b c a c b a a c b b a c a c b b a c c b a b a c c b a a c b c b a a c b b a c b a c c b a a c b a c b b a c c b a c b a b a c a c b a c b c b a b a c b a c a c b c b a c b a b a c a c b b a c a c b c b a a c b c b a b a c

The numbering starts with zero (as 3:0) rather than one (as 3:1). Although many researchers begin numbering at one, the present author used a computer program in the C++ language to perform almost all of the calculations, and in that language, array indexing begins with zero, not one. For convenience, zero-based numbering was retained for nearly all purposes.

§12A Conjugates. This is a method of converting one TQG into another. Although the two TQGs (called conjugates) will have the same set, their operations will differ. The device required for conjugation is a unary operation that permutes a TQG's set; such a permutator can be characterized as a bijection from the set to itself. A popular symbol for a permutator is π. (Incidentally, a set and a permutator are the ingredients of a unary quasigroup.)

To conjugate a TQG, the procedure is simply to apply the permutator to every input and output in the table that defines the TQG's operation. Conjugacy turns out to be an equivalence relation.

Here is a full example of conjugation, with an identifying number in each cell of each table to show how the values move around. Start with 4:15:

 4:15 #0aaa = a #1aab = b #2aac = c #3aad = d #4aba = b #5abb = a #6abc = d #7abd = c #8aca = c #9acb = d #10acc = a #11acd = b #12ada = d #13adb = c #14adc = b #15add = a #16baa = b #17bab = a #18bac = d #19bad = c #20bba = a #21bbb = b #22bbc = c #23bbd = d #24bca = d #25bcb = c #26bcc = b #27bcd = a #28bda = c #29bdb = d #30bdc = a #31bdd = b #32caa = d #33cab = c #34cac = b #35cad = a #36cba = c #37cbb = d #38cbc = a #39cbd = b #40cca = b #41ccb = a #42ccc = d #43ccd = c #44cda = a #45cdb = b #46cdc = c #47cdd = d #48daa = c #49dab = d #50dac = a #51dad = b #52dba = d #53dbb = c #54dbc = b #55dbd = a #56dca = a #57dcb = b #58dcc = c #59dcd = d #60dda = b #61ddb = a #62ddc = d #63ddd = c

Permute a, b, c and d in all the inputs and outputs according to this rule:

• π (a) = c
• π (b) = b
• π (c) = a
• π (d) = d
obtaining the next table:

 #0ccc = c #1ccb = b #2cca = a #3ccd = d #4cbc = b #5cbb = c #6cba = d #7cbd = a #8cac = a #9cab = d #10caa = c #11cad = b #12cdc = d #13cdb = a #14cda = b #15cdd = c #16bcc = b #17bcb = c #18bca = d #19bcd = a #20bbc = c #21bbb = b #22bba = a #23bbd = d #24bac = d #25bab = a #26baa = b #27bad = c #28bdc = a #29bdb = d #30bda = c #31bdd = b #32acc = d #33acb = a #34aca = b #35acd = c #36abc = a #37abb = d #38aba = c #39abd = b #40aac = b #41aab = c #42aaa = d #43aad = a #44adc = c #45adb = b #46ada = a #47add = d #48dcc = a #49dcb = d #50dca = c #51dcd = b #52dbc = d #53dbb = a #54dba = b #55dbd = c #56dac = c #57dab = b #58daa = a #59dad = d #60ddc = b #61ddb = c #62dda = d #63ddd = a

Now rearrange this table according to the usual pattern of inputs, yielding 4:55237:

 4:55237 #42aaa = d #41aab = c #40aac = b #43aad = a #38aba = c #37abb = d #36abc = a #39abd = b #34aca = b #33acb = a #32acc = d #35acd = c #46ada = a #45adb = b #44adc = c #47add = d #26baa = b #25bab = a #24bac = d #27bad = c #22bba = a #21bbb = b #20bbc = c #23bbd = d #18bca = d #17bcb = c #16bcc = b #19bcd = a #30bda = c #29bdb = d #28bdc = a #31bdd = b #10caa = c #9cab = d #8cac = a #11cad = b #6cba = d #5cbb = c #4cbc = b #7cbd = a #2cca = a #1ccb = b #0ccc = c #3ccd = d #14cda = b #13cdb = a #12cdc = d #15cdd = c #58daa = a #57dab = b #56dac = c #59dad = d #54dba = b #53dbb = a #52dbc = d #55dbd = c #50dca = c #49dcb = d #48dcc = a #51dcd = b #62dda = d #61ddb = c #60ddc = b #63ddd = a

A concise way to express the conjugacy of 4:15 and 4:55237 is:

π (4:15 (p, q, r)) = 4:55237 (π (p), π (q), π (r))

Because π is a bijection, its functional inverse exists, and can be denoted π−1. This allows one to also write:

4:15 (p, q, r) = π−1 (4:55237 (π (p), π (q), π (r)))
π (4:15 (π−1 (p), π−1 (q), π−1 (r))) = 4:55237 (p, q, r)

Conjugation preserves properties such as c'ty and a'ty. Further, if two TQG's are conjugated using the same π, d'ty of one over the other is preserved.

§12B. Because conjugation can be viewed as merely a change of symbols, two conjugate TQGs might be regarded as essentially the same. Still, some operations yield larger equivalence classes than others. The table below details the case where C = 3, with 24 operations and 6 permutations. The operations are shown in full elsewhere.

Conjugacy
for C = 3
even permutations odd permutations
π0 (a) = a
π0 (b) = b
π0 (c) = c
π2 (a) = c
π2 (b) = a
π2 (c) = b
π4 (a) = b
π4 (b) = c
π4 (c) = a
π1 (a) = a
π1 (b) = c
π1 (c) = b
π3 (a) = c
π3 (b) = b
π3 (c) = a
π5 (a) = b
π5 (b) = a
π5 (c) = c
A3:0 3:03:163:15 3:03:163:15
3:16 3:163:153:0 3:153:03:16
3:15 3:153:03:16 3:163:153:0
B3:3 3:33:123:19 3:33:123:19
3:12 3:123:193:3 3:193:33:12
3:19 3:193:33:12 3:123:193:3
C3:5 3:53:103:21 3:53:103:21
3:10 3:103:213:5 3:213:53:10
3:21 3:213:53:10 3:103:213:5
D3:6 3:63:93:22 3:63:93:22
3:9 3:93:223:6 3:223:63:9
3:22 3:223:63:9 3:93:223:6
E3:7 3:73:233:8 3:73:233:8
3:23 3:233:83:7 3:83:73:23
3:8 3:83:73:23 3:233:83:7
F3:11 3:113:113:11 3:203:203:20
3:20 3:203:203:20 3:113:113:11
G3:13 3:133:133:13 3:183:183:18
3:18 3:183:183:18 3:133:133:13
H3:14 3:143:143:14 3:173:173:17
3:17 3:173:173:17 3:143:143:14
I3:1 3:13:13:1 3:13:13:1
J3:2 3:23:23:2 3:23:23:2
K3:4 3:43:43:4 3:43:43:4

In the table:

• classes A-E contain 3 operations each;
• classes F-H contain 2 operations each;
• classes I-K contain 1 operation each.
This variation in equivalence class size demonstrates that conjugation is not a trivial operation. With larger C, matters become more complicated.

The table above incorporates both even and odd permutations. Possible instead is a definition of conjugacy which retains only the even permutations, omitting the odd. In that case, 3:11 and 3:20 for instance would no longer be conjugates. Especially when C ≥ 5, the symmetric group (that is, the group of all permutations) has many interesting subgroups. In short, a researcher need not consider all possible permutations when exploring conjugacy.

Here is another example. When C = 4, there are 28 fully associative TQGs, and they form seven equivalence classes under a conjugacy that uses all permutations:

 A: B: C: D: E: F: G: 4:0 4:220 ∼ 4:784 ∼ 4:1632 4:16271 ∼ 4:38298 ∼ 4:54511 4:16443 ∼ 4:38852 ∼ 4:55295 4:139 ∼ 4:452 ∼ 4:626 ∼ 4:16352 ∼ 4:38456 ∼ 4:54669 4:303 ∼ 4:942 ∼ 4:2187 ∼ 4:16140 ∼ 4:37907 ∼ 4:54353 4:22241 ∼ 4:25256 ∼ 4:30039 ∼ 4:41470 ∼ 4:41474 ∼ 4:52629

§12C Isotopes. More general than conjugation is isotopy, which can use a different permutation for each input, and yet another for the output:

π0 (M (p, q, r)) = N1 (p), π2 (q), π3 (r))

The 24 operations of the table above coalesce into 4 equivalence classes under isotopy:

A-ED-H-IC-G-JB-F-K
3:0
3:7
3:8
3:15
3:16
3:23
3:1
3:6
3:9
3:14
3:17
3:22
3:2
3:5
3:10
3:13
3:18
3:21
3:3
3:4
3:11
3:12
3:19
3:20

Under istopy, an operation's properties can change substantially, for instance:

• self-d'ty within class D-H-I:
• 3:1 — full
• 3:14 — interior and dexterior
• 3:6 — none
• a'ty within class C-G-J:
• 3:2 — full
• 3:13 — exterior
• 3:5 — none
• invertibility of single skews within class B-F-K:
• 3:11 — all three
• 3:12 — only two

The variation means that, for many purposes, two isotopes ought not be considered the "same as" each other.

As another example, here is a specific isotopy:

 π0 (4:37725 (p, q, r)) = 4:10248 (π1 (p), π2 (q), π3 (r)) where: π0 (a) = b π0 (b) = c π0 (c) = a π0 (d) = d π1 (a) = c π1 (b) = a π1 (c) = d π1 (d) = b π2 (a) = d π2 (b) = a π2 (c) = c π2 (d) = b π3 (a) = d π3 (b) = a π3 (c) = b π3 (d) = c

If two operations are related by isotopy, there is often a huge number of possible combinations for the set of connecting permutations { π0, π1, π2, π3 }. In the case of 4:37725 and 4:10248, that number is 13,824 = 55,296 ÷ 4.

§12D. Let S be a set of n-ary quasigroups over the elements {a, b, c …} as frequently encountered in this report. Then set S has at least one subset T such that:

• every quasigroup in S has an isotope in T;
• no two quasigroups in T are isotopes of each other.
Set T is termed a distillate of S. Because isotopy is an equivalence relation, all distillates of S must be of the same size.

If S has more than one distillate, one of them can be selected as the lexicographically minimal distillate ("LMD") with regard to operation numbers (section 11 above). Here is what that means: Suppose operations n:i and n:j are isotopes of each other. If subset T is the LMD of S, and if i < j, then n:j must not be a member of T.

n-arycard.
C
number of
operations
LMD
binary 1 1{1::0}
2 2{2::0}
3 12{3::0 … 3::1}
4 576{4::0 … 4::23}
5161,280{5::0 … 5::1343}
ternary 1 1{1:0}
2 2{2:0}
3 24{3:0 … 3:3}
4 55,296{4:0 … 4:2304}

It is surprising and convenient that the LMDs of the quasigroups studied here happen to coincide with the lowest-numbered operations. Not yet established, however, is whether this observation applies to all values of n or C.

 Page one contains sections 1 through 4. Page two contains sections 5 through 10. This is page three, with sections 11 and 12. Page four contains sections 13 through 16.