Some ternary quasigroups over small sets.
Page one contains sections 1 through 4. This is page two, with sections 5 through 10. Page three contains sections 11 and 12. Page four contains sections 13 through 16. 
§5A Identities. Many operations have identities. The notation employs an underscore, suggesting a blank spot where something could be filled in:
(A stricter definition calls for p to equal q, but that limitation is not pursued here.)
Cancellativity ensures that if O (p_{1}, q, x) = x and O (p_{2}, q, x) = x, then p_{1} = p_{2}.
Some operations have several identities, but others have none; a wealth of possibilities are illustrated in the first row of each entry throughout this chart. In general, the existence of one identity does not imply the existence of others.
TQG identities behave differently from the familiar identity elements of binary operations, because a TGQ's identity cannot stand as an input to an operation.
The classification of operations can be aided by inspection of identities, because if two operations have different quantities of identities, they cannot be isomorphic.
The three identities above can be described as onesided identities (1SIs) to distinguish them from what follows.
§5B Loops. The loop, which is normally associated with binary operations, can be extended to ternary operations. To set up an analogy, suppose a BQG has the operation F and an element p such that both of the following formulas are satisfied for all x:
Now suppose a TQG has the operation O and an ordered pair of elements (p, q) such that all three of the following formulas are satisfied for all x:
Consider an operation with 3SIs (p, q) and (q, p). These six equations must be satisfied:
If x = p is substituted, and duplicates are eliminated, consistent results are (fortunately) obtained:
Similarly when x = q:
An open question is whether the existence of (p, q) implies the existence of (q, p) for all C. The answer is yes when C ≤ 4.
§6A Skew elements. (see Dudek) For each TQG operation O, three skew operations can be defined. The notation for each skew is an element, a diamond ♦, and a number:
Example:
from 4:21255  

value  reason  value  reason  value  reason 
a♦1 = c  caa = a  a♦2 = b  aba = a  a♦3 = d  aad = a 
b♦1 = b  bbb = b  b♦2 = b  bbb = b  b♦3 = b  bbb = b 
c♦1 = b  bcc = c  c♦2 = a  cac = c  c♦3 = d  ccd = c 
d♦1 = c  cdd = d  d♦2 = a  dad = d  d♦3 = b  ddb = d 
Thanks to cancellativity, each of x♦1, x♦2, and x♦3 is guaranteed to exist and be unique. However, they might not be equal. For instance, the skews of 4:21401 have no conspicuous pattern; the skews of operation 4:43310 are as variegated as possible; and the skews of 3:8 are all equal.



A superscript notation is convenient for repeated application of skewing. For instance:
Under 4:21401, x♦1 is not an invertible operation, as b(♦^{−1})1 could equal b or c, and c(♦^{−1})1 does not equal anything. In contrast, x♦1 under 4:43310 is invertible, making x(♦^{−1})1 a valid expression.
Some operations are invertible in all of x♦1, x♦2, and x♦3. By quantity these are:
Skewness might distribute over an operation. Examples:
4:50  4:8796  4:8787  4:22802  4:10821  4:18919  4:13189  4:126  
(O (p, q, r))♦1 = O (p♦1, q♦1, r♦1)  true  true  true  true  false  false  false  false 
(O (p, q, r))♦2 = O (p♦2, q♦2, r♦2)  true  true  false  false  true  true  false  false 
(O (p, q, r))♦3 = O (p♦3, q♦3, r♦3)  true  false  true  false  true  false  true  false 
§6B. Call the skew operations defined so far singleskew. One can go further and establish doubleskew or tripleskew operations:
Depending on the choice of O and x, multipleskew operations might have single values, multiple values, or none at all. An operation that is singlevalued for all three doubleskew conditions is 4:29262, the pertinent results therefrom being:
from 4:29262  

value  reason  value  reason  value  reason 
a♦12 = b  bba = a  a♦13 = c  cac = a  a♦23 = d  add = a 
b♦12 = d  ddb = b  b♦13 = c  cbc = b  b♦23 = a  baa = b 
c♦12 = c  ccc = c  c♦13 = c  ccc = c  c♦23 = c  ccc = c 
d♦12 = d  aad = d  d♦13 = c  cdc = d  d♦23 = b  dbb = d 
When C = 3, there are 6 operations that are singlevalued under all three doubleskews, 3 of them invertible. These 6 operations are precisely the ones for which the tripleskew is not singlevalued.
When C = 4, there are 48 operations that are singlevalued under all three doubleskews, but none of them is invertible in all three. The tripleskew operation is not singlevalued for any of these 48 operations.
Because singleskewness is intimately connected with ordinary cancellativity, fullydoubleskew operations such as 4:29262 suggest that some sort of secondorder cancellativity might be defined.
Singlevalued for the tripleskew operation is 4:39155:
from 4:39155  

value  reason 
a♦123 = d  ddd = a 
b♦123 = c  ccc = b 
c♦123 = a  aaa = c 
d♦123 = b  bbb = d 
§6C. Of course, skew operations can be defined for operations with other than three inputs:
1 input  
x♦1 = p if Op = x  
2 inputs  
x♦1 = p if Opx = x
x♦2 = q if Oxq = x  x♦12 = p if Opp = x  
3 inputs — as above  
x♦1 = p if Opxx = x
x♦2 = q if Oxqx = x x♦3 = r if Oxxr = x  x♦12 = p if Oppx = x
x♦13 = q if Oqxq = x x♦23 = r if Oxrr = x  x♦123 = p if Oppp = x  
4 inputs  
x♦1 = p if Opxxx = x
x♦2 = q if Oxqxx = x x♦3 = r if Oxxrx = x x♦4 = s if Oxxxs = x  x♦12 = p if Oppxx = x
x♦13 = q if Oqxqx = x x♦14 = r if Orxxr = x x♦23 = s if Oxssx = x x♦24 = t if Oxtxt = x x♦34 = u if Oxxuu = x  x♦123 = p if Opppx = x
x♦124 = q if Oqqxq = x x♦134 = r if Orxrr = x x♦234 = s if Oxsss = x  x♦1234 = p if Opppp = x 
§7 Direct products. These are simple but lengthy to explain.
§8 Generators and subTQGs. In some cases, the set belonging to a TQG can be obtained by starting with one element as input, and repeatedly applying the operation using for inputs the original element and any derived outputs. This corresponds to the key feature of binary cyclic groups.
An example is 4:20867, where a is a generator because Oaaa = b, Oaab = c, and Oaac = d, covering all four elements. For this operation, b, c and d also happen to be generators.
With 4:34466 by contrast, a is not a generator. Consider Oaaa = c, Oaac = Oaca = Ocaa = a, Oacc = Ocac = Occa = c and Occc = a, in all of which b and d never appear. Still, a generates a ternary quasisubgroup (subTQG) of two elements; c would have generated the same subTQG, which is isomorphic to 2:1. By the same token, either of b or d generates the subTQG containing both b and d.
4:6509 has two generators, b and c. Meanwhile, each of a and d generates the subTQG containing only itself.
Although a binary cyclic group is surely commutative, a TQG generated by one element might not be. Consider 4:27496, which can be generated by any one of its elements. Explanation: Oaaa = b, Oaab = d and Oabb = c, so a is a generator. Now observe Obbb = a. Since b generates a generator, b is itself a generator. Similarly, c and d are generators because Occc = a and Oddd = b. Still, 4:27496 is noncommutative:
examples of noncommutativity of 4:27496  

sinisterior  exterior  dexterior  cyclical 
abc = a bac = d  dac = a cad = d  abc = a acb = b  bcd = a dbc = b cdb = d 
4:27496 also fails all three of the associativity criteria of section 2B.
§9A Decomposition. A ternary operation O is:
An easy way to find a decomposable O is to work backwards, first selecting F and G and then composing them. Here for instance are two binary operations over the set { a, b, c, d }:
F = 4::134  G = 4::415  

aa = a  ab = d  ac = c  ad = b  aa = c  ab = d  ac = b  ad = a  
ba = c  bb = b  bc = d  bd = a  ba = a  bb = c  bc = d  bd = b  
ca = b  cb = c  cc = a  cd = d  ca = d  cb = b  cc = a  cd = c  
da = d  db = a  dc = b  dd = c  da = b  db = a  dc = c  dd = d 
When combined as F (p, G (q, r)), 4:34618 is obtained for O (p, q, r). Hence 4:34618 exhibits dexterior decomposability.
§9B. Exemplifying indecomposable operations is 4:21401. The following demonstration first assumes that a dexterior decomposition exists, and then derives a contradiction. Expand seven entries from the operation definition table:
To reduce bulk, write Gaa for G (a, a), Gab for G (a, b) et cetera:
Since F (a, Gaa) does not equal F (a, Gab), Gaa cannot equal Gab. Further, Gaa, Gab, Gac and Gad are pairwise unequal. In contrast, Gcc must equal one of Gaa, Gab, Gac and Gad because there are only C = 4 values to pick from.
Because F (a, Gcc) = b, Gcc cannot be Gab, Gac or Gad, so by elimination Gcc equals Gaa. Now the last item in the list becomes F (b, Gaa) = c, which contradicts the nexttolast, F (b, Gaa) = d. Hence 4:21401 has no dexterior decomposition.
§9C. When C = 4, there are 576 binary quasigroups, and they can be sinisteriorly composed 331,776 = 576 × 576 ways; the dexterior number is of course the same. Of the 55,296 ternary quasigroups:
When C = 3, all ternary quasigroups decompose both ways. There are 12 binary quasigroups, and they can be sinisteriorly composed 144 = 12 × 12 ways. Each of the 24 ternary quasigroups can be sinisteriorly decomposed 6 ways. The same applies to dexteriority.
When C = 2, all ternary quasigroups decompose both ways. There are 2 binary quasigroups, and they can be sinisteriorly composed 4 = 2 × 2 ways. Each of the 2 ternary quasigroups can be sinisteriorly decomposed 2 ways. The same applies to dexteriority.
A general method for detecting decomposability is given on another page, although the computer program used to calculate the figures above simply generated all possible compositions and classified them.
§9D. More broadly, composition is simply a matter of using the output from one operation as an input to another operation — or indeed, as an input to another invocation of the same operation. If H is an nary cancellative operation over set S, and I is an mary cancellative operation over S, then a composition of H and I will be a (n+m−1)ary cancellative operation over S. In short, quasigroups are composable.
§10 Orthogonality. Consider ternary operations O, N, and M, which will accept elements p, q, r, s, t, and u as inputs.
If the ordered triples of outputs (Opqr, Npqr, Mpqr) and (Ostu, Nstu, Mstu) are unequal whenever the ordered triples of inputs (p, q, r) and (s, t, u) are unequal, then the three operations O, N, and M are said to be orthogonal.
With ordered triples, it is important to remember that a statement like (p, q, r) ≠ (s, t, u) means merely that at least one of the following statments is true: p ≠ s, q ≠ t, or r ≠ u. Not required is that all corresponding elements be unequal.
Combining the Cayley tables of two or more orthogonal ternary operations gives the threedimensional version of a GraecoLatin square, here for instance with 4:116, 4:1300 and 4:1881. Example:
4:116, 4:1300, 4:1881  skews  

aaa = a, a, a  aab = b, b, b  aac = c, c, c  aad = d, d, d  aba = b, c, d  abb = a, d, c  abc = d, a, b  abd = c, b, a  aca = c, d, b  acb = d, c, a  acc = a, b, d  acd = b, a, c  ada = d, b, c  adb = c, a, d  adc = b, d, a  add = a, c, b  a♦1 = a, a, a  a♦2 = a, a, a  a♦3 = a, a, a  
baa = d, c, b  bab = c, d, a  bac = b, a, d  bad = a, b, c  bba = c, a, c  bbb = d, b, d  bbc = a, c, a  bbd = b, d, b  bca = b, b, a  bcb = a, a, b  bcc = d, d, c  bcd = c, c, d  bda = a, d, d  bdb = b, c, c  bdc = c, b, b  bdd = d, a, a  b♦1 = c, b, d  b♦2 = d, b, c  b♦3 = d, b, d  
caa = b, d, c  cab = a, c, d  cac = d, b, a  cad = c, a, b  cba = a, b, b  cbb = b, a, a  cbc = c, d, d  cbd = d, c, c  cca = d, a, d  ccb = c, b, c  ccc = b, c, b  ccd = a, d, a  cda = c, c, a  cdb = d, d, b  cdc = a, a, c  cdd = b, b, d  c♦1 = d, c, b  c♦2 = b, c, d  c♦3 = b, c, b  
daa = c, b, d  dab = d, a, c  dac = a, d, b  dad = b, c, a  dba = d, d, a  dbb = c, c, b  dbc = b, b, c  dbd = a, a, d  dca = a, c, c  dcb = b, d, d  dcc = c, a, a  dcd = d, b, b  dda = b, a, b  ddb = a, b, a  ddc = d, c, d  ddd = c, d, c  d♦1 = b, d, c  d♦2 = c, d, b  d♦3 = c, d, c 
Each output combination (a, a, a), (a, a, b), (a, a, c), and so forth appears precisely once. On the other hand, the skew elements do not exhibit any obvious pattern.
Naturally, orthogonality also applies to binary operations. Example:
4::348, 4::450  skews  

aa = c, d  ab = b, a  ac = a, b  ad = d, c  a♦1 = d, d  a♦2 = c, b  
ba = b, c  bb = c, b  bc = d, a  bd = a, d  b♦1 = a, b  b♦2 = a, b  
ca = d, b  cb = a, c  cc = b, d  cd = c, a  c♦1 = d, d  c♦2 = d, b  
da = a, a  db = d, d  dc = c, c  dd = b, b  d♦1 = a, b  d♦2 = b, b 
Page one contains sections 1 through 4.
This is page two, with sections 5 through 10. Page three contains sections 11 and 12. Page four contains sections 13 through 16. 