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 15. Page five contains sections 16 through 18.

§5A Identities. Many operations have identities. The notation employs an underscore, suggesting a blank spot where something could be filled in:

• sinisterior identity: an ordered pair of values p and q such that O (p, q, x) = x for all x, notated pq_.
• exterior identity: an ordered pair of values p and q such that O (q, x, p) = x for all x, notated q_p.
• dexterior identity: an ordered pair of values p and q such that O (x, p, q) = x for all x, notated _pq.

(A stricter definition calls for p to equal q, but that limitation is not pursued here.)

Cancellativity ensures that if O (p₁, q, x) = x and O (p₂, q, x) = x, then p₁ = p₂.

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 one-sided 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:

• the p_ identity: F (p, x) = x
• the _p identity: F (x, p) = 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:

• the pq_ identity: O (p, q, x) = x
• the q_p identity: O (q, x, p) = x
• the _pq identity: O (x, p, q) = x

• 4:16443 has four 3SIs: (a, b), (b, a), (c, d), (d, c).
• 4:607 has the single-element 3SI (a, a) as well as some unrelated 1SIs.
• Although 4:9546 has 1SIs, it does not have any of the right combinations to form a 3SI.
• More examples.

Consider an operation with 3SIs (p, q) and (q, p). These six equations must be satisfied:

• O (p, q, x) = x
• O (q, x, p) = x
• O (x, p, q) = x
• O (q, p, x) = x
• O (p, x, q) = x
• O (x, q, p) = x

If x = p is substituted, and duplicates are eliminated, consistent results are (fortunately) obtained:

• O (p, p, q) = p
• O (p, q, p) = p
• O (q, p, p) = p

Similarly when x = q:

• O (q, q, p) = q
• O (q, p, q) = q
• O (p, q, q) = 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:

• x♦1 = p if Opxx = x
• x♦2 = q if Oxqx = x
• x♦3 = r if Oxxr = x

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, 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.

4:21401 skews a♦1 = d a♦2 = b a♦3 = d b♦1 = b b♦2 = b b♦3 = b c♦1 = b c♦2 = d c♦3 = b d♦1 = a d♦2 = c d♦3 = a

4:43310 skews a♦1 = c a♦2 = d a♦3 = b b♦1 = a b♦2 = c b♦3 = d c♦1 = d c♦2 = b c♦3 = a d♦1 = b d♦2 = a d♦3 = c

3:8 skews a♦1 = b a♦2 = b a♦3 = b b♦1 = b b♦2 = b b♦3 = b c♦1 = b c♦2 = b c♦3 = b

A superscript notation is convenient for repeated application of skewing. For instance:

• Under 4:21401, (a♦1)♦1 is more compactly written a♦²1 and equals d♦1 = a.
• Under 4:43310, x♦⁴3 equals x for any x. Equivalent is to write x(♦⁴)(3), the parentheses becoming helpful when complicated subexpressions are incorporated: h₆(♦^{max (n, m)})(1 + rem (i, j)).
• Under 3:8, a♦ⁿ2 = a only when n = 0.

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, x♦3. By quantity these are:

• C = 1: 1 of 1
• C = 2: 2 of 2
• C = 3: 12 of 24
• C = 4: 3168 of 55,296

An operation might be distributive over skewness. 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 single-skew. One can go further and establish double-skew or triple-skew operations:

• double:
  • x♦12 = p if Oppx = x
  • x♦13 = q if Oqxq = x
  • x♦23 = r if Oxrr = x
• triple:
  • x♦123 = s if Osss = x

Depending on the choice of O and x, multiple-skew operations might have single values, multiple values, or none at all. An operation that is single-valued for all three double-skew 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 single-valued under all three double-skews, 3 of them invertible. These 6 operations are precisely the ones for which the triple-skew is not single-valued.

When C = 4, there are 48 operations that are single-valued under all three double-skews, but none of them is invertible in all three. The triple-skew operation is not single-valued for any of these 48 operations.

Because single-skewness is intimately connected with ordinary cancellativity, fully-double-skew operations such as 4:29262 suggest that some sort of second-order cancellativity might be defined.

Single-valued for the triple-skew 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, 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 quasi-sub-group (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 non-commutative:

examples of non-commutativity 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:

• sinisteriorly decomposable if there exist binary operations F_s and G_s such that O (p, q, r) = F_s (G_s (p, q), r).
• dexteriorly decomposable if there exist binary operations F_d and G_d such that O (p, q, r) = F_d (p, G_d (q, r)).

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:

• aaa = O (a, a, a) = F (a, G (a, a)) = b
• aab = O (a, a, b) = F (a, G (a, b)) = c
• aac = O (a, a, c) = F (a, G (a, c)) = d
• aad = O (a, a, d) = F (a, G (a, d)) = a
• acc = O (a, c, c) = F (a, G (c, c)) = b
• baa = O (b, a, a) = F (b, G (a, a)) = d
• bcc = O (b, c, c) = F (b, G (c, c)) = c

To reduce bulk, write Gaa for G (a, a), Gab for G (a, b) et cetera:

• F (a, Gaa) = b
• F (a, Gab) = c
• F (a, Gac) = d
• F (a, Gad) = a
• F (a, Gcc) = b
• F (b, Gaa) = d
• F (b, Gcc) = c

Since F (a, Gaa) does not equal F (a, Gab), Gaa cannot equal Gab. Further, Gaa, Gab, Gac, Gad are pairwise unequal. In contrast, Gcc must equal one of Gaa, Gab, Gac, 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 next-to-last, 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:

• 3,456 have 24 sinisterior and 24 dexterior decompositions;
• 10,368 have 24 sinisterior;
• 10,368 have 24 dexterior;
• 31,104 have none.

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 n-ary cancellative operation over set S, and I is an m-ary 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, M, which will accept elements p, q, r, s, t, 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, 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 three-dimensional version of a Graeco-Latin 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 15. Page five contains sections 16 through 18.