The tables below show all 28 of the fully associative variegated ternary operations over four values:

• Operations 1-16 are fully commutative, and can be defined with 20 entries. Operation 1 is mulpos.
• Operations 17-28, which exhibit alternate but not cyclical commutativity, require 40 entries.

To save space, we write abc = d instead of operation_name (a, b, c) = d. Using this notation with atoms p, q, r, s and t:

• Full associativity means pq(rst) = p(qrs)t = (pqr)st.
• Alternate commutativity means pqr = rqp.
• Cyclical commutativity means pqr = qrp = rpq.

The product of two atoms is sometimes an identity, as listed in the rightmost column of the tables. With atoms p, q and r:

• pq is a left identity if pqr always equals r.
• pq is a right identity if rpq always equals r.

It turns out that for the 28 operations in the tables:

• The left and right identities are the same.
• When pq is an identity, so is qp, even when the operation is not fully commutative.

For operations that are not fully commutative, it is reasonable to say that pq is a central identity if prq always equals r. However, none of the operations in the second table (operations 17-28) has a central identity.

	Values for the operations that are fully commutative				Identities
1	aaa = a aab = b aac = c aad = d abb = a abc = d abd = c acc = a acd = b add = a	bbb = b bbc = c bbd = d bcc = b bcd = a bdd = b	ccc = c ccd = d cdd = c	ddd = d	aa, bb, cc, dd
2	aaa = b aab = a aac = d aad = c abb = b abc = c abd = d acc = a acd = b add = a	bbb = a bbc = d bbd = c bcc = b bcd = a bdd = b	ccc = c ccd = d cdd = c	ddd = d	ab, cc, dd
3	aaa = c aab = d aac = a aad = b abb = a abc = b abd = c acc = c acd = d add = a	bbb = b bbc = c bbd = d bcc = d bcd = a bdd = b	ccc = a ccd = b cdd = c	ddd = d	ac, bb, dd
4	aaa = d aab = c aac = b aad = a abb = a abc = d abd = b acc = a acd = c add = d	bbb = b bbc = c bbd = d bcc = b bcd = a bdd = c	ccc = c ccd = d cdd = b	ddd = a	ad, bb, cc
5	aaa = a aab = b aac = c aad = d abb = d abc = a abd = c acc = d acd = b add = a	bbb = c bbc = b bbd = a bcc = c bcd = d bdd = b	ccc = b ccd = a cdd = c	ddd = d	aa, bc, dd
6	aaa = a aab = b aac = c aad = d abb = c abc = d abd = a acc = a acd = b add = c	bbb = d bbc = a bbd = b bcc = b bcd = c bdd = d	ccc = c ccd = d cdd = a	ddd = b	aa, bd, cc
7	aaa = a aab = b aac = c aad = d abb = a abc = d abd = c acc = b acd = a add = b	bbb = b bbc = c bbd = d bcc = a bcd = b bdd = a	ccc = d ccd = c cdd = d	ddd = c	aa, bb, cd
8	aaa = b aab = a aac = d aad = c abb = b abc = c abd = d acc = b acd = a add = b	bbb = a bbc = d bbd = c bcc = a bcd = b bdd = a	ccc = d ccd = c cdd = d	ddd = c	ab, cd
9	aaa = c aab = a aac = d aad = b abb = b abc = c abd = d acc = b acd = a add = c	bbb = d bbc = a bbd = c bcc = d bcd = b bdd = a	ccc = a ccd = c cdd = d	ddd = b	ab, cd
10	aaa = d aab = a aac = b aad = c abb = b abc = c abd = d acc = d acd = a add = b	bbb = c bbc = d bbd = a bcc = a bcd = b bdd = c	ccc = b ccd = c cdd = d	ddd = a	ab, cd
11	aaa = b aab = d aac = a aad = c abb = c abc = b abd = a acc = c acd = d add = b	bbb = a bbc = d bbd = b bcc = a bcd = c bdd = d	ccc = d ccd = b cdd = a	ddd = c	ac, bd
12	aaa = c aab = d aac = a aad = b abb = c abc = b abd = a acc = c acd = d add = c	bbb = d bbc = a bbd = b bcc = d bcd = c bdd = d	ccc = a ccd = b cdd = a	ddd = b	ac, bd
13	aaa = d aab = c aac = a aad = b abb = d abc = b abd = a acc = c acd = d add = c	bbb = c bbc = a bbd = b bcc = d bcd = c bdd = d	ccc = b ccd = a cdd = b	ddd = a	ac, bd
14	aaa = b aab = c aac = d aad = a abb = d abc = a abd = b acc = b acd = c add = d	bbb = a bbc = b bbd = c bcc = c bcd = d bdd = a	ccc = d ccd = a cdd = b	ddd = c	ad, bc
15	aaa = c aab = d aac = b aad = a abb = c abc = a abd = b acc = d acd = c add = d	bbb = d bbc = b bbd = a bcc = c bcd = d bdd = c	ccc = a ccd = b cdd = a	ddd = b	ad, bc
16	aaa = d aab = c aac = b aad = a abb = d abc = a abd = b acc = d acd = c add = d	bbb = c bbc = b bbd = a bcc = c bcd = d bdd = c	ccc = b ccd = a cdd = b	ddd = a	ad, bc

	Values for the operations that are not cyclically commutative				Identities
17	aaa = a aab = b aac = c aad = d aba = b abb = a abc = d abd = c aca = d acb = c acc = a acd = b ada = c adb = d adc = b add = a	bab = a bac = d bad = c bbb = b bbc = c bbd = d bcb = d bcc = b bcd = a bdb = c bdc = a bdd = b	cac = b cad = a cbc = a cbd = b ccc = c ccd = d cdc = d cdd = c	dad = b dbd = a dcd = c ddd = d	aa, bb, cc, dd
18	aaa = a aab = b aac = c aad = d aba = c abb = a abc = d abd = b aca = b acb = d acc = a acd = c ada = d adb = c adc = b add = a	bab = d bac = a bad = c bbb = b bbc = c bbd = d bcb = c bcc = b bcd = a bdb = a bdc = d bdd = b	cac = d cad = b cbc = b cbd = a ccc = c ccd = d cdc = a cdd = c	dad = a dbd = c dcd = b ddd = d	aa, bb, cc, dd
19	aaa = a aab = b aac = c aad = d aba = d abb = a abc = b abd = c aca = c acb = d acc = a acd = b ada = b adb = c adc = d add = a	bab = c bac = d bad = a bbb = b bbc = c bbd = d bcb = a bcc = b bcd = c bdb = d bdc = a bdd = b	cac = a cad = b cbc = d cbd = a ccc = c ccd = d cdc = b cdd = c	dad = c dbd = b dcd = a ddd = d	aa, bb, cc, dd
20	aaa = a aab = b aac = c aad = d aba = b abb = a abc = d abd = c aca = d acb = c acc = b acd = a ada = c adb = d adc = a add = b	bab = a bac = d bad = c bbb = b bbc = c bbd = d bcb = d bcc = a bcd = b bdb = c bdc = b bdd = a	cac = a cad = b cbc = b cbd = a ccc = d ccd = c cdc = c cdd = d	dad = a dbd = b dcd = d ddd = c	aa, bb, cd, dc
21	aaa = a aab = b aac = c aad = d aba = d abb = c abc = b abd = a aca = c acb = d acc = a acd = b ada = b adb = a adc = d add = c	bab = a bac = d bad = c bbb = d bbc = a bbd = b bcb = c bcc = b bcd = a bdb = b bdc = c bdd = d	cac = a cad = b cbc = d cbd = c ccc = c ccd = d cdc = b cdd = a	dad = a dbd = d dcd = c ddd = b	aa, cc, bd, db
22	aaa = a aab = b aac = c aad = d aba = c abb = d abc = a abd = b aca = b acb = a acc = d acd = c ada = d adb = c adc = b add = a	bab = a bac = d bad = c bbb = c bbc = b bbd = a bcb = b bcc = c bcd = d bdb = d bdc = a bdd = b	cac = a cad = b cbc = c cbd = d ccc = b ccd = a cdc = d cdd = c	dad = a dbd = c dcd = b ddd = d	aa, dd, bc, cb
23	aaa = d aab = c aac = b aad = a aba = b abb = a abc = d abd = c aca = c acb = d acc = a acd = b ada = a adb = b adc = c add = d	bab = d bac = a bad = b bbb = b bbc = c bbd = d bcb = c bcc = b bcd = a bdb = a bdc = d bdd = c	cac = d cad = c cbc = b cbd = a ccc = c ccd = d cdc = a cdd = b	dad = d dbd = b dcd = c ddd = a	bb, cc, ad, da
24	aaa = c aab = d aac = a aad = b aba = b abb = a abc = d abd = c aca = a acb = b acc = c acd = d ada = d adb = c adc = b add = a	bab = c bac = b bad = a bbb = b bbc = c bbd = d bcb = a bcc = d bcd = c bdb = d bdc = a bdd = b	cac = c cad = d cbc = b cbd = a ccc = a ccd = b cdc = d cdd = c	dad = c dbd = b dcd = a ddd = d	bb, dd, ac, ca
25	aaa = b aab = a aac = d aad = c aba = a abb = b abc = c abd = d aca = c acb = d acc = a acd = b ada = d adb = c adc = b add = a	bab = b bac = c bad = d bbb = a bbc = d bbd = c bcb = c bcc = b bcd = a bdb = d bdc = a bdd = b	cac = b cad = a cbc = a cbd = b ccc = c ccd = d cdc = d cdd = c	dad = b dbd = a dcd = c ddd = d	cc, dd, ab, ba
26	aaa = b aab = a aac = d aad = c aba = a abb = b abc = c abd = d aca = c acb = d acc = b acd = a ada = d adb = c adc = a add = b	bab = b bac = c bad = d bbb = a bbc = d bbd = c bcb = c bcc = a bcd = b bdb = d bdc = b bdd = a	cac = a cad = b cbc = b cbd = a ccc = d ccd = c cdc = c cdd = d	dad = a dbd = b dcd = d ddd = c	ab, ba, cd, dc
27	aaa = c aab = d aac = a aad = b aba = b abb = c abc = d abd = a aca = a acb = b acc = c acd = d ada = d adb = a adc = b add = c	bab = a bac = b bad = c bbb = d bbc = a bbd = b bcb = c bcc = d bcd = a bdb = b bdc = c bdd = d	cac = c cad = d cbc = b cbd = c ccc = a ccd = b cdc = d cdd = a	dad = a dbd = d dcd = c ddd = b	ac, ca, bd, db
28	aaa = d aab = c aac = b aad = a aba = b abb = d abc = a abd = c aca = c acb = a acc = d acd = b ada = a adb = b adc = c add = d	bab = a bac = d bad = b bbb = c bbc = b bbd = a bcb = b bcc = c bcd = d bdb = d bdc = a bdd = c	cac = a cad = c cbc = c cbd = d ccc = b ccd = a cdc = d cdd = b	dad = d dbd = b dcd = c ddd = a	ad, da, bc, cb

These associative operations were found by a brute-force search. Generated were the 55,296 operations with the property that changing any one input changes the output; each was examined for associativity. Noteworthy is that 55,296 is the number of possible latin hypercubes of dimension three and order four.