## List of small integral relation algebras

#### Compiled by Peter Jipsen

This list is in the order of Roger Maddux's list (see R. Maddux, The first 198 Relation Algebras, preprint 1993)

The vector <i,j,k> gives the number of identity atoms, symmetric atoms and nonsymmetric atoms. The total number of atoms is given by i+j+2k.

The algebras are given by a list of cycles. A cycle xyz means x;y >= z, and x~ is the converse of x. From this information the operation table for the atoms can be easily reconstructed. (The cycles that include the identity atom are not given since they are determined by the type <i,j,k>.)

If an algebra with 4 atoms or less is nonrepresentable then it is labeled as such. If no label appears, the algebra is representable (for 5-atom algebras this information is incomplete). In some cases a representation in a finite group relation algebra is known and is given by the name of the group (in GAP 3), the size of the group and the blocks of group elements that give the atoms of the RA subalgebra (these were computed using GAP in 1992 and are in a group of minimal size). For smaller groups these representations were originally computed by Steve Comer around 1986.

Some of the other representable algebras are known to have no finite representation, but I am not aware of any that are known to have no group representation.

A relation algebra is minimal if it generates a variety of height 2 in the lattice of RA varieties. These algebras are labeled as "Minimal", and more information about them can be found in the paper
[1] P. Jipsen and E. Lukacs, Minimal relation algebras, Algebra Universalis, 32, (1994), 189-203.

Page edited 2006-01-30 (added information computed in '92-'93).

```Number of nonisomorphic finite relation algebras with

i = atoms below the identity,
s = symmetric atoms and
n = pairs of nonsymmetric atoms:

<i,s,n> atoms    algebras   cycles  quads  perms
---------------------------------------------------
<1,0,0>  1              1        0      0     1
<1,1,0>  2              2        1      0     1
<1,0,1>  3              3        2      1     2
<1,2,0>  3              7        4      0     2
<1,1,1>  4             37        7      8     2
<1,3,0>  4             65       10      3     6
<1,0,2>  5             83       12     29     8
<1,2,1>  5           1316       16     26     4
<1,4,0>  5           3013       20     15    24
<1,1,2>  6          47965       25     73     8
<1,3,1>  6         988464       30     64    12 (Jipsen ~1992)
<1,5,0>  6        3849920       35     45   120 (   "   ~1992)
<1,0,3>  7        4492953       38    156    48 (   "   ~1992)
<1,2,2>  7     2186435732       44    151    16 (   "    2010-02-05)
<1,4,1>  7    50265788140       50    134    48 (   "    2010-02-07)
<1,6,0>  7   292352449486       56    105   720 (   "    2010-02-08)
<1,1,3>  8 52506245239168       63    291    48 (Fahy/Jipsen 2010-03-24, Maddux 2010-04-23)
<1,3,2>  8                      70    278    48
<1,5,1>  8                      77    251   240
<1,7,0>  8                      84    210  5040
<1,0,4>  9                      88    502   384
```
The entries before <1,2,2> were computed around 1992/93 with a Pascal program (now at http://math.chapman.edu/~jipsen/relalg/ra/findra.p; compile with FreePascal from www.freepascal.org using the command fpc findra.p). Please email additions or corrections to jipsen(AT)chapman(DOT)edu.
1 atom: <1,0,0>
11'1'1' Z_1 1 ["0"] i.e., Cm(Z_1), Generates a variety term-equivalent to Boolean algebras

2 atoms: <1,1,0>
1 1'1'1' aa1' Z_2 2 ["0", "1"] i.e., Cm(Z_2)
2 1'1'1' aa1' aaa Z_3 3 ["0", "1 2"]

3 atoms: <1,0,1>
1 aaa~ Z_3 3 ["0", "1", "2"] i.e., Cm(Z_3), Minimal, C_3 in [1]
2 aaa Q countable ["0", "negative rationals", "positive rationals"], Minimal, C_2 in [1]
3 aaa~ aaa Z_7 7 ["0", "1 2 4", "3 5 6"], Minimal, C_1 in [1]

3 atoms: <1,2,0>
1 abb Z_4 4 ["0", "1 3", "2"], Minimal, B_1 in [1]
2 abb aaa Z_6 6 ["0", "1 3 5", "2 4"], Minimal, B_2 in [1]
3 abb bbb Z_6 6 ["0", "1 2 4 5", "3"], Minimal, B_3 in [1]
4 abb aaa bbb Z_9 9 ["0", "1 2 4 5 7 8", "3 6"], Minimal, B_4 in [1]
5 aab abb Z_5 5 ["0", "1 4", "2 3"], Minimal, B_5 in [1]
6 aab abb aaa Z_8 8 ["0", "2 3 5 6", "1 4 7"], Minimal, B_6 in [1]
7 aab abb aaa bbb Z_3xZ_3 9 ["00", "10 20 01 02", "11 22 12 21"], Minimal, B_7 in [1]

4 atoms: <1,1,1> (11 nonrepresentable, 26 representable)
1 abb~ Z_4 4 ["0", "1", "2", "3"] i.e., Cm(Z_4)
2 abb ab~b~ bbb~ Z_6 6 ["0", "1 4", "3", "5 2"]
3 abb ab~b~ bbb~ aaa 3x3 9 ["0", "2 6 4", "1 3 8", "5 7"]
4 abb abb~ ab~b~ aaa Z_12 12 ["0", "1 9 7", "6 2 10 4 8", "11 3 5"]
5 aab bbb~ Z_6 6 ["0", "4", "3 1 5", "2"]
6 aab bbb~ aaa 3x3 9 ["0", "2", "1", "3 6 4 8 5 7"]
7 aab abb abb~ ab~b~ aaa
8 aab abb abb~ ab~b~ bbb~ Nonrepresentable
9 aab abb abb~ ab~b~ bbb~ aaa
10 abb~ bbb~ bbb Q8 8 ["0", "3 5 7", "1", "2 4 6"]
11 abb ab~b~ bbb
12 abb ab~b~ aaa bbb
13 abb ab~b~ bbb~ bbb Z_14 14 ["0", "1 2 11 4 9 8", "7", "13 12 3 10 5 6"]
14 abb ab~b~ bbb~ aaa bbb Z_21 21 ["0","1 2 18 4 16 15 8 9 11","7 14","20 19 3 17 5 6 13 12 10"]
15 abb abb~ ab~b~ aaa bbb Nonrepresentable
16 abb abb~ ab~b~ bbb~ aaa bbb 4x4 16 ["0", "2 6 8 13 15 14", "1 4 5", "3 7 12 9 10 11"]
17 aab bbb
18 aab aaa bbb
19 aab bbb~ bbb Z_14 14 ["0", "2 4 8", "7 1 13 3 11 5 9", "12 10 6"]
20 aab bbb~ aaa bbb Z_21 21 ["0","3 6 12","1 20 2 19 4 17 5 16 7 14 8 13 10 11","18 15 9"]
21 aab abb~ bbb
22 aab abb~ aaa bbb Nonrepresentable
23 aab abb~ bbb~ bbb Nonrepresentable
24 aab abb~ bbb~ aaa bbb Nonrepresentable
25 aab ab~b~ aaa bbb
26 aab abb~ ab~b~ bbb Nonrepresentable
27 aab abb~ ab~b~ aaa bbb Nonrepresentable
28 aab abb~ ab~b~ bbb~ bbb Nonrepresentable
29 aab abb~ ab~b~ bbb~ aaa bbb
30 aab abb ab~b~ bbb Nonrepresentable
31 aab abb ab~b~ aaa bbb
32 aab abb ab~b~ bbb~ bbb Nonrepresentable
33 aab abb ab~b~ bbb~ aaa bbb
34 aab abb abb~ ab~b~ bbb Nonrepresentable
35 aab abb abb~ ab~b~ aaa bbb
36 aab abb abb~ ab~b~ bbb~ bbb Z_21 21 ["0","19 4 16 15 14 12 11","1 20 3 18 8 13","2 17 5 6 7 9 10"]
37 aab abb abb~ ab~b~ bbb~ aaa bbb 7:3 21 ["0","5 7 20 16 19 15 18","1 6 3 4 8 17","2 14 9 10 11 12 13"]

4 atoms: <1,3,0> (20 nonrepresentable, 45 representable)
1 abc
2 abc bcc bbb Z_6 6 ["0", "1 5", "3", "2 4"]
3 abc bbc bcc bbb ccc Z_10 10 ["0", "3 7 4 6", "5", "1 9 2 8"]
4 abc acc bcc aaa bbb ccc
5 abc acc bbc bcc aaa ccc
6 abc acc bbc bcc aaa bbb ccc
7 aac abc bbc ccc D12 12 ["0", "8 10 11", "6 7 9", "3 1 2 4 5"]
8 aac abc acc bbc bcc ccc
9 aac abc acc bbc bcc aaa cccMinimal, B_8 in [1]
10 aac abc acc bbc bcc aaa bbb ccc
11 abb acc bcc 2^3 8 ["0", "6 7", "2 3 4 5", "1"]
12 abb acc bcc aaa 6x2 12 ["0", "9 10 11", "3 6 4 5 7 8", "1 2"]
13 abb acc bcc bbb 6x2 12 ["0", "1 2 10 11", "3 6 4 5 7 8", "9"]
14 abb acc bcc aaa bbb 6x3 18
15 abb acc bcc ccc 6x2 12 ["0", "6 9", "1 2 4 5 7 8 10 11", "3"]
16 abb acc bcc aaa ccc D18 18
17 abb acc bcc bbb ccc 6x3 18
18 abb acc bcc aaa bbb ccc
19 abb abc acc bcc aaa bbb Nonrepresentable
20 abb abc acc bcc aaa bbb ccc Nonrepresentable
21 abb acc bbc bcc Z_10 10 ["0", "1 9 4 6", "5", "2 8 3 7"]
22 abb acc bbc bcc aaa Z_15 15
23 abb acc bbc bcc bbb 8x2 16
24 abb acc bbc bcc aaa bbb
25 abb acc bbc bcc bbb ccc 6x3 18
26 abb acc bbc bcc aaa bbb ccc
27 abb abc acc bbc bcc aaa
28 abb abc acc bbc bcc aaa bbb
29 abb abc acc bbc bcc aaa bbb ccc
30 aac abb abc bcc Z_7 7 ["0", "3 4", "2 5", "1 6"], Minimal, B_9 in [1]
31 aac abb abc bcc aaa Nonrepresentable, Minimal, B_10 in [1]
32 aac abb abc bcc aaa bbb Nonrepresentable, Minimal, B_11 in [1]
33 aac abb abc bcc aaa bbb ccc Nonrepresentable, Minimal, B_12 in [1]
34 aac abb abc bbc bcc Nonrepresentable
35 aac abb abc bbc bcc aaa Nonrepresentable
36 aac abb abc bbc bcc bbb Nonrepresentable
37 aac abb abc bbc bcc aaa bbb Nonrepresentable
38 aac abb abc bbc bcc ccc Nonrepresentable
39 aac abb abc bbc bcc aaa ccc Nonrepresentable
40 aac abb abc bbc bcc bbb ccc 10x2 20
41 aac abb abc bbc bcc aaa bbb ccc Nonrepresentable
42 aac abb acc bbc Z_10 10 ["0", "4 6", "5 1 9 3 7", "2 8"]
43 aac abb acc bbc aaa 8x2 16
44 aac abb acc bbc bbb Z_15 15
45 aac abb acc bbc aaa bbb
46 aac abb acc bbc aaa ccc 6x3 18
47 aac abb acc bbc aaa bbb ccc
48 aac abb abc acc bbc aaa ccc Nonrepresentable
49 aac abb abc acc bbc aaa bbb ccc Nonrepresentable
50 aac abb abc acc bbc bcc Nonrepresentable
51 aac abb abc acc bbc bcc aaa
52 aac abb abc acc bbc bcc bbb Nonrepresentable
53 aac abb abc acc bbc bcc aaa bbb Nonrepresentable (last nonrep 4-atom RA; long proof found in 1993)
54 aac abb abc acc bbc bcc ccc
55 aac abb abc acc bbc bcc aaa ccc
56 aac abb abc acc bbc bcc bbb ccc
57 aac abb abc acc bbc bcc aaa bbb ccc
58 aab aac abb acc bbc bcc Nonrepresentable
59 aab aac abb acc bbc bcc aaa Nonrepresentable
60 aab aac abb acc bbc bcc aaa bbb Nonrepresentable
61 aab aac abb acc bbc bcc aaa bbb ccc
62 aab aac abb abc acc bbc bcc
63 aab aac abb abc acc bbc bcc aaa
64 aab aac abb abc acc bbc bcc aaa bbb
65 aab aac abb abc acc bbc bcc aaa bbb ccc Finite group representation known from [P. Jipsen, R. D. Maddux and Z. Tuza, Small representations of the relation algebra En+1(1, 2, 3), Algebra Universalis, 33 (1995), 136-139] but probably not the smallest.

5 atoms: <1,0,2> (some nonrepresentability results are known but not included below)
1 aab~ ab~b
2 aaa~ abb ab~b~ bbb~
3 aaa~ abb ab~b~ bbb
4 aaa~ abb ab~b~ bbb~ bbb
5 abb ab~b~ bbb~ aaa
6 aaa~ abb ab~b~ bbb~ aaa
7 abb abb~ ab~b~ aaa
8 abb ab~b~ aaa bbb
9 abb ab~b~ bbb~ aaa bbb
10 aaa~ abb ab~b~ aaa bbb
11 aaa~ abb ab~b~ bbb~ aaa bbb
12 abb abb~ ab~b~ aaa bbb
13 abb abb~ ab~b~ bbb~ aaa bbb
14 abb abb~ ab~b ab~b~ aaa
15 aaa~ abb abb~ ab~b ab~b~ aaa
16 aab~ abb ab~b ab~b~ bbb~ aaa
17 abb abb~ ab~b ab~b~ aaa bbb
18 abb abb~ ab~b ab~b~ bbb~ aaa bbb
19 aaa~ abb abb~ ab~b ab~b~ aaa bbb
20 aaa~ abb abb~ ab~b ab~b~ bbb~ aaa bbb
21 aab aab~ abb abb~ ab~b ab~b~ aaa
22 aab aab~ abb abb~ ab~b ab~b~ bbb~ aaa
23 aaa~ aab aab~ abb abb~ ab~b ab~b~ aaa
24 aaa~ aab aab~ abb abb~ ab~b ab~b~ bbb~ aaa
25 aab aab~ abb abb~ ab~b ab~b~ aaa bbb
26 aab aab~ abb abb~ ab~b ab~b~ bbb~ aaa bbb
27 aaa~ aab aab~ abb abb~ ab~b ab~b~ aaa bbb
28 aaa~ aab aab~ abb abb~ ab~b ab~b~ bbb~ aaa bbb
29 aa~b abb ab~b~ aaa bbb
30 aa~b abb ab~b~ bbb~ aaa bbb
31 aa~b abb abb~ ab~b~ aaa bbb
32 aa~b abb abb~ ab~b~ bbb~ aaa bbb
33 aa~b abb abb~ ab~b ab~b~ aaa bbb
34 aa~b abb abb~ ab~b ab~b~ bbb~ aaa bbb
35 aab aab~ aa~b abb abb~ ab~b ab~b~ aaa
36 aab aab~ aa~b abb abb~ ab~b ab~b~ bbb~ aaa
37 aaa~ aab aab~ aa~b abb abb~ ab~b ab~b~ aaa
38 aaa~ aab aab~ aa~b abb abb~ ab~b ab~b~ bbb~ aaa
39 aab aab~ aa~b abb abb~ ab~b ab~b~ aaa bbb
40 aab aab~ aa~b abb abb~ ab~b ab~b~ bbb~ aaa bbb
41 aaa~ aab aab~ aa~b abb abb~ ab~b ab~b~ aaa bbb
42 aaa~ aab aab~ aa~b abb abb~ ab~b ab~b~ bbb~ aaa bbb
43 aaa~ aab~ aa~b aba abb ab~b~ bbb
44 aab~ aa~b aba abb abb~ ab~b~ bbb
45 aa~b aba abb ab~b~ aaa bbb
46 aa~b aba abb ab~b~ bbb~ aaa bbb
47 aaa~ aa~b aba abb ab~b~ bbb~ aaa bbb
48 aa~b aba abb abb~ ab~b~ aaa bbb
49 aa~b aba abb abb~ ab~b~ bbb~ aaa bbb
50 aab~ aa~b aba abb abb~ ab~b~ aaa bbb
51 aaa~ aab~ aa~b aba abb ab~b ab~b~ bbb~
52 aaa~ aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~
53 aaa~ aab~ aa~b aba abb ab~b ab~b~ bbb~ bbb
54 aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ bbb
55 aaa~ aab~ aa~b aba abb abb~ ab~b ab~b~ bbb
56 aaa~ aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ bbb
57 aa~b aba abb abb~ ab~b ab~b~ aaa
58 aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa
59 aaa~ aa~b aba abb abb~ ab~b ab~b~ aaa
60 aaa~ aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa
61 aab~ aa~b aba abb abb~ ab~b ab~b~ aaa
62 aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa
63 aaa~ aab~ aa~b aba abb abb~ ab~b ab~b~ aaa
64 aaa~ aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa
65 aa~b aba abb abb~ ab~b ab~b~ aaa bbb
66 aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa bbb
67 aaa~ aa~b aba abb abb~ ab~b ab~b~ aaa bbb
68 aaa~ aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa bbb
69 aab~ aa~b aba abb ab~b ab~b~ aaa bbb
70 aab~ aa~b aba abb ab~b ab~b~ bbb~ aaa bbb
71 aaa~ aab~ aa~b aba abb ab~b ab~b~ bbb~ aaa bbb
72 aab~ aa~b aba abb abb~ ab~b ab~b~ aaa bbb
73 aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa bbb
74 aaa~ aab~ aa~b aba abb abb~ ab~b ab~b~ aaa bbb
75 aaa~ aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa bbb
76 aaa~ aab aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~
77 aab aab~ aa~b aba abb abb~ ab~b ab~b~ bbb
78 aab aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ bbb
79 aaa~ aab aab~ aa~b aba abb abb~ ab~b ab~b~ bbb
80 aaa~ aab aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ bbb
81 aab aab~ aa~b aba abb abb~ ab~b ab~b~ aaa bbb
82 aab aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa bbb
83 aaa~ aab aab~ aa~b aba abb abb~ ab~b ab~b~ bbb~ aaa bbb