Every Steiner triple system (V,B,I) with an element e
(e not in V) yields a Steiner loop by defining a
binary operation as follows:
- xe = x, ex = x
- xx = e
- for every different x and y
there exists a unique z
such that {x,y,z} in B; define xy=z
Example.
Let (V,B,I) be this Steiner triple system:
V={1,2,3,4,5,6,7},
B={{1,2,3}, {1,4,7}, {1,5,6}, {2,5,7}, {3,4,5}, {3,6,7}, {2,4,6}},
I={<1,{1,2,3}>, <2,{1,2,3}>, <3,{1,2,3}>,
<1,{1,4,7}>, <4,{1,4,7}>, <7,{1,4,7}>,
<1,{1,5,6}>, <5,{1,5,6}>, <6,{1,5,6}>,
<2,{2,5,7}>, <5,{2,5,7}>, <7,{2,5,7}>,
<3,{3,4,5}>, <4,{3,4,5}>, <5,{3,4,5}>,
<3,{3,6,7}>, <6,{3,6,7}>, <7,{3,6,7}>,
<2,{2,4,6}>, <4,{2,4,6}>, <6,{2,4,6}>}
Then the definition gives the Steiner loop with this Cayley table:
*| e 1 2 3 4 5 6 7
------------------
e| e 1 2 3 4 5 6 7
1| 1 e 3 2 7 6 5 4
2| 2 3 e 1 6 7 4 5
3| 3 2 1 e 5 4 7 6
4| 4 7 6 5 e 3 2 1
5| 5 6 7 4 3 e 1 2
6| 6 5 4 7 2 1 e 3
7| 7 4 5 6 1 2 3 e