Editing Steiner system (section)

===Steiner triple systems===
An S(2,3,''n'') is called a '''Steiner triple system''', and its blocks are called '''triples'''. It is common to see the abbreviation STS(''n'') for a Steiner triple system of order ''n''. The total number of pairs is ''n(n-1)/2'', of which three appear in a triple, and so the total number of triples is ''n''(''n''&minus;1)/6. This shows that ''n'' must be of the form ''6k+1'' or ''6k + 3'' for some ''k''. The fact that this condition on ''n'' is sufficient for the existence of an S(2,3,''n'') was proved by [[Raj Chandra Bose]]<ref>{{Cite journal | doi=10.1111/j.1469-1809.1939.tb02219.x|title = On the Construction of Balanced Incomplete Block Designs| journal=Annals of Eugenics| volume=9| issue=4| pages=353–399|year = 1939|last1 = Bose|first1 = R. C.| doi-access=free}}</ref> and T. Skolem.<ref>T. Skolem. [http://www.mscand.dk/article/view/10551 Some remarks on the triple systems of Steiner.] Math. Scand. 6 (1958), 273–280.</ref> The projective plane of order 2 (the [[Fano plane]]) is an STS(7) and the [[Affine plane (incidence geometry)|affine plane]] of order 3 is an STS(9). Up to isomorphism, the STS(7) and STS(9) are unique, there are two STS(13)s, 80 STS(15)s, and 11,084,874,829 STS(19)s.<ref name=Handbook>{{harvnb|Colbourn|Dinitz|2007|loc=pg.60}}</ref>

We can define a multiplication on the set ''S'' using the Steiner triple system by setting ''aa'' = ''a'' for all ''a'' in ''S'', and ''ab'' = ''c'' if {''a'',''b'',''c''} is a triple. This makes ''S''  an [[idempotent]], [[commutative]] [[quasigroup]].  It has the additional property that ''ab'' = ''c'' implies ''bc'' = ''a'' and ''ca'' = ''b''.{{NoteTag|This property is equivalent to saying that (xy)y {{=}} x for all x and y in the idempotent commutative quasigroup.}}  Conversely, any (finite) quasigroup with these properties arises from a Steiner triple system. Commutative idempotent quasigroups satisfying this additional property are called ''Steiner quasigroups''.<ref>{{harvnb|Colbourn|Dinitz|2007|loc=pg. 497, definition 28.12}}</ref>