§7. Forming direct products is routine. Any ternary quasigroup operation can be extended to take three ordered n-tuples as operands, delivering an n-tuple as a result.
Toward a clearer notation, square brackets are used on this page to enclose ordered pairs, and round brackets for other groupings. Also to reduce congestion is employed an infix notation:
O ( p, q, r ) = p ◇ q ◇ r
With ordered pairs, the fundamental formula defining the direct product of two TQGs is:
O ( [ p′, p″ ], [ q′, q″ ], [ r′, r″ ] ) = [ O ( p′, q′, r′ ), O ( p″, q″, r″ ) ]
Equivalently,
[ p′, p″ ] ◇ [ q′, q″ ] ◇ [ r′, r″ ] = [ ( p′ ◇ q′ ◇ r′ ), ( p″ ◇ q″ ◇ r″ ) ]
This can readily be extended to ordered n-tuples.
§7A. The first matter is to show that cancellativity is preserved under direct multiplication. Assume:
Now form ordered pairs:
[ ( p′ ◇ q′ ◇ r′ ), ( p″ ◇ q″ ◇ r″ ) ] = [ s′, s″ ]
Apply the direct product rule:
[ p′, p″ ] ◇ [ q′, q″ ] ◇ [ r′, r″ ] = [ s′, s″ ]
Thus [ r′, r″ ] is a solution of [ p′, p″ ] ◇ [ q′, q″ ] ◇ [ r′, r″ ] = [ s′, s″ ]. With existence verified, next is to establish unicity. Consider the ordered pair [ R′, R″ ] where R′ ≠ r′ or R″ ≠ r″ or both. Substitute:
[ p′, p″ ] ◇ [ q′, q″ ] ◇ [ R′, R″ ] = [ s′, s″ ]
Decompose:
[ ( p′ ◇ q′ ◇ R′ ), ( p″ ◇ q″ ◇ R″ ) ] = [ s′, s″ ]
Separate:
p′ ◇ q′ ◇ R′ = s′
p″ ◇ q″ ◇ R″ = s″
which means that R′ = r′ and R″ = r″, contradicting the original assumption. Thus [ r′, r″ ] is the unique solution of [ p′, p″ ] ◇ [ q′, q″ ] ◇ [ r′, r″ ] = [ s′, s″ ], and cancellativity is maintained.
§7B. If a TQG is commutative when applied to atomic elements, it is also commutative when applied to ordered pairs; this is easy to show. Given an O which is sinisteriorly commutative, start with:
O ( p′, q′, r′ ) =
O ( q′, p′, r′ )
O ( p″, q″, r″ ) =
O ( q″, p″, r″ )
or by infix:
p′ ◇ q′ ◇ r′ =
q′ ◇ p′ ◇ r′
p″ ◇ q″ ◇ r″ =
q″ ◇ p″ ◇ r″
Arrange into ordered pairs:
[ p′ ◇ q′ ◇ r′,
p″ ◇ q″ ◇ r″ ]
equals
[ q′ ◇ p′ ◇ r′,
q″ ◇ p″ ◇ r″ ]
Decompose each member:
[ p′, p″ ] ◇
[ q′, q″ ] ◇
[ r′, r″ ]
equals
[ q′, q″ ] ◇
[ p′, p″ ] ◇
[ r′, r″ ]
revealing sinisterior commutativity of the ordered pairs. The other commutativities are established the same way.
§7C. Sinisterior associativity is next. Much as before, start with:
O (
O ( p′,
q′,
r′ ),
s′,
t′ ) =
O ( p′,
O ( q′,
r′,
s′ ),
t′ )
O (
O ( p″,
q″,
r″ ),
s″,
t″ ) =
O ( p″,
O ( q″,
r″,
s″ ),
t″ )
or by infix:
( p′ ◇
q′ ◇
r′ ) ◇
s′ ◇
t′ =
p′ ◇
( q′ ◇
r′ ◇
s′ ) ◇
t′
( p″ ◇
q″ ◇
r″ ) ◇
s″ ◇
t″ =
p″ ◇
( q″ ◇
r″ ◇
s″ ) ◇
t″
Arrange into ordered pairs:
[ ( p′ ◇
q′ ◇
r′ ) ◇
s′ ◇
t′,
( p″ ◇
q″ ◇
r″ ) ◇
s″ ◇
t″ ]
equals
[ p′ ◇
( q′ ◇
r′ ◇
s′ ) ◇
t′ ),
p″ ◇
( q″ ◇
r″ ◇
s″ ) ◇
t″ ) ]
Decompose:
[ ( p′ ◇
q′ ◇
r′ ),
( p″ ◇
q″ ◇
r″ ) ] ◇
[ s′,
s″ ] ◇
[ t′,
t″ ]
equals
[ p′,
p″ ] ◇
[ ( q′ ◇
r′ ◇
s′ ),
( q″ ◇
r″ ◇
s″ ) ] ◇
[ t′,
t″ ]
Decompose again:
( [ p′, p″ ] ◇
[ q′, q″ ] ◇
[ r′, r″ ] ) ◇
[ s′, s″ ] ◇
[ t′, t″ ]
equals
[ p′, p″ ] ◇
( [ q′, q″ ] ◇
[ r′, r″ ] ◇
[ s′, s″ ] ) ◇
[ t′, t″ ]
which is sinisterior associativity over ordered pairs.