Binary Operation - Pair and Tuple

Pair and Tuple

A binary operation, ab, depends on the ordered pair (a, b) and so (ab)c (where the parentheses here mean first operate on the ordered pair (a, b) and then operate on the result of that using the ordered pair ((ab), c)) depends in general on the ordered pair ((a, b), c). Thus, for the general, non-associative case, binary operations can be represented with binary trees.

However:

• If the operation is associative, (ab)c = a(bc), then the value depends only on the tuple (a, b, c).
• If the operation is commutative, ab = ba, then the value depends only on { {a, b}, c}, where braces indicate multisets.
• If the operation is both associative and commutative then the value depends only on the multiset {a, b, c}.
• If the operation is both associative and commutative and idempotent, aa = a, then the value depends only on the set {a, b, c}.