In the example above, + is an associative operation, so the final result will be the same regardless of parenthesization, although the specific way in which it is calculated will be different.
12.
Since conjunction is a commutative and associative operation, the formal setting-up of sequent theory normally includes "'structural rules "'for rewriting the sequent ? accordingly-for example for deducing
13.
A finger tree gives amortized constant time access to the " fingers " ( leaves ) of the tree, where data is stored, and also stores in each internal node the result of applying some associative operation to its descendants.
14.
There are 18 inequivalent ways to define an associative operation on three elements : while there are, altogether, a total of 3 9 = 19683 different binary operations that can be defined, only 113 of these are associative, and many of these are isomorphic or antiisomorphic so that there are essentially only 18 possibilities.
