Freely reduced word

A word in generators and their formal inverses is freely reduced if no consecutive letters are mutual inverses. Deleting such inverse pairs gives the unique reduced representative in a free group. In a general group presentation, free reduction preserves the represented group element but may not produce a shortest representative.

Free reduction removes adjacent inverse-letter pairs. A stack algorithm scans the letters, canceling the top letter when the next is its inverse. The resulting reduced word is unique: deleting an inverse pair before scanning does not change the final stack, so every cancellation sequence gives the same result. In any group presentation these deletions preserve the represented element, and in a free group the reduced word is the identity exactly when it is empty.

A freely reduced word is cyclically reduced if its first and last letters are not inverse, with the empty word also allowed. Every cyclic permutation of a nonempty cyclically reduced word is freely reduced, represents a conjugate, and has the same length. Positive powers are freely reduced because no cancellation occurs at copy boundaries.