The Lexicographic Cross-Section of the Plactic Monoid Is Regular

摘要

The plactic monoid is the quotient of the free monoid by the congruence generated by Knuth's well-celebrated rules. It is well-known that the set of Young tableaux is a cross-section of this congruence which happens to be regular. The main result of this work shows that the set of alphabetically minimal elements in the congruence classes is also regular. We give a full combinatorial characterization of these minimal elements and show that constructing them is as fast as constructing a tableau.