  In this article, we prove combinatorial and algebraic properties on the
plactic and hypoplactic congruences via a new rewriting approach on tableaux,
that we apply to the jeu de taquin and the right-bottom rectification
algorithms. The jeu de taquin is an algorithm that transforms a skew tableau
into a Young tableau by local transformation rules on the columns of the
tableaux. This algorithm has remarkable combinatorial and algorithmic
properties, in particular it is confluent, that is the resulting Young tableau
does not depend on the way the local transformations rules are applied. We
introduce the notion of string of columns rewriting system in order to study
rewriting properties of algorithms defined on glued sequences of columns. We
prove the convergence of the jeu de taquin and we show how to deduce algebraic
properties on the plactic congruence. We define the right-bottom rectification
algorithm that transforms a Young tableau into a quasi-ribbon tableau by a
string of columns rewriting system. We prove its convergence and we deduce
algebraic properties on the hypoplactic congruence.
