Languages of R-trivial monoids

AbstractWe consider the family of languages whose syntactic monoids are R-trivial. Languages whose syntactic monoids are J-trivial correspond to a congruence which tests the subwords of length n or less that appear in a given word, for some integer n. We show that in the R-trivial case the required congruence also takes into account the order in which these subwords first appear, from left to right. Characterizations of the related finite automata and regular expressions are summarized. Dual results for L-trivial monoids are also discussed