Semigroups with I-Quasi Length and Syntactic Semigroups

Abstract

In this article, we first generalize the method of [13] and give some further characterizations of semigroups with [0-]quasi length. Then we give a very simple necessary and sufficient condition for a class of semigroups, including finitely generated semigroups with [0-]quasi length, finite π-groups, finite nil-extensions of finite inverse semigroups, to be syntactic. This work completely answers the following questions: “How to characterize the syntactic semigroup (monoid) of f-disjunctive languages?” or “What kind of f-disjunctive congruences is syntactic?” These questions are asked forward and studied in [6, 12, 13].