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].

Key Words:

• f-Disjunctive congruence
• f-Disjunctive language
• Semigroup with I-quasi length
• Syntactic semigroup
• 2-Weakly reductive semigroup

2000 Mathematics Subject Classification:

• 20M35
• 68Q70

ACKNOWLEDGMENTS

The authors thank the referees for their valuable comments and suggestions.

The research is supported by NSF (China) grant #10871161 and the Natural Science Foundation of Yunnan Province (China) grant #2008ZC162 M.

Notes

Let ρ be a congruence on a semigroup S. For any w ∈ S, we use wρ to denote the ρ-class containing w. The mapping ρ^♮: S → S/ρ defined by

is a homomorphism, called the natural homomorphism (cf. [7]).

Notice that the δ-class mentioned here must all contain at most two elements since S is supposed to be 2-weakly reductive. So we have already exhausted all cases here.

Communicated by V. Gould.