Semimodularity in congruence lattices of graph inverse semigroups

Abstract

Congruences on a graph inverse semigroup were recently described in terms of the underline graph. Based on such descriptions, we show that the congruence lattice of a graph inverse semigroup is upper semimodular but not lower semimodular.

Keywords:

• Congruence triples
• graph inverse semigroups
• lower semimodular lattice
• upper semimodular lattice

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20M18

Acknowledgments

The second author thanks John Meakin for discussions concerning various aspects of this article. We would also like to express our profound gratitude to the anonymous referee for the valuable remarks and suggestions, which greatly improve the presentation of this article.

Additional information

Funding

Partially supported by NSFC (12071375) and the Fundamental Research Funds for the Central Universities (XDJK2016B038).