Semiring and involution identities of powers of inverse semigroups

Abstract

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis.

KEYWORDS:

• Additively idempotent semiring
• Clifford semigroup
• finite basis problem
• inverse semigroup
• involution semigroup
• power semiring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16Y60
• 20M18
• 08B05

Acknowledgments

The authors thank the anonymous referee for careful reading and Edmond W. H. Lee for valuable remarks.

Notes

1 A word of warning appears to be in place here: even though we retain the notation and the name “inversion”, in general, the subset $A^{-1}$ is not the inverse of A in the sense of the above definition of an inverse element!

Additional information

Funding

I. Dolinka was supported by the Personal grant F-121 of the Serbian Academy of Sciences and Arts, and, partially, by the Ministry of Science, Technological Development and Innovations of the Republic of Serbia. S. V. Gusev and M. V. Volkov were supported by the Russian Science Foundation (Grant No. 22-21-00650).