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On the complexity of inverse semigroup conjugacy

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Abstract

We investigate the computational complexity of various decision problems related to conjugacy in finite inverse semigroups. We describe deterministic algorithms requiring logarithmic space for checking if two elements are conjugate in the full inverse semigroup with respect to various notions of conjugacy. We prove the following two problems are \(\textsf{PSPACE}\)-complete: given generators for an inverse semigroup, (1) whether the generated semigroup contains a given idempotent and (2) whether two given elements are \(\sim _i\) conjugate in the generated semigroup. We show that checking if an inverse monoid is factorizable is in \(\textsf{NC}\) and is \(\textsf{NL}\)-hard. We prove that the following problems are all \(\textsf{NL}\)-complete: given generators for a partial bijection semigroup, whether the group (1) is nilpotent, (2) is \(\mathrel {\mathcal {R}}\)-trivial, and (3) has central idempotents. We prove that the problem of checking zero membership in a partial bijection semigroup given by generators is \(\textsf{L}\)-complete. We also extend several complexity results for partial bijection semigroups to inverse semigroups.

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Acknowledgements

The author would like to thank Alan Cain, António Malheiro, and Peter Mayr for their valuable comments and contributions. The author would also like to thank the referee for their significant contributions to the results and quality of this paper.

Funding

This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UIDB/00297/2020 (Centro de Matemática e Aplicações) and PTDC/MAT-PUR/31174/2017.

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  1. Illinois Wesleyan University, Bloomington, IL, USA

    Trevor Jack

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  1. Trevor Jack
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Correspondence to Trevor Jack.

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Communicated by Mikhail Volkov.

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Jack, T. On the complexity of inverse semigroup conjugacy. Semigroup Forum 106, 618–632 (2023). https://doi.org/10.1007/s00233-023-10349-y

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