Abstract
We prove that every quotient structure of the form \({\mathscr {A}}/E\), where \({\mathscr {A}}\) is a structure computable in polynomial time (\(\text {P}\)-computable), and E is a \(\text {P}\)-computable congruence in \({\mathscr {A}}\), is isomorphic to a \(\text {P}\)-computable structure. We also prove that for every \(\text {P}\)-computable group \({\mathscr {A}} = (A,\cdot ) \), there is a \(\text {P}\)-computable group \( {\mathscr {B}}\cong {\mathscr {A}} \), in which the inversion operation \(x^{-1}\) is also \(\text {P}\)-computable.
The author is grateful to S. S. Goncharov, N. A. Bazhenov, V. L. Selivanov, S. S. Ospichev, and A. V. Seliverstov for fruitful discussions that allowed to improve the quality of the article.
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Acknowledgement
The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no FWNF-2022-0011), and partially supported by RFBR according to the research project no. 20-01-00300.
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Alaev, P. (2022). Quotient Structures and Groups Computable in Polynomial Time. In: Kulikov, A.S., Raskhodnikova, S. (eds) Computer Science – Theory and Applications. CSR 2022. Lecture Notes in Computer Science, vol 13296. Springer, Cham. https://doi.org/10.1007/978-3-031-09574-0_3
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