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Representation of integral quantales by tolerances

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The central result of the paper claims that every integral quantale \(\mathbf {Q}\) has a natural embedding into the quantale of complete tolerances on the underlying lattice of \(\mathbf {Q}\). As an application, we show that the underlying lattice of any finite integral quantale is distributive in 1 and dually pseudocomplemented. Besides, we exhibit relationships between several earlier results. In particular, we give an alternative approach to Valentini’s ordered sets and show how the ordered sets are related to tolerances.

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Acknowledgements

We thank the anonymous referee for the most valuable suggestions that helped us considerably to improve the final version of the paper.

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Authors and Affiliations

  1. University of Tartu, Tartu, Estonia

    Kalle Kaarli

  2. University of Miskolc, Miskolc, Hungary

    Sándor Radeleczki

Authors
  1. Kalle Kaarli
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  2. Sándor Radeleczki
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Corresponding author

Correspondence to Sándor Radeleczki.

Additional information

Presented by M. Ploščica.

Dedicated to the memory of E. Tamás Schmidt.

This article is part of the topical collection “In memory of E. Tamás Schmidt” edited by Robert W. Quackenbush.

The research of Kalle Kaarli was partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. The research of Sándor Radeleczki started as a part of the project TÁMOP-4.2.1.B-10/2/KONV-2010-0001, supported by the European Union, co-financed by the European Social Fund 113/173/0-2. Mutual visits of the authors were made possible by the exchange agreement between the Estonian and the Hungarian Academies of Sciences.

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Kaarli, K., Radeleczki, S. Representation of integral quantales by tolerances. Algebra Univers. 79, 5 (2018). https://doi.org/10.1007/s00012-018-0484-1

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  • DOI: https://doi.org/10.1007/s00012-018-0484-1

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