Algebra universalis

, 79:5 | Cite as

Representation of integral quantales by tolerances

Part of the following topical collections:
  1. In memory of E. Tamás Schmidt


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.


Quantale Complete lattice Tolerance relation Residuated pair Join endomorphism 

Mathematics Subject Classification

Primary 06F07 Secondary 06B15 06B23 06A15 06D22 



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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of TartuTartuEstonia
  2. 2.University of MiskolcMiskolcHungary

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