Soft Computing

, Volume 22, Issue 6, pp 1773–1776 | Cite as

A representation of residuated lattices satisfying the double negation law



Every residuated lattice can be considered as an idempotent semiring. Conversely, if an idempotent semiring is finite, then it can be organized into a residuated lattice. Unfortunately, this does not hold in general. We show that if an idempotent semiring is equipped with an involution which satisfies certain conditions, then it can be organized into a residuated lattice satisfying the double negation law. Also conversely, every residuated lattice satisfying the double negation law can be considered as an idempotent semiring with an involution satisfying the mentioned conditions.


Residuated lattice Double negation law Idempotent semiring Involution 



The research of the author is supported by the project IGA PrF 2017012 Palacky University Olomouc and by the Austrian Science Fund (FWF), Project I 1923-N25, and the Czech Science Foundation (GAR): Project 15-34697L.

Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Algebra and GeometryPalacký University OlomoucOlomoucCzech Republic

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