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States on bounded commutative residuated lattices

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Mathematica Slovaca

Abstract

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,

  1. (1)

    If s is a state, then X/ker(s) is an MV-algebra.

  2. (2)

    If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent:

  1. (i)

    s is a state-morphism on X.

  2. (ii)

    ker(s) is a maximal filter of X.

  3. (iii)

    s is extremal on X.

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Authors and Affiliations

  1. School of Information Environment, Tokyo Denki University, Inzai, 270-1382, Japan

    Michiro Kondo

Authors
  1. Michiro Kondo
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Corresponding author

Correspondence to Michiro Kondo.

Additional information

Communicated by Anatolij Dvurečenskij

This work was supported by JSPS KAKENHI Grant No. 24500024.

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Kondo, M. States on bounded commutative residuated lattices. Math. Slovaca 64, 1093–1104 (2014). https://doi.org/10.2478/s12175-014-0261-3

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  • DOI: https://doi.org/10.2478/s12175-014-0261-3

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