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The Relations between Residuated Frames and Residuated Connections

Author

Listed:
  • Yong Chan Kim

    (Department of Mathematics, Gangneung-Wonju National University, Gangneung, Gangwondo 25457, Korea
    These authors contributed equally to this work.)

  • Ju-Mok Oh

    (Department of Mathematics, Gangneung-Wonju National University, Gangneung, Gangwondo 25457, Korea
    These authors contributed equally to this work.)

Abstract

We introduce the notion of (dual) residuated frames as a viewpoint of relational semantics for a fuzzy logic. We investigate the relations between (dual) residuated frames and (dual) residuated connections as a topological viewpoint of fuzzy rough sets in a complete residuated lattice. As a result, we show that the Alexandrov topology induced by fuzzy posets is a fuzzy complete lattice with residuated connections. From this result, we obtain fuzzy rough sets on the Alexandrov topology. Moreover, as a generalization of the Dedekind–MacNeille completion, we introduce R - R (resp. D R - D R ) embedding maps and R - R (resp. D R - D R ) frame embedding maps.

Suggested Citation

  • Yong Chan Kim & Ju-Mok Oh, 2020. "The Relations between Residuated Frames and Residuated Connections," Mathematics, MDPI, vol. 8(2), pages 1-24, February.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:2:p:295-:d:323573
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