### Abstract

Let S be any nonempty set. A fuzzy relation on S is a fuzzy subset of S × S. In this paper we study fuzzy relations on rings and groups. In particular we show that if μ and σ are fuzzy left (right) ideals of a ring R, then μ × σ is a fuzzy left (right) ideal of R × R and conversely if μ × σ is a fuzzy left (right) ideal of R × R, then either μ or σ is a fuzzy left (right) ideal of R. An example is given to show that if μ × σ is a fuzzy left (right) ideal of R × R, then μ and σ both need not be fuzzy left (right) ideals of R. An example is also given to show that if μ is a fuzzy left (right) ideal of R × R, then σ_{μ}, the weakest fuzzy subset of R on which μ is a fuzzy relation, need not be a fuzzy left (right) ideal of R. We obtain similar results for groups. We also show that certain results of Bhattacharya and Mukherjee (1985) are not true.

Original language | English |
---|---|

Pages (from-to) | 117-123 |

Number of pages | 7 |

Journal | Fuzzy Sets and Systems |

Volume | 43 |

Issue number | 1 |

DOIs | |

State | Published - Sep 5 1991 |

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### All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Computer Science Applications
- Computer Vision and Pattern Recognition
- Information Systems and Management
- Statistics, Probability and Uncertainty
- Electrical and Electronic Engineering
- Statistics and Probability

### Cite this

*Fuzzy Sets and Systems*,

*43*(1), 117-123. https://doi.org/10.1016/0165-0114(91)90025-L

**Fuzzy relations on rings and groups.** / Malik, Davender S.; Mordeson, John N.

Research output: Contribution to journal › Article

*Fuzzy Sets and Systems*, vol. 43, no. 1, pp. 117-123. https://doi.org/10.1016/0165-0114(91)90025-L

}

TY - JOUR

T1 - Fuzzy relations on rings and groups

AU - Malik, Davender S.

AU - Mordeson, John N.

PY - 1991/9/5

Y1 - 1991/9/5

N2 - Let S be any nonempty set. A fuzzy relation on S is a fuzzy subset of S × S. In this paper we study fuzzy relations on rings and groups. In particular we show that if μ and σ are fuzzy left (right) ideals of a ring R, then μ × σ is a fuzzy left (right) ideal of R × R and conversely if μ × σ is a fuzzy left (right) ideal of R × R, then either μ or σ is a fuzzy left (right) ideal of R. An example is given to show that if μ × σ is a fuzzy left (right) ideal of R × R, then μ and σ both need not be fuzzy left (right) ideals of R. An example is also given to show that if μ is a fuzzy left (right) ideal of R × R, then σμ, the weakest fuzzy subset of R on which μ is a fuzzy relation, need not be a fuzzy left (right) ideal of R. We obtain similar results for groups. We also show that certain results of Bhattacharya and Mukherjee (1985) are not true.

AB - Let S be any nonempty set. A fuzzy relation on S is a fuzzy subset of S × S. In this paper we study fuzzy relations on rings and groups. In particular we show that if μ and σ are fuzzy left (right) ideals of a ring R, then μ × σ is a fuzzy left (right) ideal of R × R and conversely if μ × σ is a fuzzy left (right) ideal of R × R, then either μ or σ is a fuzzy left (right) ideal of R. An example is given to show that if μ × σ is a fuzzy left (right) ideal of R × R, then μ and σ both need not be fuzzy left (right) ideals of R. An example is also given to show that if μ is a fuzzy left (right) ideal of R × R, then σμ, the weakest fuzzy subset of R on which μ is a fuzzy relation, need not be a fuzzy left (right) ideal of R. We obtain similar results for groups. We also show that certain results of Bhattacharya and Mukherjee (1985) are not true.

UR - http://www.scopus.com/inward/record.url?scp=1842467082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1842467082&partnerID=8YFLogxK

U2 - 10.1016/0165-0114(91)90025-L

DO - 10.1016/0165-0114(91)90025-L

M3 - Article

AN - SCOPUS:1842467082

VL - 43

SP - 117

EP - 123

JO - Fuzzy Sets and Systems

JF - Fuzzy Sets and Systems

SN - 0165-0114

IS - 1

ER -