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 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.
| 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 relations on rings and groups. / Malik, Davender S.; Mordeson, John N.
In: Fuzzy Sets and Systems, Vol. 43, No. 1, 05.09.1991, p. 117-123.Research output: Contribution to journal › Article
}
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.
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U2 - 10.1016/0165-0114(91)90025-L
DO - 10.1016/0165-0114(91)90025-L
M3 - Article
VL - 43
SP - 117
EP - 123
JO - Fuzzy Sets and Systems
JF - Fuzzy Sets and Systems
SN - 0165-0114
IS - 1
ER -