Fuzzy Systems and its Applications

Fuzzy Systems and its Applications

Some derivations on EQ-algebras

Document Type : Original Article

Authors
leila abbasian 1
lida Torkzadeh 2
1 Dept, of Math. Kerman Branch, Islamic Azad university, Kerman, Iran.
2 Dept. of Math. Kerman branch, Islamic Azad university, Kerman, Iran
10.22034/jfsa.2023.347345.1136
Abstract
In this paper, first the notion of (⊗,ο€ͺ)−π‘‘π‘’π‘Ÿπ‘–π‘£π‘Žπ‘‘π‘–π‘œπ‘›π‘  on 𝐸𝑄−π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Žπ‘  where ο€ͺ∈{~,→} is introduced and then they are characterized, in this way, the function 𝑑:𝐸→𝐸 on 𝐸𝑄−π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Ž 𝐸 with the bottom element 0 is a (⊗,~)((⊗,→))−π‘‘π‘’π‘Ÿπ‘–π‘£π‘Žπ‘‘π‘–π‘œπ‘› if and only if 𝑑(π‘₯)=1 ,π‘₯~𝑦=1, for all π‘₯,𝑦∈𝐸. Thus 𝑑 is an isotone, homomorphism, and closure derivation. Furthermore, (⊗,∧)−π‘‘π‘’π‘Ÿπ‘–π‘£π‘Žπ‘‘π‘–π‘œπ‘›π‘  on 𝐸𝑄−π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Žπ‘  are investigated and some related results are discussed. Indeed it is shown that if 𝑑:𝐸→𝐸 be a (⊗,∧)−π‘‘π‘’π‘Ÿπ‘–π‘£π‘Žπ‘‘π‘–π‘œπ‘› on an 𝐸𝑄−π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Ž 𝐸 such that for all π‘₯∈𝐸, 𝑑(1)⨂π‘₯≤𝑑(π‘₯), so 𝑑 is an isotone derivation. Then by studying on (⊗,∨)−π‘‘π‘’π‘Ÿπ‘–π‘£π‘Žπ‘‘π‘–π‘œπ‘›π‘  on π‘™π‘Žπ‘‘π‘‘π‘–π‘π‘’−π‘œπ‘Ÿπ‘‘π‘’π‘Ÿπ‘’π‘‘ 𝐸𝑄−π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Žπ‘ , semi-dual closure derivation is introduced and it is proved that for π‘Ž∈𝐸, π‘‘π‘Ž(π‘₯)=π‘₯β¨‚π‘Ž, for all π‘₯∈𝐸, is a semi-dual closure (⊗,∨)−π‘‘π‘’π‘Ÿπ‘–π‘£π‘Žπ‘‘π‘–π‘œπ‘› on 𝐸. Finally, by proving a property of 𝐸𝑄−π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Žπ‘ , some conditions are found under which the set of fixed points in (⊗,∨)−π‘‘π‘’π‘Ÿπ‘–π‘£π‘Žπ‘‘π‘–π‘œπ‘›π‘  on π‘™π‘Žπ‘‘π‘‘π‘–π‘π‘’−π‘œπ‘Ÿπ‘‘π‘’π‘Ÿπ‘’π‘‘ 𝐸𝑄−π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Žπ‘ , also is a π‘™π‘Žπ‘‘π‘‘π‘–π‘π‘’−π‘œπ‘Ÿπ‘‘π‘’π‘Ÿπ‘’π‘‘ 𝐸𝑄−π‘Žπ‘™π‘”π‘’π‘π‘Ÿa.
Keywords
πΈπ‘„βˆ’π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Ž
derivation
π‘™π‘Žπ‘‘π‘‘π‘–π‘π‘’βˆ’π‘œπ‘Ÿπ‘‘π‘’π‘Ÿπ‘’π‘‘ πΈπ‘„βˆ’π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Ž
fixed point
Subjects
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Fuzzy Systems and its Applications
Volume 6, Issue 1 - Serial Number 12
Open Access Statement
June 2023
Pages 83-108

  • Receive Date 15 June 2022
  • Revise Date 18 August 2022
  • Accept Date 26 April 2023