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This document discusses fantastic filters in lattice implication algebras. It begins by introducing lattice implication algebras and defining filters and fantastic filters within these algebraic structures. It then proves that fantastic filters are equivalent to ordinary filters, and that fuzzy fantastic filters are equivalent to fuzzy filters. Therefore, fantastic filters do not constitute a distinct type of filter in lattice implication algebras.
![*Corresponding Author: M. A. Hashemi, Email: m.a.hashemipnu@gmail.com
REVIEW ARTICLE
Available Online at
What is (fuzzy) fantastic filter in lattice implication algebras?
1
M. A. Hashemi*
1
*Department of Mathematics, Payame Noor University, P. O. Box: 19395-3697, Tehran, Iran
Received on: 08/03/2017, Revised on: 14/03/2017, Accepted on: 22/03/2017
ABSTRACT
In this paper we prove that the notion of (fuzzy) fantastic filters, introduced by Y. B. Jun in [1, 2]
, is
equivalent of (fuzzy)filter and it is not a different (fuzzy) filter in lattice implication algebras.
Keywords: Filter, fantastic filter. AMS:03 G10, 06B10, 54E15.
INTRODUCTION
Non-classical logic has become a considerable formal tool for computer science and artificial intelligence
to deal with fuzzy information and uncertainty information. Many-valued logic, a great extension and
development of classical logic, has always been a crucial direction in non-classical logic. In order to
research the many-valued logical system whose propositional value is given in a lattice, in 1990 Xu [3]
proposed the concept of lattice implication algebra. In lattice implication algebra, filters are important
substructures; they play a significant role in studying the structure and the properties of lattice implication
algebras [4]. In [1], Jun defined and studied the notion of fantastic filter in lattice implication algebras.
Then some other researchers worked on this filter [2, 5, 6]
. In this paper, we'll prove that the notation of
fantastic filters is equivalent of filter and it is not a different filter in lattice implication algebras.
Definition 1: Lattice implication algebra is defined to be a bounded lattice (L; ν; Λ; 0; 1) with order-
reversing involution "′" and a binary operation “→ ". In the sequel the binary operation "→" will be
denoted by juxtaposition. In a lattice implication algebra L, the following hold:
(I1) x (yz) = y(xz);
(I2) xx = 1;
(I3) xy = y′x′;
(I4) xy = yx = 1) x = y;
(I5) (xy) y = (yx) x;
(L1) (x ν y) z = (xz) ^ (yz);
(L2) (x ^ y)z = (xz) ν (yz);
For all x; y; z € L.
We can define a partial ordering ≤ on a lattice implication algebra L by x ≤ y if and only if xy = 1.
Definition 2. In a lattice implication algebra L, the following hold:
(P1) 0x = 1; 1x = x and x1 = 1.
(P2) xy ≤ (yz) (xz).
(P3) x ≤ y implies yz ≤ xz and zx ≤ zy.
(P4) x′ = x0.
(P5) x ν y = (xy)y.
(P6) ((yx) y′)′ = x ^ y = ((xy) x′) ′.
(P7) x ≤ (xy) y.
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![M. A. Hashemi et al. What is (fuzzy) fantastic filter in lattice implication algebras?
106
© 2017, AJMS. All Rights Reserved.
Definition 3. ([4]
) A subset F of L is called a filter of L if it satisfies for all x; y € L,
(F1) 1 € F,
(F2) x € F and xy € F imply y € F.
Definition 4. ([1]
) A subset F of L is called an fantastic filter of L if it satisfies (F1) and (F3) z (yx) € F
and z € F imply ((xy) y) x € F for all x; y; z € L.
Theorem 5. Fantastic filters and filters in lattice implication algebras are equivalent.
Proof. By [1]
, we need to prove that filters are fantastic filter. Assume that F is a filter in lattice
implication algebra L, then for all x; y; z € L such that z (yx) € F and z € F we have yx € F. Since (((xy)
y) x = (((yx) x) x) = yx € F, we have (((xy) y) x € F, i.e., F is fantastic filter.
Definition 6. ([2]
) A fuzzy subset µ of L is called a fuzzy filter of L if it satisfies for all x; y € L,
(FF1) for all x € L, µ (1) ≥ µ(x),
(FF2) for all x; y € L, µ(y) ≥ min {µ(xy), µ(x)}.
Definition 7. ([2]
) A fuzzy subset _ of L is called a fantastic filter of L if it satisfies (FF1) and (FF3) for
all x; y; € L, µ (((xy) y) x) ≥ min {µ (z (yx)), µ (z)}.
Theorem 8. Fuzzy fantastic filters and fuzzy filters in lattice implication algebras are equivalent.
Proof. By [1], we need to prove that fuzzy filters are fuzzy fantastic filter. Assume that µ is a fuzzy filter
of lattice implication algebra L, then for all x; y; z € L we have µ(yx) ≥ min{µ(z(yx)), µ(z)}. Since (((xy)
y)x = (((yx) x) x) = yx, we have µ((((xy)y)x) ≥ min{µ(z(yx)); µ(z)}, i.e., µ is fuzzy fantastic filter.
REFERENCES
1. Y. B. Jun, Fantastic filters of lattice implication algebras, Int. J. Math. and Math. Sci., 24, no. 4
(2000), 277-281.
2. Y. B. Jun, S. Z. Song , On fuzzy fantastic filters of lattice implication algebras, Journal of Applied
Mathematics and Computing March 2004, Volume 14, Issue 1-2,pp 137-155.
3. Y. Xu, Lattice implication algebras, J. Southwest Jiao tong Univ. 1, (1993), 20-27.
4. Y. Xu and K.Y. Qin, On fitters of lattice implication algebras, J. Fuzzy Math., 1, no.2, (1993),
251-260.
5. Zhan Jianming , Chen Yiping and Tan Zhisong, Fuzzy fantastic filters of lattice implication
algebras, Scientiae Mathematicae Japonicae Online, Vol. 9, (2003), 209-211.
6. M. Sambasiva, Transitive and absorbent filters of lattice implication algebras, J. Appl.Math. and
Informatics Vol. 32(2014), No. 3-4, pp. 323-3303.
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Mar-April,
2017,
Vol.
1,
Issue
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9. MA Hashemi.pdf
- 1. *Corresponding Author: M. A. Hashemi, Email: [email protected] REVIEW ARTICLE Available Online at What is (fuzzy) fantastic filter in lattice implication algebras? 1 M. A. Hashemi* 1 *Department of Mathematics, Payame Noor University, P. O. Box: 19395-3697, Tehran, Iran Received on: 08/03/2017, Revised on: 14/03/2017, Accepted on: 22/03/2017 ABSTRACT In this paper we prove that the notion of (fuzzy) fantastic filters, introduced by Y. B. Jun in [1, 2] , is equivalent of (fuzzy)filter and it is not a different (fuzzy) filter in lattice implication algebras. Keywords: Filter, fantastic filter. AMS:03 G10, 06B10, 54E15. INTRODUCTION Non-classical logic has become a considerable formal tool for computer science and artificial intelligence to deal with fuzzy information and uncertainty information. Many-valued logic, a great extension and development of classical logic, has always been a crucial direction in non-classical logic. In order to research the many-valued logical system whose propositional value is given in a lattice, in 1990 Xu [3] proposed the concept of lattice implication algebra. In lattice implication algebra, filters are important substructures; they play a significant role in studying the structure and the properties of lattice implication algebras [4]. In [1], Jun defined and studied the notion of fantastic filter in lattice implication algebras. Then some other researchers worked on this filter [2, 5, 6] . In this paper, we'll prove that the notation of fantastic filters is equivalent of filter and it is not a different filter in lattice implication algebras. Definition 1: Lattice implication algebra is defined to be a bounded lattice (L; ν; Λ; 0; 1) with order- reversing involution "′" and a binary operation “→ ". In the sequel the binary operation "→" will be denoted by juxtaposition. In a lattice implication algebra L, the following hold: (I1) x (yz) = y(xz); (I2) xx = 1; (I3) xy = y′x′; (I4) xy = yx = 1) x = y; (I5) (xy) y = (yx) x; (L1) (x ν y) z = (xz) ^ (yz); (L2) (x ^ y)z = (xz) ν (yz); For all x; y; z € L. We can define a partial ordering ≤ on a lattice implication algebra L by x ≤ y if and only if xy = 1. Definition 2. In a lattice implication algebra L, the following hold: (P1) 0x = 1; 1x = x and x1 = 1. (P2) xy ≤ (yz) (xz). (P3) x ≤ y implies yz ≤ xz and zx ≤ zy. (P4) x′ = x0. (P5) x ν y = (xy)y. (P6) ((yx) y′)′ = x ^ y = ((xy) x′) ′. (P7) x ≤ (xy) y. www.ajms.in Asian Journal of Mathematical Sciences 2017; 1(2):105-106
- 2. M. A. Hashemi et al. What is (fuzzy) fantastic filter in lattice implication algebras? 106 © 2017, AJMS. All Rights Reserved. Definition 3. ([4] ) A subset F of L is called a filter of L if it satisfies for all x; y € L, (F1) 1 € F, (F2) x € F and xy € F imply y € F. Definition 4. ([1] ) A subset F of L is called an fantastic filter of L if it satisfies (F1) and (F3) z (yx) € F and z € F imply ((xy) y) x € F for all x; y; z € L. Theorem 5. Fantastic filters and filters in lattice implication algebras are equivalent. Proof. By [1] , we need to prove that filters are fantastic filter. Assume that F is a filter in lattice implication algebra L, then for all x; y; z € L such that z (yx) € F and z € F we have yx € F. Since (((xy) y) x = (((yx) x) x) = yx € F, we have (((xy) y) x € F, i.e., F is fantastic filter. Definition 6. ([2] ) A fuzzy subset µ of L is called a fuzzy filter of L if it satisfies for all x; y € L, (FF1) for all x € L, µ (1) ≥ µ(x), (FF2) for all x; y € L, µ(y) ≥ min {µ(xy), µ(x)}. Definition 7. ([2] ) A fuzzy subset _ of L is called a fantastic filter of L if it satisfies (FF1) and (FF3) for all x; y; € L, µ (((xy) y) x) ≥ min {µ (z (yx)), µ (z)}. Theorem 8. Fuzzy fantastic filters and fuzzy filters in lattice implication algebras are equivalent. Proof. By [1], we need to prove that fuzzy filters are fuzzy fantastic filter. Assume that µ is a fuzzy filter of lattice implication algebra L, then for all x; y; z € L we have µ(yx) ≥ min{µ(z(yx)), µ(z)}. Since (((xy) y)x = (((yx) x) x) = yx, we have µ((((xy)y)x) ≥ min{µ(z(yx)); µ(z)}, i.e., µ is fuzzy fantastic filter. REFERENCES 1. Y. B. Jun, Fantastic filters of lattice implication algebras, Int. J. Math. and Math. Sci., 24, no. 4 (2000), 277-281. 2. Y. B. Jun, S. Z. Song , On fuzzy fantastic filters of lattice implication algebras, Journal of Applied Mathematics and Computing March 2004, Volume 14, Issue 1-2,pp 137-155. 3. Y. Xu, Lattice implication algebras, J. Southwest Jiao tong Univ. 1, (1993), 20-27. 4. Y. Xu and K.Y. Qin, On fitters of lattice implication algebras, J. Fuzzy Math., 1, no.2, (1993), 251-260. 5. Zhan Jianming , Chen Yiping and Tan Zhisong, Fuzzy fantastic filters of lattice implication algebras, Scientiae Mathematicae Japonicae Online, Vol. 9, (2003), 209-211. 6. M. Sambasiva, Transitive and absorbent filters of lattice implication algebras, J. Appl.Math. and Informatics Vol. 32(2014), No. 3-4, pp. 323-3303. AJMS, Mar-April, 2017, Vol. 1, Issue 2