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1. *Corresponding Author: M. A. Hashemi, Email: m.a.hashemipnu@gmail.com
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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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2. M. A. Hashemi et al. What is (fuzzy) fantastic filter in lattice implication algebras?
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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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