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BCI-algebras do not satisfy the fuzzy ascending chain condition
Fuzzy Sets and Systems 158 (2007) 922 – 923 www.elsevier.com/locate/fss
Nontrivial BCK/BCI-algebras do not satisfy the fuzzy ascending chain condition Andrzej Walendziaka, b,∗ a Institute of Mathematics, University of Podlasie, Siedlce PL-08110, Poland bWarsaw School of Information Technology, Warszawa PL-01447, Poland
Received 4 May 2006; received in revised form 3 November 2006; accepted 29 November 2006 Available online 22 January 2007
Abstract In [J. Meng, X. Guo, On fuzzy ideals in BCK/BCI-algebras, Fuzzy Sets and Systems 149 (2005) 509–525], Meng and Guo gave some characterizations of Noetherian BCK/BCI-algebras by fuzzy ideals. They introduced (see Definition 6.2) the notion of the fuzzy ascending chain condition (briefly, FACC) and stated that a BCK/BCI-algebra X is Noetherian if and only if X satisfies the FACC (Theorem 6.5), but it is not true. © 2006 Elsevier B.V. All rights reserved. MSC: 03G25; 06F35; 06B99 Keywords: BCK/BCI-algebra; Fuzzy ideal
1. Noetherian BCK/BCI-algebras A BCI-algebra is an algebra (X; ∗, 0) of type (2, 0) satisfying the following axioms: (A1) (A2) (A3) (A4)
((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (x ∗ (x ∗ y)) ∗ y = 0, x ∗ x = 0, x ∗ y = 0 and y ∗ x = 0 imply x = y.
A BCI-algebra X is said to be a BCK-algebra if it satisfies: (A5) 0 ∗ x = 0 for all x ∈ X. Let X be a BCK/BCI-algebra. A subset I of X is called an ideal of X if (a) 0 ∈ I , (b) x ∗ y ∈ I and y ∈ I imply x ∈ I . X is Noetherian if for every ascending sequence I1 ⊆ I2 ⊆ · · · of ideals of X there is a natural number n such that Ii = In for all i n. ∗ Corresponding author at: Institute of Mathematics, University of Podlasie, Siedlce PL-08110, Poland. Tel.: +48 226648769.
E-mail address: [email protected]. 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.11.018
A. Walendziak / Fuzzy Sets and Systems 158 (2007) 922 – 923
923
We now review some fuzzy logic concepts. A fuzzy set on X is a function : X → [0, 1]. It is a fuzzy ideal of a BCK/BCI-algebra X if (0)(x) and (x) min{(x ∗ y), (y)} for all x, y ∈ X. X is said to satisfy the FACC (the fuzzy ascending chain condition) if for every ascending sequence 1 ⊆ 2 ⊆ · · · of fuzzy ideals of X there is a natural number n such that i = n for all i n (see [1]). Let now X be a nontrivial BCK/BCI-algebra. For i ∈ N, we define a fuzzy set i in X by 1 if x = 0, 1− i (x) = i 0 if x = 0. It is easy to see that i is a fuzzy ideal of X. Obviously 1 ⊂ 2 ⊂ · · · . Then X does not satisfy the FACC. Theorem 6.5 of [1] states that for a BCK/BCI-algebra X, the following are equivalent: (i) X is Noetherian, (ii) X satisfies the FACC. However, (i) and (ii) are not equivalent. Indeed, let X be a finite BCK/BCI-algebra of more than one element. Clearly, X is Noetherian, but it does not satisfy the FACC. References [1] J. Meng, X. Guo, On fuzzy ideals in BCK/BCI-algebras, Fuzzy Sets and Systems 149 (2005) 509–525.
Nontrivial BCK/BCI-algebras do not satisfy the fuzzy ascending chain condition Andrzej Walendziaka, b,∗ a Institute of Mathematics, University of Podlasie, Siedlce PL-08110, Poland bWarsaw School of Information Technology, Warszawa PL-01447, Poland
Received 4 May 2006; received in revised form 3 November 2006; accepted 29 November 2006 Available online 22 January 2007
Abstract In [J. Meng, X. Guo, On fuzzy ideals in BCK/BCI-algebras, Fuzzy Sets and Systems 149 (2005) 509–525], Meng and Guo gave some characterizations of Noetherian BCK/BCI-algebras by fuzzy ideals. They introduced (see Definition 6.2) the notion of the fuzzy ascending chain condition (briefly, FACC) and stated that a BCK/BCI-algebra X is Noetherian if and only if X satisfies the FACC (Theorem 6.5), but it is not true. © 2006 Elsevier B.V. All rights reserved. MSC: 03G25; 06F35; 06B99 Keywords: BCK/BCI-algebra; Fuzzy ideal
1. Noetherian BCK/BCI-algebras A BCI-algebra is an algebra (X; ∗, 0) of type (2, 0) satisfying the following axioms: (A1) (A2) (A3) (A4)
((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (x ∗ (x ∗ y)) ∗ y = 0, x ∗ x = 0, x ∗ y = 0 and y ∗ x = 0 imply x = y.
A BCI-algebra X is said to be a BCK-algebra if it satisfies: (A5) 0 ∗ x = 0 for all x ∈ X. Let X be a BCK/BCI-algebra. A subset I of X is called an ideal of X if (a) 0 ∈ I , (b) x ∗ y ∈ I and y ∈ I imply x ∈ I . X is Noetherian if for every ascending sequence I1 ⊆ I2 ⊆ · · · of ideals of X there is a natural number n such that Ii = In for all i n. ∗ Corresponding author at: Institute of Mathematics, University of Podlasie, Siedlce PL-08110, Poland. Tel.: +48 226648769.
E-mail address: [email protected]. 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.11.018
A. Walendziak / Fuzzy Sets and Systems 158 (2007) 922 – 923
923
We now review some fuzzy logic concepts. A fuzzy set on X is a function : X → [0, 1]. It is a fuzzy ideal of a BCK/BCI-algebra X if (0)(x) and (x) min{(x ∗ y), (y)} for all x, y ∈ X. X is said to satisfy the FACC (the fuzzy ascending chain condition) if for every ascending sequence 1 ⊆ 2 ⊆ · · · of fuzzy ideals of X there is a natural number n such that i = n for all i n (see [1]). Let now X be a nontrivial BCK/BCI-algebra. For i ∈ N, we define a fuzzy set i in X by 1 if x = 0, 1− i (x) = i 0 if x = 0. It is easy to see that i is a fuzzy ideal of X. Obviously 1 ⊂ 2 ⊂ · · · . Then X does not satisfy the FACC. Theorem 6.5 of [1] states that for a BCK/BCI-algebra X, the following are equivalent: (i) X is Noetherian, (ii) X satisfies the FACC. However, (i) and (ii) are not equivalent. Indeed, let X be a finite BCK/BCI-algebra of more than one element. Clearly, X is Noetherian, but it does not satisfy the FACC. References [1] J. Meng, X. Guo, On fuzzy ideals in BCK/BCI-algebras, Fuzzy Sets and Systems 149 (2005) 509–525.