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Erratum to “State operators on generalizations of fuzzy structures” [Fuzzy Sets Syst. 187 (2012) 58–76]
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Fuzzy Sets and Systems 194 (2012) 97 – 99 www.elsevier.com/locate/fss
Erratum
Erratum to “State operators on generalizations of fuzzy structures” [Fuzzy Sets Syst. 187 (2012) 58–76] A. Dvureˇcenskija,b∗ , J. Rach˚unekb , D. Šalounovác a Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia b Department of Algebra and Geometry, Faculty of Sciences, Palacký University, tˇr. 17. listopadu 1192/12, CZ-771 46 Olomouc, Czech Republic c Department of Mathematical Methods in Economy, Faculty of Economics, VŠB – Technical University Ostrava,
Sokolská 33, CZ-701 21 Ostrava, Czech Republic Received 1 November 2011; received in revised form 11 December 2011; accepted 13 December 2011 Available online 31 December 2011
Abstract Recently, the authors characterized in Dvureˇcenskij et al. (2012) [1] subdirectly irreducible R -monoids with a faithful state operator. Unfortunately, the proof of one of the main theorems (Theorem 4.11) has a gap because we have used a property holding only for commutative R -monoids as we show. We now present counterexamples and a correct proof. Keywords: Algebras of fuzzy logics; Residuated lattice; State operator; Faithful state operator; Subdirectly irreducible algebra
1. Introduction For all necessary notions on state R-monoids, see [1]. The subdirectly irreducible R-monoids with a faithful state-operator were characterized in [1] as follows: Theorem 1.1 (Dvureˇcenskij et al. [1, Theorem 4.11]). If is a faithful operator of a state R-monoid (M, ), then (M, ) is subdirectly irreducible if and only if (M) is a subdirectly irreducible R-monoid. The proof was based on the left and right conjugates, f and f , of an element x ∈ M by the element f defined as follows: f (x) := f (x f ), f (x) := f → ( f x). We have stated that f (x) ≥ x and f (x) ≥ x. Unfortunately this property is not more valid for noncommutative R-monoids for all element f, x ∈ M as the following three examples show. DOI of original article: 10.1016/S0165-0114(11)00265-X.
∗ Corresponding author at: Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia.
E-mail addresses: [email protected] (A. Dvureˇcenskij), [email protected] (J. Rach˚unek), [email protected] (D. Šalounová). 0165-0114/$ - see front matter doi:10.1016/j.fss.2011.12.007
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A. Dvureˇcenskij et al. / Fuzzy Sets and Systems 194 (2012) 97 – 99
Example 1.2. Let G be the group of all matrices of the form A= , 0 1 where and are real numbers such that > 0; the group-operation is the usual multiplication of matrices. We denote A = (, ). Then A−1 = (1/, −/), and e = (1, 0) is the neutral element. We define G + := {(, ) : where (i) > 1, or (ii) = 1 and ≥ 0}. Then G with the positive cone G + is a linearly ordered -group with a strong unit U = (2, 0). Define M = (G, U ) := {(, ) : (1/2, 0) ≤ (, ) ≤ (1, 0)}, and let x y := x y ∨ U −1 , x → y := (yx −1 ) ∧ e and xy := (x −1 y) ∧ e. Then (M; , ∨, ∧, ∧, →, , U −1 , e) is a noncommutative R-monoid (in fact a pseudo-MV-algebra). If we set F = (, ) and X = (x, y), then F (X ) = (, ) → ((, ) (x, y)) = ((x, −x + y + ) ∨ (1/(2), −/(2)) ∧ (1, 0). Now if y > (x − 1)/( − 1), then F (X ) < X . The following example is from [2]. Example 1.3. Let G be an -group. We set G † := G + (G −1 × G −1 ). The elements from G + we will denote by g and elements from G −1 by g −1 assuming g ≥ 0. We define (e−1 , e−1 ) as a top element in G −1 × G −1 and e as a bottom element in G +1 . We order G † keeping the original ordering in G + and the original coordinatewise one within G −1 × G −1 and setting x ≤ y for all x ∈ G + and y ∈ G −1 × G −1 . a −1 , b−1 † c−1 , d −1 = (ac)−1 , (bd)−1 , a −1 , b−1 † u = a −1 u ∨ e, u † a −1 , b−1 = ub−1 ∨ e, u † v = e and a −1 , b−1 , † c−1 , d −1 = ac−1 ∧ e−1 , bd −1 ∧ e−1 , c−1 , d −1 →† a −1 , b−1 = ac−1 ∧ e−1 , bd −1 ∧ e−1 , a −1 , b−1 † u = ua, u →† a −1 , b−1 = e, e , u† a −1 , b−1 = e, e , a −1 , b−1 →† u = bu, u† v = e, uv −1 ∧ e−1 , v →† u = u −1 v ∧ e−1 , e . According to [2], G † is in fact a noncommutative pseudo-BL-algebra whenever G is nontrivial. Then f ( a −1 , b−1 ) = f b−1 f −1 ∨ f −1 , e−1 . For example, if G = Z, then f (−m, −n) = −n ∨ f −1 , 0 . Now it is easy to find an example such that f (x) ⱖ x. Example 1.4. Let G be a noncommutative linearly ordered -group with the neutral element e. Let M = {⊥, e} ⊕ G −1 be an ordinal sum of the two element Boolean algebra {⊥, e} and G −1 , where ⊥∈ / G −1 . Then M is a noncommutative −1 pseudo-BL-algebra. Let x, f ∈ G such that f xx f . Then either f x < x f or f x > x f , equivalently, f (x) < x or f (x) < x. These examples only confirm the following statement: Proposition 1.5. Let M be an R-monoid. Then f (x) ≥ x and f (x) ≥ x for all f, x ∈ M if and only if M is commutative.
A. Dvureˇcenskij et al. / Fuzzy Sets and Systems 194 (2012) 97 – 99
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Proof. If M is commutative, then f (x) = f (x f ) = f → (x f ) ≥ 1 → x = x when we have used [3, Lemma 2.5(18)(19)]. Conversely, let f (x) ≥ x and f (x) ≥ x for all f, x ∈ M. Then x ≤ f (x) = f x f implies by definition of that f x ≤ x f . Similarly, x ≤ f (x) = f → f x implies by definition of → that x f ≤ f x. Consequently, f x = x f for x, f ∈ M. 2. Corrigendum Now we present the correct proof of Theorem 1.1. Proof. Let F denote the least normal -filter of (M, ). If (M) ∩ F {1}, we assert that (M) ∩ F is the least nontrivial normal filter of (M). Indeed, let G be a nontrivial normal filter of (M). Since is a state operator, then ( f (x)) = ( f (x f )) = ( f )( f ∧ (x f )) = ( f )(x f ) ≥ ( f )((x) ( f )). That is ( f (x)) ≥ ( f ) ((x)).
(a)
In a similar way, we can prove ( f (x)) ≥ ( f ) ((x)).
(b)
Hence if x ∈ G ⊂ (M) and f ∈ M, then ( f (x)) ≥ ( f ) (x) ∈ G for each f ∈ M. By Lemma 4.9 and Corollary 4.10 of [1], N F(G ∪ (G)) = N F(G) ⊇ F. It is possible to show that G = N F(G) ∩ (M). Indeed, since N F(G) ∩ (M) ⊇ G, assume a ∈ N F(G) ∩ (M). Then a ≥ a1 · · · an , where a j = ( j1 ◦ · · · ◦ jm j )(x j ) for some x1 , . . . , xn ∈ G, where jt ∈ { f jt , f jt } for some f jt ∈ M and j = 1, . . . , n and t = 1, . . . , m j . Since (a) = a and (x j ) = x j , we have a = (a) ≥ (a1 · · · an ) ≥ (a1 ) · · · (an ) and by (a) and (b), (a j ) ≥ (j1 ◦ · · · ◦ jm j )(x j ) ∈ G for some x1 , . . . , xn ∈ G, where jt ∈ {( f jt ) , ( f jt ) }. This proves a ∈ G. Then G = N F(G) ∩ (M) ⊇ F ∩ (M), which proves (M) is subdirectly irreducible. If (M) ∩ F = {1}, then for all x ∈ F, (x) = 1 because (x) ∈ (M) ∩ F and F ⊆ Ker() = {1} is the trivial normal filter, a contradiction. Therefore, only the first case is possible and (M) is subdirectly irreducible. Conversely, let (M) be subdirectly irreducible and let G be the least nontrivial normal filter of (M). Then the normal -filter F = N F(G) of (M, ) generated by G is the least nontrivial normal -filter of (M, ). Indeed, if K is another nontrivial normal -filter of (M, ), then K ∩ (M) ⊇ F ∩ (M) = G. Then K contains the normal -filter generated by G, that is F ⊆ K which proves F is the least normal -filter, and (M, ) is subdirectly irreducible. Acknowledgement A.D. thanks for the support by Center of Excellence SAS – Quantum Technologies, meta-QUTE ITMS 26240120022, the Grants VEGA No. 2/0032/09, 2/0059/12 SAV, and by CZ.1.07/2.3.00/20.0051 and MSM 6198959214. J.R. was supported by the Council of Czech Government MSM 6198959214 and CZ.1.07/2.3.00/20.0051. Finally, the authors acknowledge an unknown referee for his suggestions. References [1] A. Dvureˇcenskij, J. Rach˚unek, D. Šalounová, State operators on generalizations of fuzzy structures, Fuzzy Sets Syst. 187 (2012) 58–76. [2] P. Jipsen, F. Montagna, On the structure of generalized BL-algebras, Algebra Universalis 55 (2006) 226–237. [3] G. Georgescu, L. Leu¸stean, V. Preoteasa, Pseudo-hoops, J. Multiple-Val. Logic Soft Comput. 11 (2005) 153–184.
Fuzzy Sets and Systems 194 (2012) 97 – 99 www.elsevier.com/locate/fss
Erratum
Erratum to “State operators on generalizations of fuzzy structures” [Fuzzy Sets Syst. 187 (2012) 58–76] A. Dvureˇcenskija,b∗ , J. Rach˚unekb , D. Šalounovác a Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia b Department of Algebra and Geometry, Faculty of Sciences, Palacký University, tˇr. 17. listopadu 1192/12, CZ-771 46 Olomouc, Czech Republic c Department of Mathematical Methods in Economy, Faculty of Economics, VŠB – Technical University Ostrava,
Sokolská 33, CZ-701 21 Ostrava, Czech Republic Received 1 November 2011; received in revised form 11 December 2011; accepted 13 December 2011 Available online 31 December 2011
Abstract Recently, the authors characterized in Dvureˇcenskij et al. (2012) [1] subdirectly irreducible R -monoids with a faithful state operator. Unfortunately, the proof of one of the main theorems (Theorem 4.11) has a gap because we have used a property holding only for commutative R -monoids as we show. We now present counterexamples and a correct proof. Keywords: Algebras of fuzzy logics; Residuated lattice; State operator; Faithful state operator; Subdirectly irreducible algebra
1. Introduction For all necessary notions on state R-monoids, see [1]. The subdirectly irreducible R-monoids with a faithful state-operator were characterized in [1] as follows: Theorem 1.1 (Dvureˇcenskij et al. [1, Theorem 4.11]). If is a faithful operator of a state R-monoid (M, ), then (M, ) is subdirectly irreducible if and only if (M) is a subdirectly irreducible R-monoid. The proof was based on the left and right conjugates, f and f , of an element x ∈ M by the element f defined as follows: f (x) := f (x f ), f (x) := f → ( f x). We have stated that f (x) ≥ x and f (x) ≥ x. Unfortunately this property is not more valid for noncommutative R-monoids for all element f, x ∈ M as the following three examples show. DOI of original article: 10.1016/S0165-0114(11)00265-X.
∗ Corresponding author at: Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia.
E-mail addresses: [email protected] (A. Dvureˇcenskij), [email protected] (J. Rach˚unek), [email protected] (D. Šalounová). 0165-0114/$ - see front matter doi:10.1016/j.fss.2011.12.007
98
A. Dvureˇcenskij et al. / Fuzzy Sets and Systems 194 (2012) 97 – 99
Example 1.2. Let G be the group of all matrices of the form A= , 0 1 where and are real numbers such that > 0; the group-operation is the usual multiplication of matrices. We denote A = (, ). Then A−1 = (1/, −/), and e = (1, 0) is the neutral element. We define G + := {(, ) : where (i) > 1, or (ii) = 1 and ≥ 0}. Then G with the positive cone G + is a linearly ordered -group with a strong unit U = (2, 0). Define M = (G, U ) := {(, ) : (1/2, 0) ≤ (, ) ≤ (1, 0)}, and let x y := x y ∨ U −1 , x → y := (yx −1 ) ∧ e and xy := (x −1 y) ∧ e. Then (M; , ∨, ∧, ∧, →, , U −1 , e) is a noncommutative R-monoid (in fact a pseudo-MV-algebra). If we set F = (, ) and X = (x, y), then F (X ) = (, ) → ((, ) (x, y)) = ((x, −x + y + ) ∨ (1/(2), −/(2)) ∧ (1, 0). Now if y > (x − 1)/( − 1), then F (X ) < X . The following example is from [2]. Example 1.3. Let G be an -group. We set G † := G + (G −1 × G −1 ). The elements from G + we will denote by g and elements from G −1 by g −1 assuming g ≥ 0. We define (e−1 , e−1 ) as a top element in G −1 × G −1 and e as a bottom element in G +1 . We order G † keeping the original ordering in G + and the original coordinatewise one within G −1 × G −1 and setting x ≤ y for all x ∈ G + and y ∈ G −1 × G −1 . a −1 , b−1 † c−1 , d −1 = (ac)−1 , (bd)−1 , a −1 , b−1 † u = a −1 u ∨ e, u † a −1 , b−1 = ub−1 ∨ e, u † v = e and a −1 , b−1 , † c−1 , d −1 = ac−1 ∧ e−1 , bd −1 ∧ e−1 , c−1 , d −1 →† a −1 , b−1 = ac−1 ∧ e−1 , bd −1 ∧ e−1 , a −1 , b−1 † u = ua, u →† a −1 , b−1 = e, e , u† a −1 , b−1 = e, e , a −1 , b−1 →† u = bu, u† v = e, uv −1 ∧ e−1 , v →† u = u −1 v ∧ e−1 , e . According to [2], G † is in fact a noncommutative pseudo-BL-algebra whenever G is nontrivial. Then f ( a −1 , b−1 ) = f b−1 f −1 ∨ f −1 , e−1 . For example, if G = Z, then f (−m, −n) = −n ∨ f −1 , 0 . Now it is easy to find an example such that f (x) ⱖ x. Example 1.4. Let G be a noncommutative linearly ordered -group with the neutral element e. Let M = {⊥, e} ⊕ G −1 be an ordinal sum of the two element Boolean algebra {⊥, e} and G −1 , where ⊥∈ / G −1 . Then M is a noncommutative −1 pseudo-BL-algebra. Let x, f ∈ G such that f xx f . Then either f x < x f or f x > x f , equivalently, f (x) < x or f (x) < x. These examples only confirm the following statement: Proposition 1.5. Let M be an R-monoid. Then f (x) ≥ x and f (x) ≥ x for all f, x ∈ M if and only if M is commutative.
A. Dvureˇcenskij et al. / Fuzzy Sets and Systems 194 (2012) 97 – 99
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Proof. If M is commutative, then f (x) = f (x f ) = f → (x f ) ≥ 1 → x = x when we have used [3, Lemma 2.5(18)(19)]. Conversely, let f (x) ≥ x and f (x) ≥ x for all f, x ∈ M. Then x ≤ f (x) = f x f implies by definition of that f x ≤ x f . Similarly, x ≤ f (x) = f → f x implies by definition of → that x f ≤ f x. Consequently, f x = x f for x, f ∈ M. 2. Corrigendum Now we present the correct proof of Theorem 1.1. Proof. Let F denote the least normal -filter of (M, ). If (M) ∩ F {1}, we assert that (M) ∩ F is the least nontrivial normal filter of (M). Indeed, let G be a nontrivial normal filter of (M). Since is a state operator, then ( f (x)) = ( f (x f )) = ( f )( f ∧ (x f )) = ( f )(x f ) ≥ ( f )((x) ( f )). That is ( f (x)) ≥ ( f ) ((x)).
(a)
In a similar way, we can prove ( f (x)) ≥ ( f ) ((x)).
(b)
Hence if x ∈ G ⊂ (M) and f ∈ M, then ( f (x)) ≥ ( f ) (x) ∈ G for each f ∈ M. By Lemma 4.9 and Corollary 4.10 of [1], N F(G ∪ (G)) = N F(G) ⊇ F. It is possible to show that G = N F(G) ∩ (M). Indeed, since N F(G) ∩ (M) ⊇ G, assume a ∈ N F(G) ∩ (M). Then a ≥ a1 · · · an , where a j = ( j1 ◦ · · · ◦ jm j )(x j ) for some x1 , . . . , xn ∈ G, where jt ∈ { f jt , f jt } for some f jt ∈ M and j = 1, . . . , n and t = 1, . . . , m j . Since (a) = a and (x j ) = x j , we have a = (a) ≥ (a1 · · · an ) ≥ (a1 ) · · · (an ) and by (a) and (b), (a j ) ≥ (j1 ◦ · · · ◦ jm j )(x j ) ∈ G for some x1 , . . . , xn ∈ G, where jt ∈ {( f jt ) , ( f jt ) }. This proves a ∈ G. Then G = N F(G) ∩ (M) ⊇ F ∩ (M), which proves (M) is subdirectly irreducible. If (M) ∩ F = {1}, then for all x ∈ F, (x) = 1 because (x) ∈ (M) ∩ F and F ⊆ Ker() = {1} is the trivial normal filter, a contradiction. Therefore, only the first case is possible and (M) is subdirectly irreducible. Conversely, let (M) be subdirectly irreducible and let G be the least nontrivial normal filter of (M). Then the normal -filter F = N F(G) of (M, ) generated by G is the least nontrivial normal -filter of (M, ). Indeed, if K is another nontrivial normal -filter of (M, ), then K ∩ (M) ⊇ F ∩ (M) = G. Then K contains the normal -filter generated by G, that is F ⊆ K which proves F is the least normal -filter, and (M, ) is subdirectly irreducible. Acknowledgement A.D. thanks for the support by Center of Excellence SAS – Quantum Technologies, meta-QUTE ITMS 26240120022, the Grants VEGA No. 2/0032/09, 2/0059/12 SAV, and by CZ.1.07/2.3.00/20.0051 and MSM 6198959214. J.R. was supported by the Council of Czech Government MSM 6198959214 and CZ.1.07/2.3.00/20.0051. Finally, the authors acknowledge an unknown referee for his suggestions. References [1] A. Dvureˇcenskij, J. Rach˚unek, D. Šalounová, State operators on generalizations of fuzzy structures, Fuzzy Sets Syst. 187 (2012) 58–76. [2] P. Jipsen, F. Montagna, On the structure of generalized BL-algebras, Algebra Universalis 55 (2006) 226–237. [3] G. Georgescu, L. Leu¸stean, V. Preoteasa, Pseudo-hoops, J. Multiple-Val. Logic Soft Comput. 11 (2005) 153–184.