Fuzzy Sets and Systems 121 (2001) 325–332
www.elsevier.com/locate/fss
Quotient structures of some implicative algebras via fuzzy implicative #lters Young Bae Juna;∗ , Sun Shin Ahnb , Hee Sik Kimc a Department
of Mathematics Education, Gyeongsang National University, Chinju 660-701, South Korea of Mathematics Education, Dongguk University, Seoul 100-715, South Korea c Department of Mathematics, Hanyang University, Seoul 133-791, South Korea
b Department
Received 8 October 1998; received in revised form 22 October 1999; accepted 3 November 1999
Abstract In this paper we discuss quotient implicative algebras induced by fuzzy implicative #lters, and establish the fuzzy c 2001 Elsevier Science B.V. All rights reserved. fundamental homomorphism theorem for some implicative algebras. MSC: 03G25; 04A72 Keywords: Implicative algebra; Positive implication algebra; (fuzzy) Implicative #lter
1. Introduction Implicative algebras are closely related to ordered sets with a greatest element. In [4], it was concerned with elementary properties of implicative algebras and implicative #lters. The notion of fuzzy sets was formulated by Zadeh [6] and since then fuzzy sets have been applied to various branches of mathematics and computer science. Rosenfeld [5] inspired the development of fuzzy algebraic structures. In [2], Jun and Kim considered the fuzzi#cation of an implicative #lter in an implicative algebra, and solved the problem of classifying fuzzy implicative #lters by their level implicative #lters. They also stated characteristic fuzzy implicative #lters and the same type fuzzy implicative #lters, and gave its characterization. In this paper we discuss quotient implicative algebras induced by fuzzy implicative #lters, and establish the fuzzy fundamental homomorphism theorem for a particular case of implicative algebras. 2. Preliminaries An abstract algebra A = (A; V; ⇒), where V is a 0-argument operation and ⇒ is a two-argument operation, is said to be an implicative algebra, provided the following conditions are satis#ed for all a; b; c ∈ A ∗ Corresponding author. Tel.: +82-591-751-5674; fax: +82-591-751-6117. E-mail address:
[email protected] (Y.B. Jun).
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 0 0 8 - 7
326
Y.B. Jun et al. / Fuzzy Sets and Systems 121 (2001) 325–332
[4, p. 16]: (i1 ) a ⇒ a = V , (i2 ) if a ⇒ b = V and b ⇒ c = V , then a ⇒ c = V , (i3 ) if a ⇒ b = V and b ⇒ a = V , then a = b, (i4 ) a ⇒ V = V . Let A = (A; V; ⇒) be an implicative algebra. Then the relation a6b if and only if a ⇒ b = V de#nes a partial order on A. The element V is the greatest element in the partially ordered set (A; 6). An abstract algebra A = (A; V; ⇒) with a zero-argument operation V and a two-argument operation ⇒ will be called a positive implication algebra or Hilbert algebra, provided for all a; b; c ∈ A the following conditions are satis#ed [3, p. 4]: (p1 ) a ⇒ (b ⇒ a) = V , (p2 ) (a ⇒ (b ⇒ c)) ⇒ ((a ⇒ b) ⇒ (a ⇒ c)) = V , (p3 ) if a ⇒ b = V and b ⇒ a = V , then a = b. In any positive implication algebra A = (A; V; ⇒) the following conditions are satis#ed [4, pp. 24–25]: (1) a6b ⇒ a; (2) a6b ⇒ c implies b6a ⇒ c, (3) a6(a ⇒ b) ⇒ b; (4) V ⇒ a = a, (5) if b6c then a ⇒ b6a ⇒ c, (6) if a6b then b ⇒ c6a ⇒ c, (7) a ⇒ (a ⇒ b) = a ⇒ b, (8) a ⇒ (b ⇒ c) = b ⇒ (a ⇒ c), (9) a ⇒ b6(b ⇒ c) ⇒ (a ⇒ c), (10) b ⇒ c6(a ⇒ b) ⇒ (a ⇒ c), (11) (a ⇒ a) ⇒ b = b, (12) a ⇒ (b ⇒ c) = (a ⇒ b) ⇒ (a ⇒ c), (13) (a ⇒ b) ⇒ ((b ⇒ a) ⇒ b) = (b ⇒ a) ⇒ ((a ⇒ b) ⇒ a). 3. Quotient structures Denition 3.1 (Rasiowa [4, p. 18]). A subset of the set A of all elements of an implicative algebra A = (A; V; ⇒) is said to be an implicative 7lter, provided the following conditions are satis#ed: (f1 ) V ∈ ; (f2 ) if a ∈ and a ⇒ b ∈ , then b ∈ . We state the fuzzi#cation of an implicative #lter in an implicative algebra. Denition 3.2 (Jun and Kim [2, De#nition 3.2]). A fuzzy set in an implicative algebra A = (A; V; ⇒) is said to be a fuzzy implicative 7lter provided the following conditions are satis#ed: (L 1 ) (V ) − (a)¿0, ∀a ∈ A, (L 2 ) (b) − min{(a); (a ⇒ b)}¿0; ∀a; b ∈ A. Lemma 3.3 (Jun and Kim [2, Proposition 3.4]). In an implicative algebra every fuzzy implicative 7lter is order preserving.
Y.B. Jun et al. / Fuzzy Sets and Systems 121 (2001) 325–332
327
Theorem 3.4. Let A = (A; V; ⇒) be an implicative algebra. A fuzzy set in A is a fuzzy implicative 7lter of A if and only if for every a; b; c ∈ A the identity a ⇒ (b ⇒ c) = V implies (c) − min{(a); (b)}¿0: Proof. Let be a fuzzy implicative #lter of an implicative algebra A = (A; V; ⇒). Assume that a ⇒ (b ⇒ c) = V for all a; b; c ∈ A. Then by Lemma 3.3 we have (a)6(b ⇒ c). It follows from (L 2 ) that (c) − min{(a); (b)}¿(c) − min{(b ⇒ c); (b)}¿0: Conversely, suppose that for every a; b; c ∈ A the identity a ⇒ (b ⇒ c) = V implies (c)−min{(a); (b)}¿0. Since a ⇒ (a ⇒ V ) = V , it follows that (V ) − (a) = (V ) − min{(a); (a)}¿0: Note that (a ⇒ b) ⇒ (a ⇒ b) = V , so by assumption that (b) − min{(a ⇒ b); (a)}¿0. This completes the proof. Given an implicative #lter of an implicative algebra A = (A; V; ⇒), we shall denote by t the binary relation on A de#ned as follows: for any a; b ∈ A [4, p. 19], (14) a t b if and only if a ⇒ b ∈ and b ⇒ a ∈ . The relation t is then said to be determined by . If in A the following conditions hold for any a; b; c ∈ A: (9) (a ⇒ b) ⇒ ((b ⇒ c) ⇒ (a ⇒ c)) = V , (10) (b ⇒ c) ⇒ ((a ⇒ b) ⇒ (a ⇒ c)) = V , then the relation t determined by is a congruence in A [4, p. 19]. If, moreover (15) V ⇒ a = a; for all a ∈ A, then the quotient algebra A=t , denoted later by A=, is an implicative algebra in which the equations corresponding to (9), (10) and (15) hold (see [4, p. 19]), where the elements of A= are denoted by a for a ∈ A, and a zero-argument operation and a two-argument operation are denoted by V and ⇒V , respectively. Now, we de#ne a binary relation in an implicative algebra A by using fuzzy implicative #lters. Let be a fuzzy implicative #lter of A which is not constant. De#ne a binary relation t , in A as follows: for every a; b ∈ A, (16) a t b if and only if (a ⇒ b)¿0 and (b ⇒ a)¿0. The relation t is then said to be determined by . We now give an example of an implicative algebra satisfying (9), (10) and (15) and without being a positive implication algebra. Example 3.5. Let A := {V; 1; 2; 3; 4} be a set with the Cayley table and the Hasse diagram as follows: ⇒ V 1 2 3 4
V V V V V V
1 1 V 1 V V
2 2 2 V 2 V
3 3 1 3 V 1
4 4 4 4 4 V
328
Y.B. Jun et al. / Fuzzy Sets and Systems 121 (2001) 325–332
Then A := (A; V; ⇒) is an implicative algebra satisfying conditions (9), (10) and (15), but not (p2 ), since ((4 ⇒ (1 ⇒ 3)) ⇒ ((4 ⇒ 1) ⇒ (4 ⇒ 3)) = 1 = V , i.e., it is an implicative algebra satisfying conditions (9), (10), and (15), but without being a positive implication algebra. Proposition 3.6. Let A = (A; V; ⇒) be an implicative algebra which satis7es condition (9). Then t is an equivalence relation in A. Proof. We note that (V ) − (a)¿0 for all a ∈ A. Since is not constant, it follows that (V )¿ 0. Observe that a t a for all a ∈ A, since (a ⇒ a) = (V )¿0. The symmetry of t follows directly from (16). Assume that a t b and b t c. Then (a ⇒ b)¿0, (b ⇒ a)¿0, (b ⇒ c)¿0, and (c ⇒ b)¿0. Combining (9) and Theorem 3.4, we obtain (a ⇒ c) − min{(a ⇒ b); (b ⇒ c)}¿0: It follows from (a ⇒ b)¿0 and (b ⇒ c)¿0 that min{(a ⇒ b); (b ⇒ c)}¿0 so that (a ⇒ c)¿0. Similarly we get (c ⇒ a)¿0. Hence a t c, i.e.; t is transitive. Proposition 3.7. Let A = (A; V; ⇒) be an implicative algebra in which conditions (9) and (10) are satis7ed. If a t b and c t d; then (a ⇒ c) t (b ⇒ d); i.e.; t is a congruence relation in A. Proof. Let us suppose that a t b and c t d. Then (c ⇒ d)¿0. Hence, by (L 2 ), (10) and (L 1 ), 0 6 ((a ⇒ c) ⇒ (a ⇒ d)) − min{(((c ⇒ d) ⇒ ((a ⇒ c) ⇒ (a ⇒ d))); (c ⇒ d)} = ((a ⇒ c) ⇒ (a ⇒ d)) − min{(V ); (c ⇒ d)} = ((a ⇒ c) ⇒ (a ⇒ d)) − (c ⇒ d); which implies that (17) ((a ⇒ c) ⇒ (a ⇒ d))¿0. Since a t b; we have (b ⇒ a)¿0. Consequently, by (L 2 ), (9) and (L 1 ) 0 6 ((a ⇒ d) ⇒ (b ⇒ d)) − min{((b ⇒ a) ⇒ ((a ⇒ d) ⇒ (b ⇒ d))); (b ⇒ a)} = ((a ⇒ d) ⇒ (b ⇒ d)) − min{(V ); (b ⇒ a)} = ((a ⇒ d) ⇒ (b ⇒ d)) − (b ⇒ a); and so (18) ((a ⇒ d) ⇒ (b ⇒ d))¿0. Using (L 2 ), (10), (L 1 ), (17) and (18), we have ((a ⇒ c) ⇒ (b ⇒ d)) ¿ min{((a ⇒ c) ⇒ (a ⇒ d)); (((a ⇒ c) ⇒ (a ⇒ d)) ⇒ ((a ⇒ c) ⇒ (b ⇒ d)))} ¿ min{((a ⇒ c) ⇒ (a ⇒ d)); min{((a ⇒ d) ⇒ (b ⇒ d)); (((a ⇒ d) ⇒ (b ⇒ d)) ⇒ (((a ⇒ c) ⇒ (a ⇒ d)) ⇒ ((a ⇒ c) ⇒ (b ⇒ d))))}} = min{((a ⇒ c) ⇒ (a ⇒ d)); min{((a ⇒ d) ⇒ (b ⇒ d)); (V )}} = min{((a ⇒ c) ⇒ (a ⇒ d)); ((a ⇒ d) ⇒ (b ⇒ d))} ¿ 0:
Y.B. Jun et al. / Fuzzy Sets and Systems 121 (2001) 325–332
329
Similarly we can prove that ((b ⇒ d) ⇒ (a ⇒ c))¿0. Hence we get (a ⇒ c) t (b ⇒ d); ending the proof. The following corollaries immediately follow from the Proposition 3.7. Corollary 3.8. Let A = (A; V; ⇒) be an implicative algebra in which any one of the conditions (9) and (10) is satis7ed. If; moreover; A satis7es the following condition: (8) a ⇒ (b ⇒ c) = b ⇒ (a ⇒ c) for all a; b; c ∈ A; then the relation t determined by is a congruence relation in A. Corollary 3.9. In a positive implication algebra every relation t determined by is a congruence relation. Let A = (A; V; ⇒) be an implicative algebra having conditions (9) and (10). We denote by a the equivalence class determined by a ∈ A, i.e., the set of all b ∈ A such that a t b, and we denote by A= the quotient set { a | a ∈ A}. De#ne a two-argument operation ⇒ on A= by a ⇒ b := a ⇒ b for all a; b ∈ A. By Proposition 3.7, the operation ⇒ is well de#ned. Theorem 3.10. Let A = (A; V; ⇒) be an implicative algebra in which conditions (9); (10) and (15) hold; and let be a fuzzy implicative 7lter of A which is not constant. Then the quotient A= is an implicative algebra with V and ⇒ as a zero-argument operation and a two-argument operation; respectively; in which the equations corresponding to (9); (10) and (15) hold. Proof. For any a ∈ A we have a ⇒ a = a ⇒ a = V : Assume that a ⇒ b = V
and
b ⇒ c = V
for all a; b; c ∈ A. Then a ⇒ b = V = b ⇒ c and so (a ⇒ b) t V t (b ⇒ c). It follows from (16), (i4 ) and (15) that (V )¿0, (a ⇒ b)¿0 and (b ⇒ c)¿0. Using (15), (L 2 ), (9) and (L 1 ), we have (V ⇒ (a ⇒ c)) = (a ⇒ c) ¿ min{(b ⇒ c); ((b ⇒ c) ⇒ (a ⇒ c))} ¿ min{(b ⇒ c); min{(a ⇒ b); ((a ⇒ b) ⇒ ((b ⇒ c) ⇒ (a ⇒ c)))}} = min{(b ⇒ c); min{(a ⇒ b); (V )}} = min{(b ⇒ c); (a ⇒ b)} ¿ 0: Hence (a ⇒ c) t V , i.e., a ⇒ c = a ⇒ c = V : Suppose that a ⇒ b = V
and
b ⇒ a = V :
Then (a ⇒ b) t V t (b ⇒ a); which imply that (a ⇒ b) = (V ⇒ (a ⇒ b)) ¿ 0 and (b ⇒ a) = (V ⇒ (b ⇒ a)) ¿ 0:
330
Y.B. Jun et al. / Fuzzy Sets and Systems 121 (2001) 325–332
Thus a t b and so a = b . The equation a ⇒ V = V holds by (i4 ) and property of quotient structures. Similarly we obtain that the equations ( a ⇒ b ) ⇒ (( b ⇒ c ) ⇒ ( a ⇒ c )) = V ; ( b ⇒ c ) ⇒ (( a ⇒ b )⇒ ( a ⇒ c )) = V ; V ⇒ a = a hold by (9), (10) and (15). This completes the proof. Corollary 3.11. Let A = (A; V; ⇒) be a positive implication algebra and let be a fuzzy implicative 7lter of A which is not constant. Then A= is a positive implication algebra with V and ⇒ as a zero-argument operation and a two-argument operation; respectively. For a fuzzy set in a set A and for m ∈ [0; 1], the set m := {a ∈ A | (a)¿m} is called a level subset of (see [1]). Lemma 3.12 (Jun and Kim [2, Proposition 3.5]). Let be a fuzzy set in an implicative algebra A = (A; V; ⇒). Then is a fuzzy implicative 7lter in A if and only if for every m ∈ [0; 1]; the set m is an implicative 7lter; when m = ∅. A mapping f : A → B from an implicative algebra A into an implicative algebra B is said to be a homomorphism if f(x ⇒ y) = f(x) ⇒ f(y) for any x; y ∈ A. Theorem 3.13. Let A and be as in Theorem 3:10; and consider the set m = ∅ for m ∈ (0; 1]. Then a map h : A=m → A= de7ned by h((m )a ) = a for all (m )a ∈ A=m is an epimorphism. Proof. Assume that (m )a = (m )b for all a; b ∈ A. Then a tm b and so a ⇒ b ∈ m and b ⇒ a ∈ m . It follows that (a ⇒ b)¿m ¿ 0
and
(b ⇒ a)¿m ¿ 0;
so that a t b or a = b . This proves that the map h is well de#ned. For every (m )a , (m )b ∈ A=m ; we have h((m )a ⇒m (m )b ) = h((m )a⇒b ) = a ⇒ b = a ⇒ b = h((m )a ) ⇒ h((m )b ): Therefore h is a homomorphism. Clearly h is onto, ending the proof. Theorem 3.14. Let A and be as in Theorem 3:10. If (w) = 0 whenever w ∈ m ; m ∈ (0; 1]; then A=m is isomorphic to A=. Proof. It is suNcient to show that the map h as in Theorem 3.13 is one – one. Assume that (m )a = (m )b for some a; b ∈ A. Then a ⇒ b ∈ m or b ⇒ a ∈ m . It follows from hypothesis that (a ⇒ b) = 0 or (b ⇒ a) = 0, so that a ≈ b and hence a = b . This completes the proof.
Y.B. Jun et al. / Fuzzy Sets and Systems 121 (2001) 325–332
331
Lemma 3.15. Let be a non-empty subset of an implicative algebra A = (A; V; ⇒). Then is an implicative 7lter in A if and only if the characteristic function of is a fuzzy implicative 7lter in A. Proof. Straightforward. Proposition 3.16. Let be an implicative 7lter in an implicative algebra A = (A; V; ⇒); and let a; b ∈ . Then a t b if and only if a t b; where is the characteristic function of . Proof. For any a; b ∈ we have a t b
iL a ⇒ b ∈ and iL (a ⇒ b) = 1¿0
b⇒a∈ and
(b ⇒ a) = 1¿0
iL a t b; completing the proof. Proposition 3.17. Let A = (A; V; ⇒) be as in Theorem 3:10 and let be an implicative 7lter in A. Then a = a for every a ∈ A; and A= = A= Proof. It is straightforward by using Proposition 3.16. Proposition 3.18. Let A = (A; V; ⇒) and A = (A ; V ; ⇒ ) be implicative algebras and let h : A → A be a homomorphism. If is a fuzzy implicative 7lter in A ; then ◦ h is a fuzzy implicative 7lter in A; where ◦ h is the preimage of under h. Proof. For any a ∈ A we have ( ◦ h)(V ) − ( ◦ h)(a) = (h(V )) − (h(a)) = (V ) − (h(a)) ¿ 0; because h is a homomorphism and is a fuzzy implicative #lter. Let a; b ∈ A. Then ( ◦ h)(b) − min{( ◦ h)(a); ( ◦ h)(a ⇒ b)} = (h(b)) − min{(h(a)); (h(a ⇒ b))} = (h(b)) − min{(h(a)); (h(a) ⇒ h(b))} ¿0; because h is a homomorphism and is a fuzzy implicative #lter. Hence ◦ h is a fuzzy implicative #lter in A. Theorem 3.19 (Fuzzy fundamental homomorphism theorem). Let A = (A; V; ⇒) and A = (A ; V ; ⇒ ) be implicative algebras in which conditions (9); (10) and (15) hold; h : A → A an epimorphism and a fuzzy implicative 7lter in A which is not constant. Then A=( ◦ h) is isomorphic to A =. Proof. For Proposition 3.18, A=( ◦ h) is well de#ned. De#ne a map f : A=( ◦ h) → A = by f( a ◦h ) = h(a) for all a ∈ A. Then, for every a; b ∈ A, we have f( a ◦h ⇒◦h b ◦h )
332
Y.B. Jun et al. / Fuzzy Sets and Systems 121 (2001) 325–332
= f( a ⇒ b ◦h ) = h(a ⇒ b) = h(a) ⇒ h(b) = h(a) ⇒ h(b) = f( a ◦h ) ⇒ f( b ◦h ); which shows that f is a homomorphism. Since h is onto, for each u ∈ A there exists w ∈ A such that h(w) = u; hence f( w ◦h ) = h(w) = u : Assume that a ◦h = b ◦h : Then a t◦h b does not hold, and so ( ◦ h)(a ⇒ b) = 0 or ( ◦ h)(b ⇒ a) = 0: Since h is a homomorphism, it follows that (h(a) ⇒ h(b)) = 0 or (h(b) ⇒ h(a)) = 0 so that h(a) = h(b) . This completes the proof. Acknowledgements The authors are highly grateful to the referees for their valuable comments and suggestions for improving the paper. Young Bae Jun wishes to acknowledge the #nancial support of the Korea Research Foundation, 1998 year program, Project No. 1998-015-D00006. References [1] [2] [3] [4] [5] [6]
P.S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264–269. Y.B. Jun, H.S. Kim, Fuzzy implicative #lters in implicative algebras, J. Fuzzy Math. 7 (1) (1999) 141–149. A. Monteiro, Sur les algPebres de Heyting symQetriques, Portugal. Math. 39 (1980) 1–237. H. Rasiowa, An Algebraic Approach to Non-Classical Logics, American Elsevier Publishing Co. Inc, New York, 1974. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517. L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353.