46 Y. B. JUN AND X. L. XN the section 4, weintroduce the notion of topological homomorphisms, study some properties for this notion and show that an o

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1 Scientiae Mathematicae Japonicae Online, Vol.4 (2001), ON TOPOLOGCAL BC ALGEBRAS Young Bae Jun and Xiao LongXin Received November 29, 1999 Abstract. As a continuation of [7], we introduce the notions of topological subalgebras, topological ideals and topological homomorphisms in topological BC-algebras and study some related properties. n the section 3, we investigate the compactness in a TBC-algebra X and quotient TBC-algebra X= where is a topological ideal of X. n the section 4, we introduce the notion of topological homomorphisms, study some properties for this notion and show that an open topological homomorphism f from TBC-algebra X to TBC-algebra Y gives rise to a one-to-one correspondence between the closed c-ideals of Y and the closed c-ideals of X which contains kerf. 1. ntroduction The notion of BCK-algebras was proposed by Y. ami and K. séki in n the same year, K. séki [4] introduced the notion of a BC-algebra which is a generalization of a BCKalgebra. Since then numerous mathematical papers have been written investigating the algebraic properties of the BCK/BC-algebras and their relationship with other universial structures including lattices and Boolean algebras. R. A. Alo and E. Y. Deeba [1] attempted to study the topological aspects of the BCK-structures, and they initiated the study of various topologies on BCK-algebras analogous to that which has already been studied on lattices, but no attempts have beenmade to study the topological structures making the BCK-operation continuous. n [7], Y. B. Jun et al. initiated the study of topological BC-algebras (briefly, TBC-algebras) and some properties of this structure, and gave a characterization of a TBC-algebra in terms of neighborhoods, and showed that a TBCalgebra X is Hausdorff if and only if f0g is closed in X. They also gave a filter base Ω generating a BC-topology, and maked a BC-algebra X into a TBC-algebra for which Ω is a fundamental system of neighborhoods of 0. As acontinuation of [7], we introduce the notions of topological subalgebras, topological ideals and topological homomorphisms in topological BC-algebras and study some related properties. n the section 3, we get the following results: (i) f is a topological ideal of a TBC-algebra X, then the natural projection Φ from X to X= is open and continuous. (ii) Every topological subalgebra of a compact (locally compact) TBC-algebra is compact (locally compact). (iii) Let be a topological ideal of a compact TBC-algebra X, then and X= are compact. (iv) Let be a topological c-ideal of a locally compact TBC-algebra X, thenx= is locally compact. (v) Let be a compact topological ideal of a transitive open TBC-algebra X, fi = f x j x 2 Xg be a base for topological T and x compact for each x 2 X. Then we have thatifq X= is compact, so is P =Φ (Q); and if X= is locally compact, then X is locally compact. n 2000 Mathematics Subject Classification. 06F35, 08A05, 54H10. Key words and phrases. TBC-algebra, topological subalgebra, p-semisimple TBC-algebra, topological (c-)ideal, topological homomorphism, transitive open. The first author was supported by Korea Research Foundation Grant (KRF D00003).

2 46 Y. B. JUN AND X. L. XN the section 4, weintroduce the notion of topological homomorphisms, study some properties for this notion and show that an open topological homomorphism f from TBC-algebra X to TBC-algebra Y gives rise to a one-to-one correspondence between the closed c-ideals of Y and those closed c-ideals of X which contains kerf. 2. Preliminaries n this section we include some elementary aspects that are necessary for this paper. Recall that a BC-algebra axioms for every x; y; z 2 X, () ((x Λ y) Λ (x Λ z)) Λ (z Λ y) =0; () (x Λ (x Λ y)) Λ y =0; () x Λ x =0; (V) x Λ y =0andy Λ x = 0 imply x = y. is an algebra (X; Λ; 0) of type (2, 0) satisfying the following A partial ordering» on X can be defined by x» y if and only if x Λ y =0. n a BC-algebra X, the following hold: (1) x Λ 0=x: (2) (x Λ y) Λ z =(x Λ z) Λ y: (3) 0 Λ (x Λ y) =(0Λ x) Λ (0 Λ y): (4) x Λ y = 0 implies (x Λ z) Λ (y Λ z) = 0 and (z Λ y) Λ (z Λ x) =0. (5) x Λ 0 = 0 implies x =0. A nonempty subset S of a BC-algebra X is called a BC-subalgebra of X if x Λ y 2 S whenever x; y 2 S. A non-empty subset of a BC-algebra X is called a BC-ideal of X if (i) 0 2, (ii) x Λ y 2 and y 2 imply x 2. A BC-ideal of a BC-algebra X is said to be closed if 0 Λ x 2 whenever x 2. Here we call this a c-ideal of X. 3. Compactness in topological BC-algebras n this paper we shorten the statement U is an open set containing x" to the phrase U is a neighborhood of x". Definition 3.1. (Jun et al. [7]) A topology T on a BC-algebra X is said to be a BCtopology, and X, furnished with T, is called a topological BC-algebra (briefly, TBC-algebra) if (x; y) 7! x Λ y is continuous from X X, furnished with the cartesian product topology defined by T, to X. Moreover if X is a p-semisimple BC-algebra, we say that X, furnished with T,is a p-semisimple TBC-algebra. Example 3.2. (Jun et al. [7]) (1) A BC-algebra with discrete or indiscrete topology is a TBC-algebra. (2) Consider a BC-algebra X = f0;a;b;c;dg with Cayley table (Table 1) and Hasse diagram (Figure 1):

3 ON TOPOLOGCAL BC ALGEBRAS 47 Λ 0 a b c d 0 a b c d d a 0 0 a d b ffl b b 0 b d a ffl ffl. c c c c 0 d d d d d 0 ffl ffl 0 d Table 1 Figure 1 Note that T := f;;x; fbg; fcg; fdg; f0;ag; fb; cg; fb; dg; fc; dg; f0;a;bg; f0;a;cg; f0;a;dg; fb; c; dg; f0; a; b; cg; f0; a; b; dg; f0; a; c; dgg is a BC-topology. By routine calculations we know thatx furnished with T is a TBC-algebra. We first give a characterization of a TBC-algebra in terms of neighborhoods. Lemma 3.3. (Jun et al. [7]) Let T be a BC-topology for a BC-algebra X. Then X, furnished with T, is a TBC-algebra if and only if for each x and y in X and each neighborhood W of x Λ y there are neighborhoods U of x and V of y such thatu Λ V W. Lemma 3.4. (Jun et al. [7]) Let X be a TBC-algebra. f A is both an open subset in topological space X and an ideal of BC-algebra X; then it is a closed subset in topological space X. Lemma 3.5. (Jun et al. [7]) Let X be a TBC-algebra. Then f0g is closed subset if and only if X is Hausdorff in topological space X. Definition 3.6. A nonempty subset S of a TBC-algebra X is called a topological subalgebra of X if (i) S is a BC-subalgebra of X considered as an algebraic BC-algebra, (ii) S is a closed set in the topological space X. Definition 3.7. A nonempty subset of a TBC-algebra X is called a topological ideal of X if (i) is a BC-ideal of X considered as an algebraic BC-algebra, (ii) is open in the topological space X. Definition 3.8. A nonempty subset of a TBC-algebra X is called a topological c-ideal of X if (i) is a c-ideal of X considered as an algebraic BC-algebra, (ii) is open in the topological space X. Example 3.9. Let X = f0;a;b;c;dg be a TBC-algebra as in Example 3.2(2). By routine calculations we know that A = f0;cg is a topological subalgebra which is not a topological ideal, and B = f0;ag is a topological ideal. Proposition Every topological ideal of a TBC-algebra X is a closed subset of X considered as a topological space. Proof. Follows from Lemma 3.4. Λ Proposition Let X be a TBC-algebra and a BC-ideal of X considered as an algebraic BC-algebra. Define a map Φ : X! X= by Φ (x) = x for each x 2 X where x is the equivalence class containing x. Then the set T X= := fo X=jΦ 1 (O) is open in Xg

4 48 Y. B. JUN AND X. L. XN is a topology on X=, which is called the quotient topology on X=. Proof. Straightforward. Λ Proposition f is a topological ideal of a TBC-algebra X, then the mapping Φ as in Proposition 3.11 is open and continuous. Proof. Clearly Φ is continuous. n order to prove Φ is open, let W be an open set in X and W 0 =Φ (W ). We shall show that W 0 is open in X= or equivalently Φ 1 (W 0 )isopen in X. Let y 2 Φ 1 (W 0 ). Then Φ (y) 2 W 0 =Φ (W ); whence Φ (y) = Φ (w) for some w 2 W. t follows that y Λ w 2 and w Λ y 2. Note that f : X X! X, (x; y) 7! x Λ y, is continuous. Then B := f(a; b) 2 X Xja Λ b 2 S g = f(a; b) 2 X Xjf(a; b) 2 g = f 1 () is open in X X since is open. Hence B = (U i V i ) for some open sets U i and V i in X where Λ is an index set and i 2 Λ. t follows from y Λ w 2 that there exists i 0 2 Λsuch that y 2 U i0, w 2 V i0 and U i0 Λ V i0. Similarly w Λ y 2 implies that there exists j 0 2 Λ such that w 2 U j0, y 2 V j0 and U j0 Λ V j0. Thus y 2 U i0 V j0 and w 2 U j0 V i0. Note that (U i0 V j0 ) Λ (U j0 V i0 ) U i0 Λ V i0 and (U j0 V i0 ) Λ (U i0 V j0 ) U j0 Λ V j0. Let x 2 G where G = U i0 V j0. t follows that x Λ w 2 and w Λ x 2. Hence x = w =Φ (w) 2 Φ (W )=W 0 ; and so G Φ 1 (W 0 ). Note that y 2 G Φ 1 (W 0 )we have Φ 1 (W 0 )isopeninx. Λ Theorem f is a topological ideal of a TBC-algebra X with topology T X, then X= is a TBC-algebra with topology T X=. Proof. t is sufficient toprove that( x ; y ) 7! x Λ y is continuous. Let x; y 2 X and let W be a neighborhood of x Λ y. Then Φ 1 (W )isopeninx and x Λ y 2 Φ 1 (W ). Since X is a TBC-algebra, there exists neighborhoods U 0 of x and V 0 of y such thatu 0 Λ V 0 Φ 1 (W ) by Lemma 3.3. Put U := Φ (U 0 )andv := Φ (V 0 ): Then U and V are open in X= since Φ is open. Note that x 2 U; y 2 V and i2λ U Λ V = fφ (u) Λ Φ (v)ju 2 U 0 and v 2 V 0 g = fφ (u Λ v)ju 2 U 0 and v 2 V 0 g =Φ (U 0 Λ V 0 ) W so from Lemma 3.3 that ( x ; y ) 7! x Λ y is continuous. Theorem Let X be a TBC-algebra. Then (i) f X is compact (resp. locally compact), then every topological subalgebra S of X is compact (resp. locally compact). (ii) f X is compact and is a topological ideal, then and X= are compact. (iii) f X is locally compact, then every topological ideal is locally compact. (iv) f X is locally compact and is topological c-ideal, then X= is locally compact. Proof. (i) Note that S is closed so that S is compact if X is compact. f X is locally compact, then 8x 2 S 9 a neighborhood G of x whose closure is compact in X. Then U := G S is a neighborhood of x in S and U G S = G S. Since G S is compact in S, it follows that U is compact in S. Hence S is locally compact. (ii) Let X be compact and a topological ideal. Then is closed by Proposition Hence is compact. Note that Φ is a continuous mapping from X onto X=. Therefore X= is compact. (iii) t is similar to the proof of (i). (iv) Assume that X is locally compact and is a topological c-ideal. Let a 2 X, A = Φ (a) and let U be a neighborhood of a in X such that U is compact. Since is Λ

5 ON TOPOLOGCAL BC ALGEBRAS 49 a c-ideal of X, we have Φ () = f 0 g which is an open set in X= because is open in X and Φ is open. Note that f 0 g is a BC-ideal of X= so that f 0 g is a closed set in X= by Lemma 3.4. t follows from Lemma 3.5 that X= is Hausdorff. Since U is compact, the continuous image Φ (U) is compact, and hence Φ (U) is closed in X=. From Φ (U) Φ (U) it follows that Φ (U) Φ (U) so that Φ (U) is compact. On the other hand, put U Λ = f x j x U 6= ;g. Since a 2 a U = A U, we geta 2 U Λ. Now we claim that U Λ =Φ (U) and then U Λ is a neighborhood of A. Let x 2 U Λ. Then x U 6= ; and so 9y 2 x U: Hence x = y 2 Φ (U) and therefore U Λ Φ (U). Conversely let x =Φ (x) 2 Φ (U). Then x 2 U and thus x 2 x U, i.e., x U 6= ; which means that x 2 U Λ. Consequently U Λ =Φ (U) and U Λ = Φ (U) is a compact set in X=. Therefore X= is locally compact. This completes the proof. Λ For a fixed element a of a TBC-algebra X, define a self-map f a : X! X by f a (x) =aλx for all x 2 X. Definition A TBC-algebra X is said to be transitive open if for each a 2 X the self-map f a is open and continuous. Lemma Let U be an open set in a transitive open TBC-algebra X and let a 2 X. Then (i) a Λ U is open. (ii) f a 1 (U) =fx 2 X j a Λ x 2 Ug is open. (iii) A Λ U is open for every subset A of X. Proof. Note that f a (U) =aλu, f a 1 (U) =V and A Λ U = [ a2a (a Λ U): Hence we get the desired results. Λ Theorem Every p-semisimple TBC-algebra X is transitive open. Proof. Let a 2 X. We shall prove that f a is open and continuous. Let x 2 X and let W beaneighborhood of f a (x) =a Λ x. Then there exist neighborhoods U and V of a and x, respectively,suchthatuλv W by Lemma 3.3. t follows that f a (V )=aλv U ΛV W; which means that f a is continuous. Now letg be an open set in X. We claim that f a (G) is open. Let x 2 f a (G). Then x = f a (u) =a Λ u for some u 2 G, andsou = a Λ (a Λ u) =aλx since X is p-semisimple. t follows that a Λ x 2 G so that there exist neighborhoods P and Q of a and x, respectively, such that P Λ Q G, which implies f a (Q) =a Λ Q P Λ Q G. Thus f a (q) 2 G for all q 2 Q, and so q = a Λ (a Λ q) =a Λ f a (q) 2 f a (G), i.e., Q f a (G). Therefore f a ia an open map, ending the proof. Λ Proposition Let 4 0 be a collection of closed subsets of a TBC-algebra X having the finite intersection property. Then there is a maximal collection 4 0 of closed subsets of X having the finite intersection property suchthat Proof. Let F = f4 j 4 is a collection of closed subsets of X having the finite intersection property and 4 0 4g. We shall prove that F has a maximal element with respect to the partial order " on F. Let ::: 4 n ::: be a chain in F and let 4 = [ 1 i=1 4 i. Then 4 0 4: We also can show that4 has the finite intersection property. n fact, let U 1 ;U 2 ; ;U n 24; then there is an integer m such that U 1 ;U 2 ; ;U n 24 m and so n i=1 U i 6= ; since 4 m 2F. t follows that 42F and hence it is a upper bound of the chain f4 i : i =1; 2;::: ;n;:::g: By Zorn's Lemma, F has a maximal element 4 0,itis the desired one.

6 50 Y. B. JUN AND X. L. XN Theorem Let X be a transitive open TBC-algebra and a compact topological ideal of X. Let fi := f x j x 2 Xg be a base for topology T ; where x is compact for each x 2 X: Then (i) f Q X= is compact, then so is P =Φ 1 (Q). (ii) f X= is compact (resp. locally compact), then X is compact (resp. locally compact). Proof. (i) By Proposition 3.18, we only need to prove the theorem for a maximal collection 4 of the closed subsets in the subspace P having the finite intersection property. We shall show that it also has non-empty intersection. n the space Q, consider the collection 4 Λ of all sets of the form Φ (F )suchthatf 24. Since 4 has the finite intersection property, so is the system 4 Λ. Let 4 Λ denote the collection fφ (F ) Q j F 24g: Then 4 Λ is a collection of closed subsets in the space Q having the finite intersection property. By hypothesis, Q is compact, and so there exist a common point A 2 Φ (F ) for each F 2 4. Now let U be an arbitrary neighborhood of zero in X and U Λ a set of all equivalence classes contained in A Λ U, i.e.,u Λ = f a 2 X= j a A Λ Ug. By Lemma 3.16, A Λ U is open in X and hence Φ (A Λ U) is open in X= since Φ is an open mapping. Clearly U Λ Φ (A Λ U): S On the other hand, by hypothesis, f x j x 2 Xg is a base for topology T and so A Λ U S = xi for some index set Λ. Thus for i2λ any a =Φ (a) 2 Φ (AΛU); we have a 2 AΛU = xi and hence a 2 for some i xi Λ. Therefore a = A Λ U and so xi 0 a 2 U Λ ; which shows that Φ (A Λ U) =U Λ : Hence U Λ is open in X=. Since A A Λ U; we get A 2 U Λ and consequently U Λ meets every set of the system 4 Λ : We claim that A Λ U meets every set of 4: Let F 24. Then Φ (F ) 24 Λ and so U Λ Φ (F ) 6= ;. Thus we can find a 2 F such that a 2 Φ (F ) and a 2 U Λ : Note that a 2 U Λ implies a A Λ U and hence a 2 A Λ U: Therefore (A Λ U) F 6= ;: t follows that A Λ U meets every set of 4. Put FU 1 := fx 2 X j x Λ U 2 F for some u 2 Ug. Since (A Λ U) F 6= ;, wehave FU 1 A 6= ; for each F 24. ndeed let a 2 A Λ U F; then there is x 2 A and u 2 U such that a = x Λ u 2 A Λ U and a = x Λ u 2 F: Hence x 2 FU 1 A. Now let4 0 = ffu 1 A j F 24;U is neighborhood of the zero in Xg. From the above factwehave that the system 4 0 also has the finite intersection property. n fact, if F 1 U 1 1 A and F 2 U 1 2 A are in 4 0, then taking F = F 1 F 2 and U = U 1 U 2 we have FU 1 A 24 0 since F 24by the maximal property of4. Note that i2λ ;6= FU 1 A (F 1 U 1 1 A) (F 2 U 1 2 A): Therefore 4 0 has the finite intersection property. Since x is compact for each x 2 X by hypothesis, A (= x0 for some x 0 2 X) is compact in X and so there exists a 2 FU 1 A for each FU 1 A For any neighborhood V of zero in X, we have thata Λ V and (aλv )ΛU areopenby Lemma Hence FU 1 A and aλv have nonempty intersection and consequently FU 1 A and a Λ V have nonempty intersection. t follows that F meets the open set (a Λ V ) Λ U. By Lemma 3.3, we have that for any neighborhood W a of a, there exist neiborhoods V a and U 0 of a and 0, respectively, suchthatv a Λ U 0 W a. Put V 0 = fx 2 X j a Λ x 2 V a g; then V 0 is an open set by Lemma 3.16 and thus V 0 is a neighborhoot of zero. Since a Λ V 0 V a ; we have (a Λ V 0 ) Λ U 0 V a Λ U 0 W a. By the above argument wehave ((a Λ V 0 ) Λ U 0 ) F 6= ;; and hence W a F 6= ;: t follows that a 2 F and so a 2 F since F is closed. So a is common to all the sets of the system 4, which shows that P is compact. (ii) t follows from (i) that if X= is compact, so is X. Now letx= be locally compact and a 2 X. Then x 2 X= and thus there is a neighborhood U of x such that it's closure U is compact. Put P =Φ 1 (U): Then P is a neighborhood of x. Noticing that P Φ 1 (U);

7 ON TOPOLOGCAL BC ALGEBRAS 51 we have P Φ 1 (U) sinceφ 1 (U) is closed. Since Φ 1 (U) is compact by (i) and since P is a closed subset of Φ 1 (U), therefore P is compact. Hence X is locally compact. Λ Corollary Let X be a p-semisimple TBC-algebra, a compact topological c-ideal and the system fi := f x j x 2 Xg a base for the topology T. f Q X= is compact, then P =Φ 1 (Q) is compact in X. n particular, if X= is compact (locally compact), then X is compact (locally compact). Proof. By Theorem 3.17, X is transitive open and then for each a 2 X, f a is continuous and open. Since X is a p-semisimple, we canseethatf a is bijective. ndeed, if f a (x) =f a (y); then a Λ x = a Λ y and so x = a Λ (a Λ x) = a Λ (a Λ y) =y; which implies f a is injective. Moerover for each x 2 X, f a (a Λ x) = a Λ (a Λ x) = x and so f a is surjective. Therefore f a is a homeomorphism. Now we claim that a = f a (), for each a 2 X. Let x 2 f a (): Then x = f a (y) for some y 2. Since is c-ideal, x Λ a = (a Λ y) Λ a = 0 Λ y 2. Note that a Λ x = a Λ (a Λ y) = y 2 : Then we have x 2 a. Conversely if x 2 a, then x Λ a 2 and a Λ x 2. Thus there exists y 2 such that y = a Λ x. Hence x = a Λ (a Λ x) =aλy = f a (y) 2 f a (): Therefore a = f a () and hence a is compact. This shows that X satisfies the hypothesis of Theorem 3.19 and so Corollary 3.20 holds. Λ 4. Topological homomorphisms in topological BC-algebras Definition 4.1. Let X and Y be two TBC-algebras. A mapping g : X! Y is called a topological homomorphism if (i) g is a homomorphism from X to Y as BC-algebras, (ii) g is a continuous mapping in the topological spaces. A topological homomorphism g from X to Y is said to be open if g is an open mapping of the topological spaces. Definition 4.2. Let X and Y be two TBC-algebras. A mapping f : X! Y is called a topological isomorphism if (i) f is an isomorphism of the BC-algebras, (ii) f is a homeomorphism of the topological spaces. f X = Y, the topological isomorphism of X into Y is called a topological automorphism. Two TBC-algebras are said to be topological isomophic if there exists a topological isomorphism of X into Y. Proposition 4.3. Let X and Y be transitive open TBC-algebras and g be a homomorphism of a BC-algebra X into BC-algebra Y. (i) f for each neighborhood U Λ of 0 Λ in Y, there exists a neighborhood U of 0 in X such that g(u) U Λ ; then g is continuous. (ii) f for each neighborhood V of 0 in X, there exists a neighborhood V Λ of 0 Λ in Y such thatg(v ) V Λ ; then g is open. Proof. (i) Let a 2 X and g(a) =a Λ in Y. Assume that W Λ is an arbitrary neighborhood of a Λ. Then U Λ = fy Λ 2 Y j a Λ Λ y Λ 2 W Λ g is an open set by Lemma 3.16 and 0 Λ 2 U Λ, i.e., U Λ is a neighborhood of 0 Λ. Hence by hypothesis, there exists a neighborhood U of 0 in X such thatg(u) U Λ. Thus the open set a Λ U contains a and g(a Λ U) =g(a) Λ g(u) = a Λ Λ g(u) a Λ Λ U Λ W Λ. Therefore g is continuous. (ii) Let a 2 X and g(a) =a Λ. Assume that W is an arbitrary neighborhood of a. Then V = fx 2 X j a Λ x 2 W g is an open set containing 0 by Lemma By hypothesis there exists a neighborhood V Λ of 0 Λ in Y such that g(v ) V Λ : Note that a Λ Λ V Λ is an open set

8 52 Y. B. JUN AND X. L. XN containing a Λ and a Λ Λ V Λ g(a) Λ g(v )=g(a Λ V ) g(w ): t follows that g(w )isopen and so g is an open mapping. Λ Theorem 4.4. Let X be a TBC- algebra and a topological c-ideal of X. Then the natural projection Φ is an open topological homomorphism of X onto X= and f 0 g is an open set in X=. Proof. Obviouly Φ is a homomorphism of BC-algebras, and it is open and continuous by Proposition Hence Φ is an open topological homomorphism of X onto X=. Since is a topological c-ideal, we have Φ () =f 0 g and so f 0 g is open in X=. Λ The following theorem provides the converse of Theorem 4.4. Theorem 4.5. Let X and Y be two TBC-algebras. Assume g is an open topological homomorphism of X onto Y having kerg =, and the zero element f0 0 g of Y is an open set in Y. Then we have (i) is a topological ideal of a TBC- algebra X. (ii) if we define f : X=! Y by f( x )=g(x), then f is a topological isomorphism, i.e., X= is topological isomorphic to Y. Proof. (i) t is easy to see that is a BC-ideal of X. Moreover since g is continuous and f0 0 g is open in Y,wehave that = kerg = g 1 (0 0 )isopeninx. Therefore is a topological ideal of X. (ii) Note that f( x Λ y ) = f( xλy ) = g(x Λ y) = g(x) Λ g(y) = f( x ) Λ f( y ) for all x ; y 2 X=. Thus f is a homomorphism of the BC-algebra X= onto Y. Next for each y 2 Y; since g is onto, there is x 2 X such that g(x) = y. Hence f( x ) = g(x) = y, which implies that f is onto. Finally if f( x )=f( y ); then g(x) =g(y). t follows that g(x Λ y) = g(x) Λ g(y) = 0 0 and g(y Λ x) = g(y) Λ g(x) = 0 0 and hence x Λ y 2 kerg = and y Λ x 2 kerg =. Hence x = y. This shows that f is injective. Combining the above arguments we have that f is an isomorphism of the BC-algebra X= onto the BCalgebra Y. Finally we show that f is a homeomorphism of the topological space X= into the topological space Y. We shall verify that both f and f 1 are continuous. Let A = a 2 X= and f(a) =a 0 : Assume that U Λ is an arbitrary neyghborhood of a 0. Then g(a) =f( a )=f(a) =a 0. Since g is continuous, there exists a neighborhood V of a such that g(v ) U Λ. Denote by V Λ the neighborhood of A in X= consisting of all x with x 2 V. Then f(v Λ )=ff( x ) j x 2 V Λ g = fg(x) j x 2 V g = g(v ) U Λ, and hence f is continuous. Next let a 0 2 Y and f 1 (a 0 )=A: Denote by U Λ an arbitrary neighborhood of A in the space X=. Then U := Φ 1 (U Λ ) is an open set in X. Let a be an element in U such that A = a. Then g(a) =f( a )=f(a) =a 0. Since g is open, there exists a neighborhood V Λ of a 0 such thatv Λ g(u). Now we claim that f 1 (V Λ ) U Λ. ndeed, for each f 1 (x) 2 f 1 (V Λ ), we have x 2 V Λ and so x = g(u) for some u 2 U. Thus f 1 (x) =f 1 (g(u)) = f 1 (f( u )) = u 2 U Λ which shows that f 1 (V Λ ) U Λ. Therefore f 1 is continuous, ending the proof. Λ >From Theorem 4.5 we have the following corollary. Corollary 4.6. Let X and Y be two TBC-algebras with f0 0 g being open set in Y. f g is an open topological homomorphism of X onto Y having kerg = f0g, then g is a topological isomorphism and thus X and Y are topological isomorphic. Theorem 4.7. Let X and Y be two TBC-algebras and f an open topological homomorphism of X onto Y. Denote by N 0 the kernel of f. Then f gives rise to a one-to-one correspondence between the closed c-ideals of Y and the closed c-ideals of X which contain N 0 as follows:

9 ON TOPOLOGCAL BC ALGEBRAS 53 (i) if N Λ is a closed c-ideal of Y, then the closed c-ideal N of X corresponding to it is just the inverse image N = f 1 (N Λ ); (ii) if N is a closed c-ideal of X containing N 0, then the closed c-ideal N Λ of Y corresponding to it is just the image N Λ = f(n). (iii) The two correspondences thus defined are mutually inverse to one another. Moreover topological ideals correspond to one another. Finally if N and N Λ are two topological ideals corresponding to one another in this fashion, then the quotient TBC-algebras X=N and Y=N Λ are topological isomorphic. Proof. We consider first the correspondence of N to N Λ. Let N Λ be a closed c-ideal of Y. Then N = f 1 (N Λ ) is also closed and contains N 0 since f is continuous. Moreover for each y; x Λ y 2 N, wehave f(y);f(x) Λ f(y) =f(x Λ y) 2 N Λ. Since N Λ is an ideal of BC-algebra Y; it follows that f(x) 2 N Λ and hence x 2 f 1 (N Λ )=N. This shows that N is an ideal of BC-algebra X. Finally let x 2 N, then f(x) 2 N Λ and thus f(0 Λ x) = f(0) Λ f(x) = 0 0 Λ f(x) 2 N Λ since N Λ is a c-ideal, where 0 0 is the zero element in Y. Therefore 0 Λ x 2 f 1 (N Λ ) = N and N is a closed c-ideal of X. Now we show that the topological ideals correspond to one another and the quotient TBC-algebras are topological isomorphic. f N Λ is a topological ideal of Y andifφ N Λ denotes the natural projection of Y onto Y=N Λ,thenh:= Φ N Λ ffi f is an open topological homomorphism of X onto Y=N Λ, with kernel N, andfn Λ g =Φ N Λ(N Λ ) is an open set in Y=N Λ. By Theorem 4.5, N = kerh is a topological ideal of X and X=N is topological isomorphic to Y=N Λ. Conversely let N be a closed c-ideal of X containing N 0 and consider the corresponence of N Λ to N where N Λ = f(n) and N 0 N. We first show thattheinverse image of N Λ under the mapping f coincides with N. ndeed if f(a) 2 N Λ, then there exists b 2 N such that f(a) =f(b). Thus f(a Λ b) =f(a) Λ f(b) =0 0,i.e.,aΛb2N 0 N so that a 2 N since N is a BC-ideal and b 2 N. Hence f 1 (N Λ ) N and so f 1 (N Λ )=N. From this fact it follows that f(x n N) =Y n N Λ. Since f is open and X n N is an open set, Y n N Λ is also open and hence the set N Λ is closed in Y. Finally we showthatn Λ is a c-ideal of Y. Let y 0 ;x 0 Λ y 0 2 N Λ, then there exist x; y 2 X such that f(x) =x 0 and f(y) =y 0. t follows that f(y) =y 0 2 N Λ and f(x Λ y) =f(x) Λ f(y) =x 0 Λ y 0 2 N Λ so that y; x Λ y 2 N. Since N is a BC-ideal of X, wehave x 2 N, which shows that f(x) =x 0 2 N Λ and thus N Λ is an ideal of BC-algebra Y. Moreover for each x 0 2 N Λ ; there exists x 2 N such thatf(x) =x 0 and hence 0 0 Λ x 0 = f(0) Λ f(x) =f(0 Λ x) 2 f(n) =N Λ. t follows that N Λ is a c-ideal of BC-algebra Y. Therefore N Λ is a closed c-ideal of Y, ending the proof. Λ References [1] R. A. Alo and E. Y. Deeba, Topologies of BCK-algebras, Math. Japon. 31(6) (1986), [2] T. H. Choe, Zero-dimensional compact associative distributive universial algebras, Proc. Amer. Math. Soc. 42(2) (1974), [3] G. Gierz and A. Stralka, The Zariski topology for distributive lattices, Rocky Mountain J. Math. 17(2) (1987), [4] K. séki, An algebrarelated with a propositional calculus, Proc. Japan Acad. 42 (1966), [5] K. séki, On BC-algebras, Math. Seminar Notes 8 (1980), [6] K. séki, Some examples of BC-algebras, Math. Seminar Notes 8 (1980), [7] Y. B. Jun, X. L. Xin and D. S. Lee, On topological BC-algebras, nform. Sci. 116 (1999), [8] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoonsa, Seoul, Korea, [9] L. S. Pontryagin, Topological groups, Gordon and Breach, Science Publishers, nc., [10] S. L. Warner, Topological fields, North-Holland, Amsterdam-New York-Oxford-Tokyo, Department of Mathematics Education, Gyeongsang National University, Chinju , Korea ybjun@nongae.gsnu.ac.kr

10 54 Y. B. JUN AND X. L. XN Department of Mathematics, Northwest University, Xian , P. R. China