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The socle and fuzzy socle of a BCI-algebra
sets and systems ELSEVIER
Fuzzy Sets and Systems 72 (1995) 197 204
The socle and fuzzy socle of a BCI-algebra C.S. Hoo Department of Mathematical Sciences, University of Alberta, Edmonton. Alberta, Canada T6G 2GI
Abstract We introduce the notion of orthogonal complement or annihilator in general BCI-algebras, and the corresponding notion in the fuzzy ideals. Various properties are proved, and the related notions of the socle and fuzzy socle of a BCI-algebra are investigated. It is shown that a non-zero weakly commutative, artinian BCI-algebra has non-zero socle. Keywords: Fuzzy essential closed ideal; Fuzzy socle; Artinian; Finitely cogenerated
1. Introduction In [12] the notion of an essential closed ideal of a BCI-algebra was introduced and some ideals were identified as having this property. In [10] we studied the situation further, particularly in MValgebras. In this paper, we introduce such notions here that generalise those of [10]. Also some results of [6] are generalized here. We assume knowledge of BCI and BCK-algebras and refer the reader to [3,6-8, 14, 15] for full details. Also the basic ideas of fuzzy logic are assumed and we refer the reader to [9, 10, 13, 17, 18] for complete details. However, we will review briefly here the concepts of BCI- and BCK-algebras and fuzzy logic. A BCI-algebra is a non-empty set X with a constant 0 and a binary operation • satisfying the axioms (1) { ( x * y ) * ( x . z ) } * ( z . y ) = 0,(2) {x* (x*y)}.y=0, (3) x . x = 0 , (4) x . y = 0 and y.x=0 imply that x = y , and (5) x * 0 = 0 im-
plies that x = 0. We define a partial ordering ~< on X by putting x ~< y if and only if x * y = 0. If further every x satisfies 0 ~< x then X is a BCK-algebra. If a BCK-algebra satisfies the axiom x * ( x * y ) = y * (y * x) then it is called commutative. In this case x * (x * y) = y * (y * x) is the greatest lower bound x ^ y of x and y. We shall write [ x l , x 2 . . . . . x,] for ( . . . ( ( ( x l * x 2 ) * x 3 ) * ) * . . . ) * x , . We define Ix, y] ° = x , and for n > 0 , Ix, y]" = [ x , y . . . . . y], where y occurs n times. If X and Y are BCI-algebras, a homomorphism f : X ~ Y means a function such that f ( x l * x2) = f ( x l ) * f(x2) for all xl, x2 in X. An ideal of a BCI-algebra is a subset I containing 0 and such that if x * y and y are in I then so is x. We say that an ideal I is closed if whenever x belongs to I then so does 0* x. All ideals of a BCKalgebra are automatically closed. A closed ideal I is an essential closed ideal if for every non-zero closed ideal J we have l n J v~ {0} (see [10, 12]). A subalgebra is a subset containing 0 and closed under the operation *.
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C.S. Hoo / Fuzzy Sets and Systems 72 (1995) 197-204
In general, an ideal is not a subalgebra. However, it is a subalgebra if and only if it is closed. The p-radical of a BCI-algebra X is X÷ = {x • X[ 0 ~< x}. This is an ideal of X and is clearly closed. In fact X+ is a BCI-subalgebra of X. A BCIalgebra X is p-semisimple if X+ = {0}. It has a natural abelian group structure given by x+y=x*(0.y), -x=0*x, and 0 is the group identity. The p-semisimple part of a BCI-algebra X is X = {x I 0 • (0 • x) = x}. This is a subalgebra of X, in fact a p-semisimple BCI-subalgebra of X. It is in general not an ideal of X. For a general BCIalgebra X we have X c~ X+ = {0}. Observe that the partial ordering defined above is trivial for a p-semisimple BCI-algebra, and that X = {x • X ly ~< x =~ y = x}. An ideal I is positive if I c X+. A positive ideal I is an essential positive ideal if for every non-zero positive ideal J we have I n J # {0}. Clearly a positive ideal is a closed ideal. We shall frequently use the following three identities for computations: (1) ( x * y ) * z = (x*z)*y, (2) O * ( x * y ) = ( 0 * x ) * (0 * y), and (3) y * {y * (y * x)} = y * x. Also useful is [11, L e m m a 5.5]: in any BCI-algebra we have the identity [(a * b) • c] • [a * (b * c)] = (0 * c) * c. If A is a non-empty subset of a BCI-algebra X, let (A> denote the set {x • XI [x, aa ..... a,] = 0 for some aa,a2 ..... a. in A}. If A ~ X + # O, then (A> is an ideal of X, namely the ideal generated by A, i.e. it is the smallest ideal of X containing A. We shall denote ({a}> by w {0}. In general, these ideals are not closed. To obtain the closed ideal generated by A, let O . A = { O . a l a • A } and 0 * ( 0 * A ) = { 0 . ( 0 . a) I a • A}. Then if A c~X+ # 0, the closed ideal generated by A is (A w (0 • A) u {0 • (0 • A)} > while if A n X + = 0, we adjoint 0 to this set. These are the smallest closed ideals containing A (see [8]). In general, denote the closed ideal generated by A by [A]. Observe that if A is a subalgebra of X or if A c X+, then [A] = . Given a closed ideal I o f a BCI-algebra X, we can define a quotient BCI-algebra X / I as follows. We define an equivalence relation ~ on X by putting x ~ y if and only ifx * y and y * x are in I. It is easily checked that ~ is an equivalence relation. Let X / I denote the set of all equivalence classes. If we denote the equivalence class containing x by :~, then
the BCI-algebra operation on X/I is defined by x • y = x • y. The zero is 0. If X is BCK, then so is X/I. The (closed) ideals J of X / I are precisely the (closed) ideals J of X containing I such that J/I = J, where J/I = { x l x • J}. A BCK-algebra is bounded if it has a largest element which is usually denoted by 1. In a bounded commutative BCK-algebra, every two elements x and y have a least upper bound given by xvy=l*{(l*x)^(l*y)}. A bounded commutative BCK-algebra is a distributive lattice with the operations v, ^, and 0 and 1 are the universal bounds. The category of bounded commutative BCK-algebras is naturally equivalent to the category of MV-algebras (see [1-4, 6]). All our results for BCI-algebras are, of course, true also for BCKand MV-algebras. We now review some fuzzy logic concepts, referring the reader to [9, 10, 13, 17, 18] for further details. Let [0, 1] denote the unit closed interval. If a and b are in [0, 1], let a ^ b be the minimum of a and b, and let a v b be the m a x i m u m of a and b. A fuzzy subset of a BCI-algebra X is a function /~:X ~ [0, 1]. It is a fuzzy ideal if #(0) ~> #(x) and /~(x) >//~(x * y) ^ p(y) for all x, y in X. Given a fuzzy subset p and t • [0, 1], let p, = {x • X lp(x) >~ t}. This could be empty. It is shown in [18] Theorem 3 that p is a fuzzy ideal if and only if each P, is either empty or is an ideal. These ideals Pt are the level ideals of p. Thus, given a fuzzy ideal # we have an ideal X~, = {x • X lp(x) = p(0)} of X. If A is a non-empty subset of X, we have the characteristic function XA of A. The characteristic function Zx of an ideal I is a fuzzy ideal. If p is a fuzzy ideal with p(0) = 1, we have Zxu ~< p. In general we have Xz, = I if I is an ideal. A fuzzy ideal p is normalised if p(0) = 1. Let ~ ( X ) denote the set of all normalised fuzzy ideals of X. Given a fuzzy ideal # of X, its normalisation /~ is given by ~(x) = p(x) + 1 - p(0). Then/~ is normalised fuzzy ideal. We can define a partial ordering ~< on the set of all fuzzy ideals of X by a ~< fl if and only if a(x) ~< fl(x) for all x • X. Then for each fuzzy ideal p of X we have p ~~ #(x) for all x • X (see [10]). If p is fuzzy closed then X, is a closed ideal, and if I is a closed ideal then Zt is a fuzzy closed ideal.
C.S. Hoo / Fuzzy Sets and Systems 72 (1995) 197-204
Let ~ >~ 0 be a real number. If t e [0, 1] let t" be the positive root in case ct < 1. If p is a fuzzy ideal of X, then /~':X ~ [0, 1] given by # ' ( x ) = (~t(x))" is a fuzzy ideal, and Xu~ = X, if ~ > 0 (see [10]). We see that # is fuzzy closed if and only if so is /t ~. A fuzzy closed ideal p is a fuzzy essential closed ideal if for all fuzzy closed ideals ct of X with X~ 4: {0} we have X~, n X, ~ {0}. Thus # is a fuzzy essential closed ideal if and only if X~, is an essential closed ideal, and I is an essential closed ideal if and only if Z~ is a fuzzy essential closed ideal (see [10, L e m m a 3.2]). Clearly if # is fuzzy essential closed, then so is/~. O b s e r v e that # is fuzzy essential closed if and only if so is p~ for all ~ ~> 0. A useful result is L e m m a 3.3 of [10] which we re-state as the following result. L e m m a 1.1. Suppose that la is a f u z z y essential closed ideal of X and c~ is a fuzzy closed ideal of X such that It ~ ~ and kt(O) = ~(0). Then cc is a fuzzy essential closed ideal of X.
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The following result is easily established. L e m m a 2.3. (i) I f A is a non-empty subset of Y, then A r i A ± = {0}. (ii) l f A c B then B ± C A ±. (iii) I f A is a non-empty subset of Y, then A ± = n,~A{a} ±. (iv) I f Y+ = A then A ± = {0}.
(v) {0}±= Y+. (vi) I f A contains a non-zero positive element of Y, then A ± is a proper ideal of Y+. (vii) I f I is an essential closed (essential positive) ideal of Y then I ± = {0}. Lemma2.4. If xe? ( I 7 ) ± = Y+.
then {x} ± = Y+ and hence
Proof. By [8] Proposition 2.25 we have x * y = x for all x e I7 and all y e Y+. Hence if x e I7 we have {x} ± = Y+. This means that (17)± = Nx~¢ {x} ± = Y+. [] Corollary 2.5. l f A c Y, then A ± = (A - 17)±.
2. Annihilators in BCl-algebras In the rest of the p a p e r Y is a general BCIalgebra and /~ is a general fuzzy ideal of Y. Any special properties possessed by them will be explicitly stated.
Definition 2.1. If A is a n o n - e m p t y subset of Y, let AJ-={yeY+la=a*y for all a e A } . Define ( 0 ) ± = Y+. R e m a r k . If Y is a c o m m u t a t i v e BCI-algebra or an MV-algebra, this gives the same definition as in [1,6].
Theorem 2.2. I f A is a subset of Y, then A j- is a positive ideal of Y. Proof. We need only consider the case A 4: 0. Clearly A ± c Y÷. Obviously, 0 e A ±. Suppose that x , y, y e A ±. Then a * y = a and a * (x * y) = a for all a e A . N o w a * x = ( a * y ) * x = ( a , x ) , y < < , a * ( x * y ) = a by [11, L e m m a 5.5]. []
Proof. We need only consider the case A ~ I7. Then A ± = N=~A{a} ± = n,~A_~{a} ± = ( A - 17)±. [] Theorem 2.6. I f {Aa l2 e A} is a family of subsets of Y, then (Ua~aAa) ± = Na~A A~. Proof. Since for each 2 e A we have A~ c U),~A A~,, it follows that (U,tEAA;~)±cN;~6AA~. N o w if xeN~AA~, then x e A ~ = N r ~ A ~ { y } ± for all 2 e A, and hence x e {y}, for all y e U~A A~. Thus x e (U~A A)~)±, proving the result. [] Definition 2.7. Y is weakly c o m m u t a t i v e if for all a,b in Y+, a e {b} ± implies b e {a} ± Definition 2.8. Y satisfies condition D if for all a e Y+ we have { y e Y I Y <<-a } n { a } ± = {0}. Clearly c o m m u t a t i v e B C K - a l g e b r a s are weakly c o m m u t a t i v e . It is shown in [7] T h e o r e m 2.9 that Y is weakly c o m m u t a t i v e if and only if Y satisfies condition D.
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Theorem 2.9. I f Y is weakly commutative, A c Y and C c Y÷, then the follow±n9 are true: (i) AcaC c A "±. (ii) (A ca C) ± = (A ca C) ±±±. (iii) (Ac~C)" =
(ii) Let A ~ = A ca C. Then A 1 ca C = A 1 and by (i) we have A~ = A± caC c A~ ±. Hence A1x±" c A~-. But A~-= A~ca Y+ c Ai~±± by (i) also. Thus we have A~- = A~ ±±, i.e. (AcaC) ± = (AcaC) ±±±. (iii) We have A c a C c ( A c a C ) and hence ( A c a C ) ± c (AcaC) ±. Suppose that a e (AcaC) ±. Then for all c e A c a C we have a e { c } ' . Since a, c e Y+ this means that for all c e A ca C we have c e {a} ±, i.e. AcaC c {a} ±. Hence ( a c a c > c {a}" and hence by (i) we have {a} c {a} ±" c ( A c a C ) ±. This means that (AcaC) ± c ( A c a C ) ' , proving the result. (iv) This is proved in [7] T h e o r e m 2.10. []
Theorem 2.10. Let A, B and C be ideals of Y with A, C c Y÷. Then A c a B c a C = { 0 } if and only if A c (B ca C)'. Proof. Clearly if A c ( B n C ) " then AcaBcaC = {0}. Conversely suppose that A n B c a C = { O } . Let a e A . Then for each b e ( B c a C ) we have b * ( b * a ) <~a and b , ( b * a ) <<.b. Hence b , ( b * a ) e A c a B c a C = { O } . This means that b = b , a , i.e., a e { b } ±. Thus a e O b ~ a ~ c { b } ± = ( B c a C ) ". This proves that A c (Bca C) ±. [ ]
Definition 2.11. Let #± : Y --, [0, 1] be given by /t ± = ;(ty,,)~.
Proof. We first show that la ~< Zr,. We need only consider the case la(x) = 1. In this case x e Y, and hence ;~r,(x) = 1. On the other hand, to show that Xr, ~ # we also need only consider the case ;~r,(x) = 1, i.e. x e Yu. Then la(x) = 1. []
Corollary 2.13. I f la ~ constant takes on only the values {0, 1} then Y, = {0} if and only i f # = X{ol. Theorem 2.14. Suppose that la ~ r~^ul = {0}.
constant. Then
Proof. Let y e Y z ^ ~ , i.e. # ( y ) ^ l a ± ( y ) = l a ( 0 ) ^ la'(0) = la(0) since la±(0) = 1. Thus la(y) = la(0) and #(0) ~< la±(y) = X{Y,r(Y)- This means that y e Y~. Since la 4 constant, it follows that la(0) > 0. Hence Ztr, r(Y) > 0 which means that Z{r~r(Y)= 1. Thus y e (YA" and hence y e y, ca(y~)l = {0}. []
Definition 2.15. # is fuzzy positive if Y~ c Y+. Remark. Observe that la is fuzzy positive if and only if so is la~ for all ~ > 0.
Definition 2.16. la is fuzzy essential positive if it is fuzzy positive and Yu is an essential positive ideal. Remark. Observe that # is fuzzy essential positive if and only if so is la~ for all ~ > 0. L e m m a 1.1 also holds if we replace "closed" by "positive".
Theorem 2.17. I f Y is weakly commutative and Ia is fuzzy positive then laxx± = laX. Proof. p±J-± = Xr~ll = Z~r.}. . . . T h e o r e m 2.9(ii). []
Z~r,r = P±
using
Theorem 2.18. I f Y is weakly commutative and # is fuzzy essential positive, then # . x = Zr+ and P± = X{ol.
Remark. Observe that # ' e ~-(Y) and takes on only the values {0, 1}. Also Y,~ = (Y,)± and Y#~ = (Y,0" = (Y,)±±. Finally, observe that la±± = Z r , ~ =
Proof. By L e m m a 2.3(vii) we have (Y~)" = {0}. Hence we have I1+ --- {0} ± = (Y,)'± which means that p±± = X~y~r~= Xr~. Thus la± = # ± " = (Zr+) ± = Z{r+)l = Z{o}- []
Lemma 2.12. Suppose that # takes on only the values {0, 1 }. Then la = Zy,,.
Theorem 2.19. Y#^#~ = {0}. Hence if Y # {0} we have #" ^ p±± = X{ol.
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Proof. If x • Y,~ ^ ~ then #~(x) ^ #±±(x) = la±(0) ^ la±±(0) = 1. Thus x • Yui = (Yu)± and x • Y,~ = (Y~)±±. This means that x • (Yu)±c~(Y~)±± = {0}. If Y # {0} then we can find x ~ 0 in Y. This means that (la" ^ #±')(x) :/: 1. Since la± ^ la±± takes on only the values {0, 1}, we have (la± ^ la±±)(x) = 0. Hence la±^la±± is non-constant. By Corollary 2.13, we have la± ^ l a ' ± = Z~o~. [] Definition 2.20. An element a of Y is an atom of Y if a > 0 and if whenever x ~ < a then x = 0 or x = a. Let A t ( Y ) denote the set of all atoms of Y. Theorem 2.21. I f la is fuzzy essential closed (essential positive) then A t ( Y ) c Y~. may assume that A t ( Y ) ~ O . Let a • A t ( Y ) and consider ( a ) . This is the positive ideal generated by a and is non-zero. Hence Y~,c~ ( a ) ~ {0}. Let x # 0 be any element of Y,c~ ( a ) . Then Ix, a]" = 0 for some n > 0. We may suppose that n is the smallest positive integer for which [x, a]" = 0. Then we have [x, a] "- t ~ a and hence [x,a] " - l = a . Thus a * x = [ x , a ] " - l * x = O , i.e. a ~< x. Hence a • Y~. This proves the theorem. [] Proof. We
Corollary 2.22. I f I is an essential closed (essential positive) ideal of Y then At(Y) c I.
need only consider the case Xr,(X)= 1, i.e. x Y~,. Then 0* x • Y, and hence Xr,(0* x ) = 1. This proves that Zr, is fuzzy closed. Since Yx,~ = Yu it follows that if # is fuzzy positive then so is Zr,. Also since Yx~, = Y,, we see that if la is fuzzy essential closed (essential positive) then so is Zy. []
Corollary 2.25. Suppose that Y is weakly commutative and la is fuzzy positive, l f la is fuzzy essential closed (essential positive) then so is la±±. Proof. By T h e o r e m 2.23(i) we have ~r, ~< la±±, The result follows from L e m m a 2.24, L e m m a 1.1 and the remark following Definition 2.16. [] We conclude this section with the following results.
Lemma 2.26. If p is a fuzzy closed ideal of Y such that Y~,c~ Y+ = {0} then iax = Xr+. Proof. By [8] T h e o r e m 2.8 we have Y~ c 17. Hence Y+=(17)±c(Y,)±=Y+, i.e. Y , ~ = ( Y , ) ± = Y + . Then by L e m m a 2.12, la± = XY,,~= ZY+- []
Corollary 2.27. I f I is a closed ideal of Y such that I n Y+ = {0} then I ± = Y+. Proof. Let p = X~ and apply L e m m a 2.26.
Proof. T a k e la = Xt and apply T h e o r e m 2.21.
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[]
[]
The following result is easily established.
3. The socle and fuzzy socle
Theorem 2.23. (i) I f Y is weakly commutative and la is fuzzy positive then Xr, <<,#±±. (ii) Suppose that ~ and fl are fuzzy ideals of Y such that ~<,fl and ~(O)=fl(O). Then fl± < ~1. (iii) Suppose that ~, fl • ~ ( Y ) are fuzzy positive. Then Y~^~ = {0} if and only if Y~ c (y~)±, and Zr~^~ = Z/o} if and only if zy ~ <~Z
Definition 3.1. Y is finitely (finitely positively) cogenerated if for every family { I j i j • J} of closed (positive) ideals of Y such that (3 {ljlj • J} = {0} there exists a finite subset K of d such that 0 {I~lj• K} -- {0}.
Lemma 2.24. l f p is fuzzy essential closed (essential positive), then so is Xr,.
Lemma 3.2. Suppose that Y is finitely (finitely positively) cogenerated, l f {I~ IJ • J} is a family of essential closed (essential positive) ideals of Y, then (3 {I~lj • J } is an essential closed (essential positive) ideal of Y.
Proof. Suppose that p is fuzzy closed. Then so is gr,, i.e. Xr,(x) <~Xr,(O* x) for all x • Y. In fact, we
Remark. If Y is finitely cogenerated then it is also finitely positively cogenerated.
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Proof. If each lj is a closed (positive) ideal, then so is 0 { I j l j • J}. Suppose that K is a closed (positive) ideal of Y and O { l i l j e J } n K = {0}. Since Y is finitely (finitely positively) cogenerated, it follows that we can find finitely many ideals I1 .... , I, in the family such that l ~ n l 2 n ... c~I,c~K = {0}. Since I~, . . . , I , are essential closed (essential positive) it follows easily that K = {0}. This proves the lemma. [] Lemma 3.3. I f Y is finitely cogenerated, then we can find Xx .... ,x~ in Y such that {xl . . . . . x~} ± = {O} for some n >10. I f Y is finitely positively cogenerated, we can find the xi • Y+. Proof. We have 0 = Y ± = N ~ y { x } ±. Hence the result follows if Y is finitely cogenerated. In case Y is finitely positively cogenerated, we use the fact t h a t 0 = ( Y + ) ± . []
Definition 3.4. The socle of Y is Soc(Y) = ~ {I I I is an essential closed ideal of Y}. The positive socle of Y is P Soc(Y) = N {/11 is an essential positive ideal of Y}. The fuzzy socle of Y is F S o c ( Y ) = A{/~ • ~ ( Y ) I # is a fuzzy essential closed ideal of Y} and the fuzzy positive socle of Y is FP Soc(Y) = A {# • ~ ( Y ) ] / ~ is a fuzzy essential positive ideal of Y}. Remark. Clearly Soc(Y) is a closed ideal of Y and P Soc(Y) is a positive ideal of Y. Similarly F Soc(Y) and F P S o c ( Y ) are fuzzy ideals of Y in ~ ( Y ) and are fuzzy closed and fuzzy positive respectively.
Theorem 3.5. I f Y is finitely (finitely positively) co#enerated, then S o c ( Y ) ( P S o c ( Y ) ) is an essential closed (essential positive) ideal of Y, in fact, the smallest essential closed (essential positive) ideal of Y, and F S o c ( Y ) ( F P S o c ( Y ) ) is a fuzzy essential closed (essential positive) ideal of Y in ~ ( Y ) such that Soc(Y) = Yrso¢~r~ and P S o c ( Y ) = YrPso~r~. Proof. Suppose that K is a closed (positive) ideal of Y such that S o c ( Y ) n K = {0} ( P S o c ( Y ) n K = {0}). Since Y is finitely (finitely positively) cogenerated, we can find finitely many essential closed (essential positive) ideals I1 . . . . . I, of Y such that 11 n ..- n I, n K = {0}. Since 11 n ..- n I, is an essential closed (essential positive) ideal of Y, we
have K = {0}, proving that Soc(Y)(P Soc(Y)) is an essential closed (essential positive) ideal of Y. It is clearly the smallest such ideal of Y. Observe that YFsoc~r~ = A { Y~I/~ • ~ ( Y ) and/z is a fuzzy essential closed ideal of Y}. This can be established directly or by appealing to [10] Lemma 4.2. Since { Y~I/~ • ~ ( Y ) and/z is a fuzzy essential closed ideal of Y } c {II I is an essential closed ideal of Y } we have Soc(Y) c Yrso~r~. This means that YFso¢~r~is an essential closed ideal of Y and hence that F Soc(Y) is a fuzzy essential closed ideal of Y. On the other hand, we have {II I is an essential closed ideal of Y } = { Yx, [I is an essential closed ideal of Y} c { Y ~ l / ~ • f f ( Y ) and # is a fuzzy essential closed ideal of Y}. Hence Yrso¢tr) c Soc(Y), proving that Soc(Y) = YFsoctr~. The proof is similar for PSoc(Y) = Y~eso~r~. []
Theorem 3.6. I f Y is finitely (fnitely positively) cogenerated, then F Soc(Y)(FP Soc(Y)) takes on only the values {0,1} and in fact FSoc(Y)=Zso¢~r~, FP Soc(Y) = ZpSo~Y~. Proof. For all ct/> 0, (F Soc(Y)) ~ is a fuzzy essential closed ideal of Y and is an element of ~ ( Y ) . Hence F S o c ( Y ) < ~ ( F S o c ( Y ) ) ~. On the other hand, if /> 1, we have (FSoc(Y)) ~ ~< FSoc(Y). Hence for ~> 1, we have F Soc(Y) = (F Soc(Y))L If for some x • Y we have F S o c ( Y ) ( x ) = t where 0 < t < 1, then t = F S o c ( Y ) ( x ) = (FSoc(Y))~(x)= t ~ for all > 1, a contradiction. Hence FSoc(Y) takes on over the values {0, 1}. Then by Lemma 2.12 we have F S o c ( Y ) = ZrFsoc~Y,= ZSoctY). The proof for F P S o c ( Y ) and PSoc(Y) is similar. [] Remark. In general, if I is an essential closed ideal of Y then I n Y÷ is an essential positive ideal of Y. Hence { I n Y + l I is an essential closed ideal of Y } c {J I J is an essential positive ideal of Y }. This means that S o c ( Y ) n Y+ ~ PSoc(Y). Also by Corollary 2.22 we have A t ( Y ) c P S o c ( Y ) and A t ( Y ) c Soc(Y) if Y is finitely positively (finitely) cogenerated, respectively. Hence we have the following result.
Theorem 3.7. I f Y is fnitely cogenerated, we have A t ( Y ) c PSoc(Y) c S o c ( Y ) n Y+.
C.S. Hog / Fuzzy Sets and Systems 72 (1995) 197-204
In order to clarify the notion of being finitely cogenerated, we consider the situation further. We are motivated by 1-16].
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"positive ideals" instead of "closed ideals", and Theorem 3.9 is still true.
Definition 3.10. Y is simple if Y :~ {0} and if {0} Definition 3.8. Y is artinian if every descending
and Y are the only closed ideals of Y.
chain of closed ideals stops.
Theorem 3.9. The following are equivalent: (i) Y is artinian. (ii) Every non-empty family of closed ideals of Y has a minimal element. (iii) Every quotient algebra of Y by a closed ideal is finitely cogenerated. Proof. Suppose that Y is artinian. Let ~ be a nonempty family of closed ideals of Y. Suppose that f " does not have a minimal element. This means that for each I e ~-, the set {I' e ~ 1 I ' ~ I } is nonempty. Then by the axiom of choice we have a function I ~ I' with I' ~ I for each I e ~ . Fixing I e ~we therefore obtain an infinite descending chain I ~ I' ~ I" ~ ... of closed ideals of Y, contradicting the fact that Y is artinian. This proves that (i) =~ (ii). N o w suppose that (ii) is true. Let / be a closed ideal of Y. To show that Y / I is finitely cogenerated. We need only show that if {J~l • e A} is a family of closed ideals of Y with N~A J ~ - - I then I -- G~B Ja for some finite subset B of A. Let X = {G~TJtl T c A is finite}. Then X is a nonempty family of closed ideals of Y. By (ii), X has a minimal element, say Na~nJ~ with B c A and B is finite. Then N~BJp = I. In fact, Na~nJ~ ~ I . If Na~sJa :~ I, then we can find x such that x e J~ for all f l e B and xq~J7 for some V e A - B. But B w { 7 } is a finite subset of A and (Na~nJ~)nJr c Na~sJ~. Hence by the minimality of Na~BJa we have (n~nJt~)nJ.i = n ~ s J ~ , that is N~BJ~ ~ J~, a contradiction. This proves that (ii) => (iii). Finally, suppose that (iii) is true. Let 11 ~ Iz ~ ... be a descending chain of closed ideals of Y. Put I = N~=l I,. Then I is a closed ideal of Y and by (iii) Y/1 is finitely cogenerated. Hence I = lk for some k >/ 1. Thus Ik+~ = Ik for all i = 1,2 . . . . This proves that (iii) =:, (i). [] Remark. In the above and in the definitions of "artinian" and "finitely cogenerated", we may replace the concepts by those using "subalgebras" or
Corollary 3.11. If Y ~ {0} is artinian, then Y has a closed ideal which is a simple subalgebra. Proof. Apply Theorem 3.9 for the non-empty family of all non-zero closed ideals of Y. Similarly we have the following result.
[]
Corollary 3.12. If Y is artinian and Y+ ~ {0}, then Y has a positive ideal which is a simple subalgebra of Y. Proof. Apply Theorem 3.9 for the non-empty family of all non-zero positive ideals of Y.
Corollary 3.13. If Y is weakly commutative and artinian and Y+ ~ {0} then P S o c ( Y ) q: {0} and hence
Soc(Y)
{0}.
Proof. Consider the non-empty family of all essential positive ideals of Y. Then apply Theorems 3.5 and 3.9. []
Definition 3.14. Y is noetherian if every ascending chain of closed ideals of Y stops.
Definition 3.15. A h o m o m o r p h i s m f: X ~ Y between BCI-algebras X and Y is closed if f ( X ) is a closed ideal of Y.
Lemma 3.16. Suppose that X, Y , Z are BCI-algebras and f : X ~ Y and g: Y ~ Z are closed homomorphisms and g is injective. Then gf: X ~ Z is a closed homomorphism. Proof. Clearly gf is a homomorphism. Let z l , z 2 e Z and suppose that z l * z 2 =gf(a) and z2 = gf(b) where a, b e X. Since g is closed, it follows that zl = g(c) for some c ~ Y. This means that g(c)* g f ( b ) = gf(a). Since g is injective, we have c . f (b)=f(a). But f is closed and hence c = f ( d ) for some d e X. Then z~ = gf(d) proving that g f ( X ) is a closed ideal of Z.
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Theorem 3.17. (i) I f Y is artinian and f: Y ~ Y is a closed homomorphism which is injective then f is surjective. (ii) I f Y is noetherian and f: Y ~ Y is a surjective homomorphism then f is injective. Proof. (i) Suppose that f ( Y ) :A Y. Then we obtain an infinite descending chain Y ~ f ( Y ) ~ f 2 ( y ) ~ ... of closed ideals of Y, which cannot happen since Y is artinian. (ii) Suppose that k e r f # {0}. Let n >I 1. We claim that k e r f " # k e r f 2~. In fact, suppose that kerf~ = k e r f 2~. Let x e k e r f ". Since f* is surjective, we have x = f n ( y ) for some y e Y. Hence y e k e r f 2n = k e r f ~, which means that x = 0, i.e. k e r f ~= {0}. This also means that k e r f = {0}, a contradiction. Thus for each n 1> 1, k e r f ~ k e r f 2~ which cannot happen since Y is noetherian. []
References 1-1] L.P. Belluce, Semisimple algebras of infinite valued logic and Bold fuzzy set theory, Can. J. Math. 38 (1986) 1356-1379. [2] L.P. Belluce, Semisimple and complete MV-algebras, Alg. Univ. 29 (1992) 1-9.
[3] C.C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958) 467-490. [4] C.C. Chang, A new proof of the completeness of the Lukasiewicz axioms, Trans. Amer. Math. Soc. 93 (1959) 74-80. [5] C.S. Hoo, Ideals in BCK-algebras, Math. Japon. 33 (1988) 529-533. I-6] C.S. Hoo, MV-algebras, ideals and semisimplicity, Math. Japon. 34 (1989) 563-583. 1-7] C.S. Hoo, Closed ideals and p-semisimple BCI-algebras, Math. Japon. 35 (1990) 1103-1112. [8] C.S. Hoo, Filters and ideals in BCI-algebras, Math. Japon. 36 (1991) 987-997. [9] C.S. Hoo, Fuzzy ideals of BCI and MV-algebras, Fuzzy Sets and Systems 62 (1994) 111-114. [10] C.S. Hoo, Fuzzy implicative and Boolean ideals of MValgebras, Fuzzy Sets and Systems 66 (1994) 315-327. [11] C.S. Hoo, Extensions of BCI-algebras, submitted. 112] C.S. Hoo and Y.B. Jun, Essential ideals and homomorphisms in BCI-algebras, Math. Japon. 38 (1993) 597-602. [13] C.S. Hoo and S. Sessa, Fuzzy maximal ideals of BCI and MV-algebras, Inform. Sci. 80 (1994) 299-309. [14] K. Is6ki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japon. 21 (1976) 351-366. [15] K. Is6ki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978) 1-26. [16] G. Karpilovsky, Group Representations, Vol. 1 (NorthHolland, Amsterdam, 1992). [17] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [18] O. Xi, Fuzzy BCK-algebra, Math. Japon. 36 (1991) 935-942.
Fuzzy Sets and Systems 72 (1995) 197 204
The socle and fuzzy socle of a BCI-algebra C.S. Hoo Department of Mathematical Sciences, University of Alberta, Edmonton. Alberta, Canada T6G 2GI
Abstract We introduce the notion of orthogonal complement or annihilator in general BCI-algebras, and the corresponding notion in the fuzzy ideals. Various properties are proved, and the related notions of the socle and fuzzy socle of a BCI-algebra are investigated. It is shown that a non-zero weakly commutative, artinian BCI-algebra has non-zero socle. Keywords: Fuzzy essential closed ideal; Fuzzy socle; Artinian; Finitely cogenerated
1. Introduction In [12] the notion of an essential closed ideal of a BCI-algebra was introduced and some ideals were identified as having this property. In [10] we studied the situation further, particularly in MValgebras. In this paper, we introduce such notions here that generalise those of [10]. Also some results of [6] are generalized here. We assume knowledge of BCI and BCK-algebras and refer the reader to [3,6-8, 14, 15] for full details. Also the basic ideas of fuzzy logic are assumed and we refer the reader to [9, 10, 13, 17, 18] for complete details. However, we will review briefly here the concepts of BCI- and BCK-algebras and fuzzy logic. A BCI-algebra is a non-empty set X with a constant 0 and a binary operation • satisfying the axioms (1) { ( x * y ) * ( x . z ) } * ( z . y ) = 0,(2) {x* (x*y)}.y=0, (3) x . x = 0 , (4) x . y = 0 and y.x=0 imply that x = y , and (5) x * 0 = 0 im-
plies that x = 0. We define a partial ordering ~< on X by putting x ~< y if and only if x * y = 0. If further every x satisfies 0 ~< x then X is a BCK-algebra. If a BCK-algebra satisfies the axiom x * ( x * y ) = y * (y * x) then it is called commutative. In this case x * (x * y) = y * (y * x) is the greatest lower bound x ^ y of x and y. We shall write [ x l , x 2 . . . . . x,] for ( . . . ( ( ( x l * x 2 ) * x 3 ) * ) * . . . ) * x , . We define Ix, y] ° = x , and for n > 0 , Ix, y]" = [ x , y . . . . . y], where y occurs n times. If X and Y are BCI-algebras, a homomorphism f : X ~ Y means a function such that f ( x l * x2) = f ( x l ) * f(x2) for all xl, x2 in X. An ideal of a BCI-algebra is a subset I containing 0 and such that if x * y and y are in I then so is x. We say that an ideal I is closed if whenever x belongs to I then so does 0* x. All ideals of a BCKalgebra are automatically closed. A closed ideal I is an essential closed ideal if for every non-zero closed ideal J we have l n J v~ {0} (see [10, 12]). A subalgebra is a subset containing 0 and closed under the operation *.
0165-0114/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0165-01 14(94)00351-3
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In general, an ideal is not a subalgebra. However, it is a subalgebra if and only if it is closed. The p-radical of a BCI-algebra X is X÷ = {x • X[ 0 ~< x}. This is an ideal of X and is clearly closed. In fact X+ is a BCI-subalgebra of X. A BCIalgebra X is p-semisimple if X+ = {0}. It has a natural abelian group structure given by x+y=x*(0.y), -x=0*x, and 0 is the group identity. The p-semisimple part of a BCI-algebra X is X = {x I 0 • (0 • x) = x}. This is a subalgebra of X, in fact a p-semisimple BCI-subalgebra of X. It is in general not an ideal of X. For a general BCIalgebra X we have X c~ X+ = {0}. Observe that the partial ordering defined above is trivial for a p-semisimple BCI-algebra, and that X = {x • X ly ~< x =~ y = x}. An ideal I is positive if I c X+. A positive ideal I is an essential positive ideal if for every non-zero positive ideal J we have I n J # {0}. Clearly a positive ideal is a closed ideal. We shall frequently use the following three identities for computations: (1) ( x * y ) * z = (x*z)*y, (2) O * ( x * y ) = ( 0 * x ) * (0 * y), and (3) y * {y * (y * x)} = y * x. Also useful is [11, L e m m a 5.5]: in any BCI-algebra we have the identity [(a * b) • c] • [a * (b * c)] = (0 * c) * c. If A is a non-empty subset of a BCI-algebra X, let (A> denote the set {x • XI [x, aa ..... a,] = 0 for some aa,a2 ..... a. in A}. If A ~ X + # O, then (A> is an ideal of X, namely the ideal generated by A, i.e. it is the smallest ideal of X containing A. We shall denote ({a}> by
the BCI-algebra operation on X/I is defined by x • y = x • y. The zero is 0. If X is BCK, then so is X/I. The (closed) ideals J of X / I are precisely the (closed) ideals J of X containing I such that J/I = J, where J/I = { x l x • J}. A BCK-algebra is bounded if it has a largest element which is usually denoted by 1. In a bounded commutative BCK-algebra, every two elements x and y have a least upper bound given by xvy=l*{(l*x)^(l*y)}. A bounded commutative BCK-algebra is a distributive lattice with the operations v, ^, and 0 and 1 are the universal bounds. The category of bounded commutative BCK-algebras is naturally equivalent to the category of MV-algebras (see [1-4, 6]). All our results for BCI-algebras are, of course, true also for BCKand MV-algebras. We now review some fuzzy logic concepts, referring the reader to [9, 10, 13, 17, 18] for further details. Let [0, 1] denote the unit closed interval. If a and b are in [0, 1], let a ^ b be the minimum of a and b, and let a v b be the m a x i m u m of a and b. A fuzzy subset of a BCI-algebra X is a function /~:X ~ [0, 1]. It is a fuzzy ideal if #(0) ~> #(x) and /~(x) >//~(x * y) ^ p(y) for all x, y in X. Given a fuzzy subset p and t • [0, 1], let p, = {x • X lp(x) >~ t}. This could be empty. It is shown in [18] Theorem 3 that p is a fuzzy ideal if and only if each P, is either empty or is an ideal. These ideals Pt are the level ideals of p. Thus, given a fuzzy ideal # we have an ideal X~, = {x • X lp(x) = p(0)} of X. If A is a non-empty subset of X, we have the characteristic function XA of A. The characteristic function Zx of an ideal I is a fuzzy ideal. If p is a fuzzy ideal with p(0) = 1, we have Zxu ~< p. In general we have Xz, = I if I is an ideal. A fuzzy ideal p is normalised if p(0) = 1. Let ~ ( X ) denote the set of all normalised fuzzy ideals of X. Given a fuzzy ideal # of X, its normalisation /~ is given by ~(x) = p(x) + 1 - p(0). Then/~ is normalised fuzzy ideal. We can define a partial ordering ~< on the set of all fuzzy ideals of X by a ~< fl if and only if a(x) ~< fl(x) for all x • X. Then for each fuzzy ideal p of X we have p ~~ #(x) for all x • X (see [10]). If p is fuzzy closed then X, is a closed ideal, and if I is a closed ideal then Zt is a fuzzy closed ideal.
C.S. Hoo / Fuzzy Sets and Systems 72 (1995) 197-204
Let ~ >~ 0 be a real number. If t e [0, 1] let t" be the positive root in case ct < 1. If p is a fuzzy ideal of X, then /~':X ~ [0, 1] given by # ' ( x ) = (~t(x))" is a fuzzy ideal, and Xu~ = X, if ~ > 0 (see [10]). We see that # is fuzzy closed if and only if so is /t ~. A fuzzy closed ideal p is a fuzzy essential closed ideal if for all fuzzy closed ideals ct of X with X~ 4: {0} we have X~, n X, ~ {0}. Thus # is a fuzzy essential closed ideal if and only if X~, is an essential closed ideal, and I is an essential closed ideal if and only if Z~ is a fuzzy essential closed ideal (see [10, L e m m a 3.2]). Clearly if # is fuzzy essential closed, then so is/~. O b s e r v e that # is fuzzy essential closed if and only if so is p~ for all ~ ~> 0. A useful result is L e m m a 3.3 of [10] which we re-state as the following result. L e m m a 1.1. Suppose that la is a f u z z y essential closed ideal of X and c~ is a fuzzy closed ideal of X such that It ~ ~ and kt(O) = ~(0). Then cc is a fuzzy essential closed ideal of X.
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The following result is easily established. L e m m a 2.3. (i) I f A is a non-empty subset of Y, then A r i A ± = {0}. (ii) l f A c B then B ± C A ±. (iii) I f A is a non-empty subset of Y, then A ± = n,~A{a} ±. (iv) I f Y+ = A then A ± = {0}.
(v) {0}±= Y+. (vi) I f A contains a non-zero positive element of Y, then A ± is a proper ideal of Y+. (vii) I f I is an essential closed (essential positive) ideal of Y then I ± = {0}. Lemma2.4. If xe? ( I 7 ) ± = Y+.
then {x} ± = Y+ and hence
Proof. By [8] Proposition 2.25 we have x * y = x for all x e I7 and all y e Y+. Hence if x e I7 we have {x} ± = Y+. This means that (17)± = Nx~¢ {x} ± = Y+. [] Corollary 2.5. l f A c Y, then A ± = (A - 17)±.
2. Annihilators in BCl-algebras In the rest of the p a p e r Y is a general BCIalgebra and /~ is a general fuzzy ideal of Y. Any special properties possessed by them will be explicitly stated.
Definition 2.1. If A is a n o n - e m p t y subset of Y, let AJ-={yeY+la=a*y for all a e A } . Define ( 0 ) ± = Y+. R e m a r k . If Y is a c o m m u t a t i v e BCI-algebra or an MV-algebra, this gives the same definition as in [1,6].
Theorem 2.2. I f A is a subset of Y, then A j- is a positive ideal of Y. Proof. We need only consider the case A 4: 0. Clearly A ± c Y÷. Obviously, 0 e A ±. Suppose that x , y, y e A ±. Then a * y = a and a * (x * y) = a for all a e A . N o w a * x = ( a * y ) * x = ( a , x ) , y < < , a * ( x * y ) = a by [11, L e m m a 5.5]. []
Proof. We need only consider the case A ~ I7. Then A ± = N=~A{a} ± = n,~A_~{a} ± = ( A - 17)±. [] Theorem 2.6. I f {Aa l2 e A} is a family of subsets of Y, then (Ua~aAa) ± = Na~A A~. Proof. Since for each 2 e A we have A~ c U),~A A~,, it follows that (U,tEAA;~)±cN;~6AA~. N o w if xeN~AA~, then x e A ~ = N r ~ A ~ { y } ± for all 2 e A, and hence x e {y}, for all y e U~A A~. Thus x e (U~A A)~)±, proving the result. [] Definition 2.7. Y is weakly c o m m u t a t i v e if for all a,b in Y+, a e {b} ± implies b e {a} ± Definition 2.8. Y satisfies condition D if for all a e Y+ we have { y e Y I Y <<-a } n { a } ± = {0}. Clearly c o m m u t a t i v e B C K - a l g e b r a s are weakly c o m m u t a t i v e . It is shown in [7] T h e o r e m 2.9 that Y is weakly c o m m u t a t i v e if and only if Y satisfies condition D.
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Theorem 2.9. I f Y is weakly commutative, A c Y and C c Y÷, then the follow±n9 are true: (i) AcaC c A "±. (ii) (A ca C) ± = (A ca C) ±±±. (iii) (Ac~C)" =
(ii) Let A ~ = A ca C. Then A 1 ca C = A 1 and by (i) we have A~ = A± caC c A~ ±. Hence A1x±" c A~-. But A~-= A~ca Y+ c Ai~±± by (i) also. Thus we have A~- = A~ ±±, i.e. (AcaC) ± = (AcaC) ±±±. (iii) We have A c a C c ( A c a C ) and hence ( A c a C ) ± c (AcaC) ±. Suppose that a e (AcaC) ±. Then for all c e A c a C we have a e { c } ' . Since a, c e Y+ this means that for all c e A ca C we have c e {a} ±, i.e. AcaC c {a} ±. Hence ( a c a c > c {a}" and hence by (i) we have {a} c {a} ±" c ( A c a C ) ±. This means that (AcaC) ± c ( A c a C ) ' , proving the result. (iv) This is proved in [7] T h e o r e m 2.10. []
Theorem 2.10. Let A, B and C be ideals of Y with A, C c Y÷. Then A c a B c a C = { 0 } if and only if A c (B ca C)'. Proof. Clearly if A c ( B n C ) " then AcaBcaC = {0}. Conversely suppose that A n B c a C = { O } . Let a e A . Then for each b e ( B c a C ) we have b * ( b * a ) <~a and b , ( b * a ) <<.b. Hence b , ( b * a ) e A c a B c a C = { O } . This means that b = b , a , i.e., a e { b } ±. Thus a e O b ~ a ~ c { b } ± = ( B c a C ) ". This proves that A c (Bca C) ±. [ ]
Definition 2.11. Let #± : Y --, [0, 1] be given by /t ± = ;(ty,,)~.
Proof. We first show that la ~< Zr,. We need only consider the case la(x) = 1. In this case x e Y, and hence ;~r,(x) = 1. On the other hand, to show that Xr, ~ # we also need only consider the case ;~r,(x) = 1, i.e. x e Yu. Then la(x) = 1. []
Corollary 2.13. I f la ~ constant takes on only the values {0, 1} then Y, = {0} if and only i f # = X{ol. Theorem 2.14. Suppose that la ~ r~^ul = {0}.
constant. Then
Proof. Let y e Y z ^ ~ , i.e. # ( y ) ^ l a ± ( y ) = l a ( 0 ) ^ la'(0) = la(0) since la±(0) = 1. Thus la(y) = la(0) and #(0) ~< la±(y) = X{Y,r(Y)- This means that y e Y~. Since la 4 constant, it follows that la(0) > 0. Hence Ztr, r(Y) > 0 which means that Z{r~r(Y)= 1. Thus y e (YA" and hence y e y, ca(y~)l = {0}. []
Definition 2.15. # is fuzzy positive if Y~ c Y+. Remark. Observe that la is fuzzy positive if and only if so is la~ for all ~ > 0.
Definition 2.16. la is fuzzy essential positive if it is fuzzy positive and Yu is an essential positive ideal. Remark. Observe that # is fuzzy essential positive if and only if so is la~ for all ~ > 0. L e m m a 1.1 also holds if we replace "closed" by "positive".
Theorem 2.17. I f Y is weakly commutative and Ia is fuzzy positive then laxx± = laX. Proof. p±J-± = Xr~ll = Z~r.}. . . . T h e o r e m 2.9(ii). []
Z~r,r = P±
using
Theorem 2.18. I f Y is weakly commutative and # is fuzzy essential positive, then # . x = Zr+ and P± = X{ol.
Remark. Observe that # ' e ~-(Y) and takes on only the values {0, 1}. Also Y,~ = (Y,)± and Y#~ = (Y,0" = (Y,)±±. Finally, observe that la±± = Z r , ~ =
Proof. By L e m m a 2.3(vii) we have (Y~)" = {0}. Hence we have I1+ --- {0} ± = (Y,)'± which means that p±± = X~y~r~= Xr~. Thus la± = # ± " = (Zr+) ± = Z{r+)l = Z{o}- []
Lemma 2.12. Suppose that # takes on only the values {0, 1 }. Then la = Zy,,.
Theorem 2.19. Y#^#~ = {0}. Hence if Y # {0} we have #" ^ p±± = X{ol.
CS. Hoo / Fuzzy Sets and Systems 72 (1995) 197 204
Proof. If x • Y,~ ^ ~ then #~(x) ^ #±±(x) = la±(0) ^ la±±(0) = 1. Thus x • Yui = (Yu)± and x • Y,~ = (Y~)±±. This means that x • (Yu)±c~(Y~)±± = {0}. If Y # {0} then we can find x ~ 0 in Y. This means that (la" ^ #±')(x) :/: 1. Since la± ^ la±± takes on only the values {0, 1}, we have (la± ^ la±±)(x) = 0. Hence la±^la±± is non-constant. By Corollary 2.13, we have la± ^ l a ' ± = Z~o~. [] Definition 2.20. An element a of Y is an atom of Y if a > 0 and if whenever x ~ < a then x = 0 or x = a. Let A t ( Y ) denote the set of all atoms of Y. Theorem 2.21. I f la is fuzzy essential closed (essential positive) then A t ( Y ) c Y~. may assume that A t ( Y ) ~ O . Let a • A t ( Y ) and consider ( a ) . This is the positive ideal generated by a and is non-zero. Hence Y~,c~ ( a ) ~ {0}. Let x # 0 be any element of Y,c~ ( a ) . Then Ix, a]" = 0 for some n > 0. We may suppose that n is the smallest positive integer for which [x, a]" = 0. Then we have [x, a] "- t ~ a and hence [x,a] " - l = a . Thus a * x = [ x , a ] " - l * x = O , i.e. a ~< x. Hence a • Y~. This proves the theorem. [] Proof. We
Corollary 2.22. I f I is an essential closed (essential positive) ideal of Y then At(Y) c I.
need only consider the case Xr,(X)= 1, i.e. x Y~,. Then 0* x • Y, and hence Xr,(0* x ) = 1. This proves that Zr, is fuzzy closed. Since Yx,~ = Yu it follows that if # is fuzzy positive then so is Zr,. Also since Yx~, = Y,, we see that if la is fuzzy essential closed (essential positive) then so is Zy. []
Corollary 2.25. Suppose that Y is weakly commutative and la is fuzzy positive, l f la is fuzzy essential closed (essential positive) then so is la±±. Proof. By T h e o r e m 2.23(i) we have ~r, ~< la±±, The result follows from L e m m a 2.24, L e m m a 1.1 and the remark following Definition 2.16. [] We conclude this section with the following results.
Lemma 2.26. If p is a fuzzy closed ideal of Y such that Y~,c~ Y+ = {0} then iax = Xr+. Proof. By [8] T h e o r e m 2.8 we have Y~ c 17. Hence Y+=(17)±c(Y,)±=Y+, i.e. Y , ~ = ( Y , ) ± = Y + . Then by L e m m a 2.12, la± = XY,,~= ZY+- []
Corollary 2.27. I f I is a closed ideal of Y such that I n Y+ = {0} then I ± = Y+. Proof. Let p = X~ and apply L e m m a 2.26.
Proof. T a k e la = Xt and apply T h e o r e m 2.21.
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[]
[]
The following result is easily established.
3. The socle and fuzzy socle
Theorem 2.23. (i) I f Y is weakly commutative and la is fuzzy positive then Xr, <<,#±±. (ii) Suppose that ~ and fl are fuzzy ideals of Y such that ~<,fl and ~(O)=fl(O). Then fl± < ~1. (iii) Suppose that ~, fl • ~ ( Y ) are fuzzy positive. Then Y~^~ = {0} if and only if Y~ c (y~)±, and Zr~^~ = Z/o} if and only if zy ~ <~Z
Definition 3.1. Y is finitely (finitely positively) cogenerated if for every family { I j i j • J} of closed (positive) ideals of Y such that (3 {ljlj • J} = {0} there exists a finite subset K of d such that 0 {I~lj• K} -- {0}.
Lemma 2.24. l f p is fuzzy essential closed (essential positive), then so is Xr,.
Lemma 3.2. Suppose that Y is finitely (finitely positively) cogenerated, l f {I~ IJ • J} is a family of essential closed (essential positive) ideals of Y, then (3 {I~lj • J } is an essential closed (essential positive) ideal of Y.
Proof. Suppose that p is fuzzy closed. Then so is gr,, i.e. Xr,(x) <~Xr,(O* x) for all x • Y. In fact, we
Remark. If Y is finitely cogenerated then it is also finitely positively cogenerated.
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Proof. If each lj is a closed (positive) ideal, then so is 0 { I j l j • J}. Suppose that K is a closed (positive) ideal of Y and O { l i l j e J } n K = {0}. Since Y is finitely (finitely positively) cogenerated, it follows that we can find finitely many ideals I1 .... , I, in the family such that l ~ n l 2 n ... c~I,c~K = {0}. Since I~, . . . , I , are essential closed (essential positive) it follows easily that K = {0}. This proves the lemma. [] Lemma 3.3. I f Y is finitely cogenerated, then we can find Xx .... ,x~ in Y such that {xl . . . . . x~} ± = {O} for some n >10. I f Y is finitely positively cogenerated, we can find the xi • Y+. Proof. We have 0 = Y ± = N ~ y { x } ±. Hence the result follows if Y is finitely cogenerated. In case Y is finitely positively cogenerated, we use the fact t h a t 0 = ( Y + ) ± . []
Definition 3.4. The socle of Y is Soc(Y) = ~ {I I I is an essential closed ideal of Y}. The positive socle of Y is P Soc(Y) = N {/11 is an essential positive ideal of Y}. The fuzzy socle of Y is F S o c ( Y ) = A{/~ • ~ ( Y ) I # is a fuzzy essential closed ideal of Y} and the fuzzy positive socle of Y is FP Soc(Y) = A {# • ~ ( Y ) ] / ~ is a fuzzy essential positive ideal of Y}. Remark. Clearly Soc(Y) is a closed ideal of Y and P Soc(Y) is a positive ideal of Y. Similarly F Soc(Y) and F P S o c ( Y ) are fuzzy ideals of Y in ~ ( Y ) and are fuzzy closed and fuzzy positive respectively.
Theorem 3.5. I f Y is finitely (finitely positively) co#enerated, then S o c ( Y ) ( P S o c ( Y ) ) is an essential closed (essential positive) ideal of Y, in fact, the smallest essential closed (essential positive) ideal of Y, and F S o c ( Y ) ( F P S o c ( Y ) ) is a fuzzy essential closed (essential positive) ideal of Y in ~ ( Y ) such that Soc(Y) = Yrso¢~r~ and P S o c ( Y ) = YrPso~r~. Proof. Suppose that K is a closed (positive) ideal of Y such that S o c ( Y ) n K = {0} ( P S o c ( Y ) n K = {0}). Since Y is finitely (finitely positively) cogenerated, we can find finitely many essential closed (essential positive) ideals I1 . . . . . I, of Y such that 11 n ..- n I, n K = {0}. Since 11 n ..- n I, is an essential closed (essential positive) ideal of Y, we
have K = {0}, proving that Soc(Y)(P Soc(Y)) is an essential closed (essential positive) ideal of Y. It is clearly the smallest such ideal of Y. Observe that YFsoc~r~ = A { Y~I/~ • ~ ( Y ) and/z is a fuzzy essential closed ideal of Y}. This can be established directly or by appealing to [10] Lemma 4.2. Since { Y~I/~ • ~ ( Y ) and/z is a fuzzy essential closed ideal of Y } c {II I is an essential closed ideal of Y } we have Soc(Y) c Yrso~r~. This means that YFso¢~r~is an essential closed ideal of Y and hence that F Soc(Y) is a fuzzy essential closed ideal of Y. On the other hand, we have {II I is an essential closed ideal of Y } = { Yx, [I is an essential closed ideal of Y} c { Y ~ l / ~ • f f ( Y ) and # is a fuzzy essential closed ideal of Y}. Hence Yrso¢tr) c Soc(Y), proving that Soc(Y) = YFsoctr~. The proof is similar for PSoc(Y) = Y~eso~r~. []
Theorem 3.6. I f Y is finitely (fnitely positively) cogenerated, then F Soc(Y)(FP Soc(Y)) takes on only the values {0,1} and in fact FSoc(Y)=Zso¢~r~, FP Soc(Y) = ZpSo~Y~. Proof. For all ct/> 0, (F Soc(Y)) ~ is a fuzzy essential closed ideal of Y and is an element of ~ ( Y ) . Hence F S o c ( Y ) < ~ ( F S o c ( Y ) ) ~. On the other hand, if /> 1, we have (FSoc(Y)) ~ ~< FSoc(Y). Hence for ~> 1, we have F Soc(Y) = (F Soc(Y))L If for some x • Y we have F S o c ( Y ) ( x ) = t where 0 < t < 1, then t = F S o c ( Y ) ( x ) = (FSoc(Y))~(x)= t ~ for all > 1, a contradiction. Hence FSoc(Y) takes on over the values {0, 1}. Then by Lemma 2.12 we have F S o c ( Y ) = ZrFsoc~Y,= ZSoctY). The proof for F P S o c ( Y ) and PSoc(Y) is similar. [] Remark. In general, if I is an essential closed ideal of Y then I n Y÷ is an essential positive ideal of Y. Hence { I n Y + l I is an essential closed ideal of Y } c {J I J is an essential positive ideal of Y }. This means that S o c ( Y ) n Y+ ~ PSoc(Y). Also by Corollary 2.22 we have A t ( Y ) c P S o c ( Y ) and A t ( Y ) c Soc(Y) if Y is finitely positively (finitely) cogenerated, respectively. Hence we have the following result.
Theorem 3.7. I f Y is fnitely cogenerated, we have A t ( Y ) c PSoc(Y) c S o c ( Y ) n Y+.
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In order to clarify the notion of being finitely cogenerated, we consider the situation further. We are motivated by 1-16].
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"positive ideals" instead of "closed ideals", and Theorem 3.9 is still true.
Definition 3.10. Y is simple if Y :~ {0} and if {0} Definition 3.8. Y is artinian if every descending
and Y are the only closed ideals of Y.
chain of closed ideals stops.
Theorem 3.9. The following are equivalent: (i) Y is artinian. (ii) Every non-empty family of closed ideals of Y has a minimal element. (iii) Every quotient algebra of Y by a closed ideal is finitely cogenerated. Proof. Suppose that Y is artinian. Let ~ be a nonempty family of closed ideals of Y. Suppose that f " does not have a minimal element. This means that for each I e ~-, the set {I' e ~ 1 I ' ~ I } is nonempty. Then by the axiom of choice we have a function I ~ I' with I' ~ I for each I e ~ . Fixing I e ~we therefore obtain an infinite descending chain I ~ I' ~ I" ~ ... of closed ideals of Y, contradicting the fact that Y is artinian. This proves that (i) =~ (ii). N o w suppose that (ii) is true. Let / be a closed ideal of Y. To show that Y / I is finitely cogenerated. We need only show that if {J~l • e A} is a family of closed ideals of Y with N~A J ~ - - I then I -- G~B Ja for some finite subset B of A. Let X = {G~TJtl T c A is finite}. Then X is a nonempty family of closed ideals of Y. By (ii), X has a minimal element, say Na~nJ~ with B c A and B is finite. Then N~BJp = I. In fact, Na~nJ~ ~ I . If Na~sJa :~ I, then we can find x such that x e J~ for all f l e B and xq~J7 for some V e A - B. But B w { 7 } is a finite subset of A and (Na~nJ~)nJr c Na~sJ~. Hence by the minimality of Na~BJa we have (n~nJt~)nJ.i = n ~ s J ~ , that is N~BJ~ ~ J~, a contradiction. This proves that (ii) => (iii). Finally, suppose that (iii) is true. Let 11 ~ Iz ~ ... be a descending chain of closed ideals of Y. Put I = N~=l I,. Then I is a closed ideal of Y and by (iii) Y/1 is finitely cogenerated. Hence I = lk for some k >/ 1. Thus Ik+~ = Ik for all i = 1,2 . . . . This proves that (iii) =:, (i). [] Remark. In the above and in the definitions of "artinian" and "finitely cogenerated", we may replace the concepts by those using "subalgebras" or
Corollary 3.11. If Y ~ {0} is artinian, then Y has a closed ideal which is a simple subalgebra. Proof. Apply Theorem 3.9 for the non-empty family of all non-zero closed ideals of Y. Similarly we have the following result.
[]
Corollary 3.12. If Y is artinian and Y+ ~ {0}, then Y has a positive ideal which is a simple subalgebra of Y. Proof. Apply Theorem 3.9 for the non-empty family of all non-zero positive ideals of Y.
Corollary 3.13. If Y is weakly commutative and artinian and Y+ ~ {0} then P S o c ( Y ) q: {0} and hence
Soc(Y)
{0}.
Proof. Consider the non-empty family of all essential positive ideals of Y. Then apply Theorems 3.5 and 3.9. []
Definition 3.14. Y is noetherian if every ascending chain of closed ideals of Y stops.
Definition 3.15. A h o m o m o r p h i s m f: X ~ Y between BCI-algebras X and Y is closed if f ( X ) is a closed ideal of Y.
Lemma 3.16. Suppose that X, Y , Z are BCI-algebras and f : X ~ Y and g: Y ~ Z are closed homomorphisms and g is injective. Then gf: X ~ Z is a closed homomorphism. Proof. Clearly gf is a homomorphism. Let z l , z 2 e Z and suppose that z l * z 2 =gf(a) and z2 = gf(b) where a, b e X. Since g is closed, it follows that zl = g(c) for some c ~ Y. This means that g(c)* g f ( b ) = gf(a). Since g is injective, we have c . f (b)=f(a). But f is closed and hence c = f ( d ) for some d e X. Then z~ = gf(d) proving that g f ( X ) is a closed ideal of Z.
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Theorem 3.17. (i) I f Y is artinian and f: Y ~ Y is a closed homomorphism which is injective then f is surjective. (ii) I f Y is noetherian and f: Y ~ Y is a surjective homomorphism then f is injective. Proof. (i) Suppose that f ( Y ) :A Y. Then we obtain an infinite descending chain Y ~ f ( Y ) ~ f 2 ( y ) ~ ... of closed ideals of Y, which cannot happen since Y is artinian. (ii) Suppose that k e r f # {0}. Let n >I 1. We claim that k e r f " # k e r f 2~. In fact, suppose that kerf~ = k e r f 2~. Let x e k e r f ". Since f* is surjective, we have x = f n ( y ) for some y e Y. Hence y e k e r f 2n = k e r f ~, which means that x = 0, i.e. k e r f ~= {0}. This also means that k e r f = {0}, a contradiction. Thus for each n 1> 1, k e r f ~ k e r f 2~ which cannot happen since Y is noetherian. []
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