Characterization theorem of lattice implication algebras

M.Kondo and W.A.Dudek∗†

Far East J. Math. Sci. 13 (2004), 325− 342.

Abstract

In this paper we show the characterization theorem of lattice implication algebraswhich is presented by [3] in 1993. Our theorem means that the class of all latticeimplication algebras coincides with the class of all bounded commutative BCK-algebrasand hence it is categorically equivalent to the class of MV -algebras and to the class ofWajsberg algebras.

1 Introduction

In [3], the notion of lattice implication algebras is introduced and some fundamental prop-erties are proved. Since then many papers about those algebras are established. In [4],the class of pseudo lattice implication algebras coincides with the class of bounded commu-tative BCK-algebras. The algebras are extensions of the BCK/BCI-algebras which areproposed by Y. Imai and K. Iseki in 1966. In [1] Iseki and Tanaka developped the theoryof BCK-algebras. They proved some important and fundamental results. Also, in [4], Xuand Qin discussed the properties of lattice H implication algebras, and gave some equivalentconditions about lattice H implication algebras.

In this paper we show the characterization theorem of lattice implication algebras, thatis,

(1) The class LIA of lattice implication algebras is the same as the class bcBCKof bounded commutative BCK-algebras, that is,

LIA = bcBCK.

This implies thatLIA = bcBCK = MV = W,

where MV means the class of all MV -algebras and W indicates the class of allWajsberg algebras;

(2) The class LHIA of lattice H implication algebras coincides with the classBA of Booelan algebras, that is,

LHIA = BA.

(3) For every filter F of a lattice implication algebra L, the following are equiv-alent:

∗2000 Mathematics Subject Classification: 03G10, 06B10, 54E15†Keywords: lattice implication algebra, bounded commutative BCK-algebra

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(a) F is an implicative filter of L,(b) L/F is a lattice H implication algebra,(c) L/F is a Boolean algebra.

2 Preliminaries

According to [3, 4], we define a lattice implication algebra. By a lattice implication algebrawe mean a structure (L;∧,∨,→,′ , 0, 1) such that

(1) (L;∧,∨, 0, 1) is a bounded lattice,

(2) a unary operator ′ is an order-reversing involution, that is,

(a) x′′ = x and

(b) if x ≤ y then y′ ≤ x′,

(3) a binary operation → satisfies the following conditions:

For all x, y, z ∈ L,

(I1) x → (y → z) = y → (x → z),(I2) x → x = 1,(I3) x → y = y′ → x′,(I4) x → y = y → x = 1 implies x = y,(I5) (x → y) → y = (y → x) → x,(L1) (x ∨ y) → z = (x → z) ∧ (y → z),(L2) (x ∧ y) → z = (x → z) ∨ (y → z).

By LIA we mean the class of all lattice implication algebras.Remark. In [3, 4], a definition of lattice implication algebra L requires that L is a lattice.But we can show that L is already the lattice under the conditions (I1)-(I5). So we do notrequire to be a lattice at the definition of lattice implication algebras.

Example. Let L = {0, a, b, c, 1} with linear order 0 < a < b < c < 1. Define the unaryoperation ′ and the binary operation → on L as follows:

x x′

0 1a cb bc a1 0

→ 0 a b c 1

0 1 1 1 1 1a c 1 1 1 1b b c 1 1 1c a b c 1 11 0 a b c 1

It is obvious that x ∧ y = min{x, y} and x ∨ y = max{x, y} and (L;∧,∨,→,′ , 0, 1) is alattice implication algebra.

In the following the binary operation → will be denoted by juxtaposition if no confusionarises. Thus we write xy and x(yz) instead of x → y and x → (y → z), respectively.

In a lattice implication algebra L, the following hold (see [3]):

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(1) 0x = 1, 1x = x and x1 = 1,

(2) x′ = x0,

(3) xy ≤ (yz)(xz),

(4) x ∨ y = (xy)y,

(5) ((yx)y′)′ = x ∧ y = ((xy)x′)′,

(6) x = y implies yz = xz and zx = zy,

(7) x ≤ y implies yz ≤ xz.

By lattice H implication algebra L we mean a lattice implication algebra L satisfyingthe condition:

H : x ∨ y ∨ (x ∧ y → z) = 1 for all x, y, z ∈ L.

It is easy to show that for any lattice H implication algebra L we have x ∨ (x → y) = 1 forall x, y ∈ L and hence x∨ x′ = 1 for all x ∈ L. By LHIA we mean the class of all lattice Himplication algebras.

We note that the order vL induced by the lattice structure (x ≤L y ⇐⇒ x ∨ y = y)is the same as the order ≤, defined by x v y ⇐⇒ x → y = 1, induced by implication.For, suppose that x vL y. This means that x ∨ y = y and hence (xy)y = y. Then we have1 = (x ∨ y)y = (xy) ∧ (yy) = xy and thus x ≤ y. Conversely assume that x ≤ y, that is,xy = 1. In this case we have y = 1y = (xy)y = x ∨ y. Hence x vL y.

A subset F of a lattice implication algebra L is called a filter of L if for all x, y ∈ F itsatisfies

(F1) 1 ∈ F ,

(F2) x ∈ F and xy ∈ F imply y ∈ F .

For every filter F , it is easy to prove that if x ∈ F and x ≤ y then y ∈ F for all x, y ∈ L.Since any lattice implication algebra L is also a lattice with respect to the order ≤, we

can define a lattice filter FL of L as usual:

(LF0) F 6= ∅,(LF1) x ∈ FL and x ≤ y imply y ∈ FL,

(LF2) x ∈ FL and y ∈ FL imply x ∧ y ∈ FL.

In this case it is a natural question whether filters and lattice filters are the same. We canshow that they are not identical. Because, let F be a filter of a lattice implication algebraL. Suppose that x, y ∈ F . It is clear that xy ∈ F by y ≤ xy. Since x(x ∧ y) = x(x′ ∨ y′)′ =(x′ ∨ y′)x′ = (x′x′) ∧ (y′x′) = y′x′ = xy ∈ F , we have x ∧ y ∈ F . Thus every filter F is alsoa lattice filter. But the converse does not hold in general. There is a counter-example:

x x′

0 1a a1 0

→ 0 a 1

0 1 1 1a a 1 11 0 a 1

This is a lattice implication algebra and FL = {a, 1} is a lattice filter. But it is not afilter, because a ∈ FL and a0 = a′ = a ∈ FL but 0 /∈ FL.

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We define a new class wLIA of weak lattice implication algebras. By a weak latticeimplication algebra we mean an algebraic structure (L;→,′ , 1) of type (2,1,0) such that forall x, y, z ∈ L,

(1) x′′ = x and(2) the operation → satisfies the conditions:

(I0) 1 → x = x,

(I1) x → (y → z) = y → (x → z),

(I2) x → x = 1,

(I3) x → y = y′ → x′,

(I5) (x → y) → y = (y → x) → x.

As the above we denote simply xy instead of x → y. In any weak lattice implicationalgebra L we define a constant 0 by 0 = 1′. Then we have

x0 = 0′x′ = 1x′ = x′.

It is easy to show thatwLIA ⊇ LIA.

In the following we shall show that the converse relation holds, that is,

wLIA ⊆ LIA.

In order to do, at first, we have to show that any weak lattice implication algebra L isa lattice. Firstly, we define an order on any weak lattice implication algebra. Let L be anyweak lattice implication algebra in this section. For all x, y ∈ L, we define a relation v onL by

x v y ⇐⇒ x → y = 1.

Proposition 1. For all x, y, z ∈ L, we have

(xy)((yz)(xz)) = 1.

Proof. It is easy to prove from the following equations:

(xy)((yz)(xz)) = (xy)(x((yz)z))= (xy)(x((zy)y))= (xy)((zy)(xy))= (zy)((xy)(xy))= (zy)1= 1

It follows from the above that the relation v is a partial order.

Proposition 2. v is a partial order on L.

Proof. We only show the case of transitivity of v. Suppose that x v y and y v z. From theabove we have

1 = (xy)((yz)(xz)) = 1(1(xz)) = xz.

This means that x v z. Thus v is the partial order.

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We note from the above that(xy) v (yz)(xz).

For this order v we have following results.

Proposition 3. If x v y then yz v xz and zx v zy.

Proof. Suppose x v y, that is, xy = 1. Since

(yz)(xz) w xy = 1,

we haveyz v xz.

We also have

(zx)(zy) = 1{(zx)(zy)}= (xy){(zx)(zy)}= (zx){(xy)(zy)}= 1.

This means that zx v zy.

Next we can show that L is a lattice under this order v.

Lemma 1. For all x, y ∈ L, we have

x ∨ y = sup{x, y} = (xy)y, x ∧ y = inf{x, y} = (x′ ∨ y′)′.

Proof. Since x((xy)y) = (xy)(xy) = 1 and (I5), we have

x, y v (xy)y.

Let u ∈ L be any element such that x, y v u. Since

((xy)y)u = ((xy)y)(1u)= ((xy)y)((yu)u)= ((xy)y)((uy)y)w (uy)(xy)w xu = 1,

we haveu w (xy)y.

This means thatx ∨ y = sup{x, y} = (xy)y.

It is easy to prove the case of x ∧ y = inf{x, y} = (x′ ∨ y′)′.

Of course, since x1 = 1 and 0x = x′ 0′ = x′ 1 = 1, we have 0 v x v 1 for all x ∈ L.Hence any weak lattice implication algebra (L;→,′ , 1) is a bounded lattice.

We can also prove that x v y implies y′ v x′. In fact, Suppose that x v y. Sincey′ x′ = xy = 1, we get that y′ v x′.

In order to show thatwLIA ⊆ LIA,

we need to prove the axioms (I4), (L1) and (L2) hold in any weak lattice implication algebra.We shall prove them one by one.

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Proposition 4. (I4) holds in any weak lattice implication algebra.

Proof. Suppose that xy = yx = 1. From (I0) and (I5), we have

x = 1x = (yx)x = (xy)y = 1y = y.

Proposition 5. (L1) holds in any weak lattice implication algebra.

Proof. It is obvious that(x ∨ y)z v xz, yz.

Let u ∈ L be any element such that

u v xz and u v yz.

Since x(uz) = u(xz) = 1 and y(uz) = u(yz) = 1, we have x, y v uz. Hence

x ∨ y v uz.

This implies that1 = (x ∨ y)(uz) = u((x ∨ y)z)

and hence thatu v (x ∨ y)z.

Thus we can conclude that

(x ∨ y)z = inf{xz, yz} = (xz) ∧ (yz).

The axiom (L2) is left to be proved. We need to prepare some lemmas to show (L2).

Lemma 2. (x ∧ y)z = (xy)(xz)

Proof. It is proved from the equations:

(x ∧ y)z = (y ∧ x)z= z′ (y ∧ x)′

= z′ (y′ ∨ x′)= z′ ((y′ x′)x′)= (y′ x′)(z′ x′)= (xy)(xz).

We shall define a new operation ¯ on any weak lattice implication algebra L by

x¯ y = (xy′)′.

For the operation we have fundamental results.

Proposition 6. For all x, y, z ∈ L, we have

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(a) x¯ y = y ¯ x,

(b) x¯ y v x, y,

(c) x v y =⇒ x¯ z v y ¯ z,

(d) x¯ y v z ⇐⇒ x v yz,

(e) x¯ (xy) = x ∧ y.

Proof. We only show the cases of (d) and (e). For the case of (d), we have

x¯ y v z ⇐⇒ (xy′)′ v z

⇐⇒ z′ v xy′

⇐⇒ z′(xy′) = 1⇐⇒ x(z′ y′) = 1⇐⇒ x(yz) = 1⇐⇒ x v yz.

Concerning to the case of (e), we have

x¯ (xy) = (xy)¯ x

= ((xy)x′)′

= ((xy)(x0))= ((x ∧ y) 0)′

= (x ∧ y)′′

= x ∧ y.

Moreover we can show that the operator ¯ is distributive with respect to the operation∨.

Lemma 3. x¯ (y ∨ z) = (x¯ y) ∨ (x¯ z)

Proof. Since y, z v y ∨ z, we have x ¯ y, x ¯ z v x ¯ (y ∨ z). For any element u ∈ L suchthat x¯ y, x¯ z v u, since y ¯ x, z ¯ x v u, we have

y v xu, z v xu.

This implies that y ∨ z v xu and hence

x¯ (y ∨ z) v u.

Thus we obtain that

x¯ (y ∨ z) = sup{x¯ y, x¯ z} = (x¯ y) ∨ (x¯ z).

It follows from the above that any weak lattice implication algebra is a distributivelattice.

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Lemma 4. For all x, y, z ∈ L, we have

x ∧ (y ∨ z) = (x ∧ y) ∨ ((x ∧ z).

Proof. It is sufficient to show that

x ∧ (y ∨ z) v (x ∧ y) ∨ ((x ∧ z),

because the converse relation holds in general. It follows from the below.

x ∧ (y ∨ z) = (y ∨ z) ∧ x

= (y ∨ z)¯ ((y ∨ z)x)= {y ¯ ((y ∨ z)x)} ∨ {z ¯ ((y ∨ z)x)}v {y ¯ (yx)} ∨ {z ¯ (zx)}= (y ∧ x) ∨ (z ∧ x)= (x ∧ y) ∨ (x ∧ z)

Now we can show that the axiom (L2) holds for any weal lattice implication algebra.

Lemma 5. (x ∧ y)z = (xz) ∨ (yz)

Proof. This follows from the below

(xz) ∨ (yz) = ((xz)(yz))(yz)= {y((xz)z)}(yz)= (y(x ∨ z))(yz)= (y ∧ (x ∨ z))z= ((y ∧ x) ∨ (y ∧ z))z= ((y ∧ x)z) ∧ ((y ∧ z)z)= ((y ∧ x)z) ∧ 1= (y ∧ x)z= (x ∧ y)z.

Therefore we can prove that every weak lattice implication algebra is also a latticeimplication algebra, that is, wLIA ⊆ LIA and hence

wLIA = LIA.

3 BCK-algebras

We recall a definition of a BCK-algebra < X; ∗, 0 >. An algebra < X; ∗, 0 > of type (2,0)is called a BCK-algebra when for every x, y, z ∈ X it satisfies the conditions:

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(B1) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0,

(B2) (x ∗ (x ∗ y)) ∗ y = 0,

(B3) x ∗ x = 0,

(B4) 0 ∗ x = 0,

(B5) x ∗ y = y ∗ x = 0 implies x = y.

A BCK-algebra X is called commutative if it satisfies

(Com) : x ∗ (x ∗ y) = y ∗ (y ∗ x) for all x, y ∈ X.

We denote x ≤B y by x ∗ y = 0. Clearly the structure < X;≤B , 0 > forms a partiallyordered set. Let X be a BCK-algebra. If there is an element 1 ∈ X such that x ≤B 1 forall x ∈ X then the BCK-algebra is called bounded. By bcBCK we mean the class of allbounded commutative BCK-algebras.

Now we shall show that any weak lattice implication algebra (L;→,′ , 1) is a boundedcommutative BCK-algebra, that is,

wLIA ⊆ bcBCK.

Let (L;→,′ , 1) be a weak lattice implication algebra. For all x, y ∈ L, we define an operation∗ and constants 0B , 1B respectively by

x ∗ y = y → x, 0B = 1, 1B = 1′ = 0.

Then it is easy to show that the structure (L; ∗, 0B , 1B) is a bounded commutative BCK-algebra.

Conversely, we can show that each bounded commutative BCK-algebra (X; ∗, 0) is aweak lattice implication algebra. Suppose that (X; ∗, 0) is a bounded commutative BCK-algebra. We define an operation → and a constant 1L respectively by

x → y = y ∗ x, 1L = 0.

Then the algebraic structure (X;→, 1L) is a weak lattice implication algebra. This meansthat

bcBCK ⊆ wLIA.

Therefore we can conclude that

Theorem 1. The class wLIA of all weak lattice implication algebras is categorically equiv-alent to the class bcBCK of all bounded commutative BCK-algebras, that is,

wLIA = bcBCK.

It is also well-known that bcBCK is categorically equivalent to the class MV of allMV -algebras and that so is the class W of all Wajsberg algebras. Hence we have

Corollary 1. wLIA = pLIA = bcBCK = MV = W

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4 Lattice H-implication algebras and Boolean algebras

In this section we shall prove that

LHIA = BA,

where BA means the class of all Boolean algebras. It is easy to show that every Booleanalgebra is a lattice H-implication algebra. Since the class of all lattice H-implication algebrasis characterized by the class wLIA of weak lattice implication algebras with the condition

(H) : x ∨ y ∨ (x ∧ y → z) = 1 for all x, y, z.

This condition (H) is equivalent to x ∨ (x → y) = 1 for all x, y and to x ∨ x′ = 1 for allx in any weak lattice implication algebra L. Therefore the class LHIA of all lattice H-implication algebras is the same as the class wLIA of all weak lattice implication algebraswith the condition x ∨ x′ = 1.

We have already proved that each weak lattice implication algebra L ∈ wLIA is also adistributive lattice. This implies that it is a Boolean algebra. Thus we can conclude that

Theorem 2. LHIA = BA

From the above we can characterize an implicative filter of a lattice implication algebras.Let L be a weak lattice implication algebra. By an implicative filter of L we mean a subsetF of L such that

(IF1) 1 ∈ F ,

(IF2) x(yz) ∈ F and xy ∈ F imply xz ∈ F .

For any filter F of L we define a relation ∼F by

x ∼F y ⇐⇒ xy ∈ F and yx ∈ F .

Then it is easy to prove that the relation ∼F is a congruence on L. We denote thequotient algebra modulo ∼F simply by L/F . That is,

L/F = {x/F | x ∈ F}, x/F = {y ∈ L | x ∼F y}.

We define an operation → on L/F by

x/F → y/F = (x → y)/F.

Since wLIA is the variety, L/F is also a weak lattice implication algebra. Moreover ifF is an implicative filter then we can show that L/F is a lattice H-implication algebra andhence that L/F is a Boolean algebra. To prove this result, we note that the H condition isequivalent to the condition x ∨ x′ = 1 for all x ∈ L for any lattice implication algebra L.

Remark. In any BCK-algebra X, a subset I of X is called an ideal if

(I1) 0 ∈ I,

(I2) x ∈ I and y ∗ x ∈ I imply y ∈ I.

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It is clear that the notion of ideal is a dual form of the notion of filter in the theory of latticeimplication algebras. Moreover an ideal I is called a positive implicative ideal of X if

x ∗ (y ∗ z), x ∗ y ∈ I =⇒ x ∗ z ∈ I.

Clearly this is a dual notion of implicative filter of the lattice implication algebra. So, thename of implicative filter is not good.

In the following we show the characterization theorem of implicative filter of a weaklattice implication algebra. We need some lemmas.

Lemma 6. Let L be a weak lattice implication algebra. Then the following conditions areequivalent:

(1) x(xy) = xy,

(2) x(yz) = (xy)(xz),

(3) x ∨ x′ = 1.

Proof. (1) =⇒ (2): Since yz v (xy)(xz), we have

x(yz) v x((xy)(xz))= (xy)(x(xz))= (xy)(xz).

On the other hand we also have

((xy)(xz))(x(yz)) = ((xy)(xz)(y(xz))= y (((xy)(xz)) (xz))= y (((xz)(xy)) (xy))= ((xz)(xy))(y(xy))= ((xz)(xy))1= 1

It follows that (xy)(xz) v x(yz) and hence that

x(yz) = (xy)(xz).

(2) =⇒ (3): Sincexx′ = x(x0) = (xx)(x0) = 1x′ = x′,

we getx ∨ x′ = (xx′)x′ = x′x′ = 1.

(3) =⇒ (1): It follows from 0 v y that x′ = x v xy and hence that

1 = x ∨ x′ v x ∨ xy = (x(xy))(xy).

This means thatx(xy) v xy.

On the other hand it is obvious that xy v x(xy). So we have

x(xy) = xy.

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Lemma 7. Let F be a filter of a weak lattice implication algebra L. Then the following areequivalent:

(1) F is an implicative filter,(2) x(xy) ∈ F implies xy ∈ F for all x, y ∈ F ,(3) x(yz) ∈ F implies (xy)(xz) ∈ F for all x, y, z ∈ F ,(4) (x(yz)) ((xy)(xz)) ∈ F for all x, y, z ∈ F .

Proof. It is easy to prove this lemma according to the dual case of positive implicative idealin the bounded commutative BCK-algebra. So we omit the proof.

We note that the condition (4) above does not have a conditional form. This plays animportant role to show the next result.

Lemma 8. A filter F of a weak lattice implication algebra L is an implicative filter if andonly if L/F satisfies the condition u(vw) = (uv)(uw) for all u, v, w ∈ L/F.

Proof. Suppose that x(yz) ∈ F and xy ∈ F . For the sake of simplicity we take u = x/F ,v = y/F and w = z/F in L/F . It follows from supposition that u(vw) = uv = 1/F. So wehave

1/F = u(vw) = (uv)(uw) = uw.

This implies that xz ∈ F and hence that F is the implicative filter.Conversely, we assume that F is the implicative filter. From the lemma above we have

(x(yz)) ((xy)(xz)) ∈ F.

On the other hand we get that

((xy)(xz)) (x(yz)) = ((xy)(xz)) (y(xz))= y (((xy)(xz)) (xz))= y (((xz)(xy)) (xy))= ((xz)(xy)) (y(xy))= ((xz)(xy)) 1= 1

This means that in L/F we have u(vw) = (uv)(uw).

From these we obtain the characterization theorem of implicative filters.

Theorem 3. Let F be a filter of a weak lattice implication algebra. Then the following areequivalent:

(1) F is an implicative filter,(2) L/F satisfies u(vw) = (uv)(uw) for all u, v, w ∈ L/F ,(3) L/F satisfies u ∨ u′ = 1/F for all u ∈ L/F ,(4) L/F is a lattice H-implication algebra,(5) L/F is a Boolean algebra.

Acknowledgements. One of authors’ (M.Kondo) visit to Poland in August 2003 was sup-ported by Institute of Mathematics (IM), Wroclaw University of Technology. The instituteprovided comfortable time to our joint research. We are grateful to IM for everything.

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5 Comments

An algebra < X; ∗, 0 > of type (2,0) is called a ÃLukasiewicz algebra (see [11]), if for everyx, y, z ∈ X it satisfies the conditions:

(1) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0,

(2) x ∗ (x ∗ y) = y ∗ (y ∗ x),

(3) (x ∗ y) ∗ (y ∗ x) = x ∗ y,

(4) (x ∗ y) ∗ x = 0,

(5) x ∗ 0 = x.

Any ÃLukasiewicz algebra is a commutative BCK-algebra [7]. In fact, it is a subdirectproduct of totally ordered BCK-algebras [7].

From result of this paper follows that the class of bounded ÃLukasiewicz algebras is thesame as the class of lattice implication algebras satisfying the identity (x∗y)∗ (y ∗x) = x∗y.

The second important class of BCK-algebras is a class of positive implicative BCK-algebras, i.e. BCK-algebras with the identity (x∗y)∗z = (x∗z)∗(y∗z). Such BCK-algebrascan be used to the description of Hilbert algebras

An algebra < X; •, 1 > of type (2,0) is called a Hilbert algebra if for every x, y, z ∈ X thefollowing conditions hold:

(H1) (x • (y • z)) • ((x • y) • (x • z)) = 1,

(H2) x • (y • x) = 1,

(H3) x • y = y • x = 1 implies x = y.

Any Hilbert algebra satisfies the following identities (see [6])

x • ((x • y) • y) = 1,

(x • y) • ((y • z) • (x • z)) = 1,

(y • z) • ((x • y) • (x • z)) = 1.

Any (bounded) Hilbert algebra is dual to a (bounded) positive implicative BCK-algebra[8], i.e. an algebra < X; •, 1 > of type (2,0) is a Hilbert algebra if and only if an algebra< X; ∗, 0 > where 0 = 1 and x • y = y ∗ x is a positive implicative BCK-algebra. InHilbert algebras the same role as ideals in BCK-algebras play so-called deductive systems(cf. [5, 9, 10]) which, in fact, are a dual form of ideals.

References

[1] K. Iseki and S. Tanaka, An introduction tho the theory of BCK-algebras, Math. Japon-ica, vol.23 (1978), 1-26.

[2] E. Turunen, Mathematics behind fuzzy logic, Physica-Verlag, Springer 1999.

[3] Y. Xu, Lattice implication algebras (Chinese), J. Southwest Jiaotong Univ. (1993), 20-27.

[4] Y. Xu and K. Y. Qin, Lattice H implication algebras and lattice implication algebraclasses (Chinese), J. Hebin Mining and Civil Engineering Institute (1992), 139-143.

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[5] D. Busneag, A note on deductive systems of a Hilbert algebra, Kobe J. Math. vol.2(1985), 29-35.

[6] A. Diego, Sur les algebres de Hilbert, Collection de Logique Math. ser. A (Ed. Hermann,Paris), vol.21 (1966), 1-52.

[7] W. A. Dudek, On the axioms system for BCI-algebras, Matematichni Studii (Lvov)vol.3 (1994), 5-9.

[8] W. A. Dudek, On embedding Hilbert algebras in BCK-algebras, Math. Moravica vol.3(1999), 25-28.

[9] W. A. Dudek, On ideals in Hilbert algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat.Math. vol.38 (1999), 31-34.

[10] Y. B. Jun, Deductive systems of Hilbert algebras, Math. Japonicae vol.43 (1996), 51-54.

[11] M. PaÃlasinski, Some remarks on BCK-algebras, Math. Sem. Notes Kobe Univ. vol.8(1980), 137-144.

M. Kondo: School of Information Environment,Tokyo Denki University,Inzai, 270-1382, Japan.E-mail: kondo@sie.dendai.ac.jp

W. A. Dudek: Institute of Mathematics,Wroclaw University of TechnologyWybrzeze Wyspianskiego 2750-370 Wroclaw, PolandE-mail: dudek@im.pwr.wroc.pl

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