Stonean Hilbert algebra

1 Introduction

The variety of Hilbert algebras is an important tool for investigations in intuitionistic logic and other non-classical logics. Hilbert algebras represent the algebraic counterpart of the implicative fragment of Intuitionistic Propositional Logic. The notion of a Hilbert algebra was studied by A. Diego [7] and developed by D. Busneag [1, 2], D. Buşneag and M. Ghita [3], S. A. Celani and D. Montangie [5], S. A. Celani [4], S. M. Hong and Y. B. Jun [10] and Figallo et al. [9]. Hilbert algebras with infimum are the algebraic counterpart of a propositional calculus weaker than the {→ , ∧}-fragment of the intuitionistic propositional calculus. The associated order of Hilbert algebras with infimum is a meet-semilattice. Also Hilbert algebras with supremum are Hilbert algebras in which the associated order is a join- semilattice. This class of algebras is a particular class of BCK-algebras with lattice operations and was studied by Idziak [11]. In continuing our study in Hilbert algebra, we define a Stonean Hilbert algebra as an algebraic structure which is weaker than Hilbert algebra and is a good step for better understanding this algebraic structure. The structure of the paper is as follows: in Section 2, some definitions and properties about Hilbert algebras are recalled. In Section 3, a Stonean implicative filter in Hilbert algebra is introduced and some of its properties are investigated. In Section 4, the notion of a Stonean Hilbert algebra is introduced and we have proved H is a Stonean Hilbert algebra if and only if R (H) is a Stonean Hilbert algebra. Also, we prove that F, is a Stonean implicative filter of the Hilbert algebra H if and only if H/F is a Stonean Hilbert algebra.

2 Preliminaries

We recall some basic definitions and results that are necessary for this paper.

A Hilbert algebra is an algebra (H, → , 1) of type (2, 0) such that the following axioms hold, for all x, y, z ∈ H [7]

x → (y → x) =1;

For a Hilbert algebra H, (H, ≤) is a poset by defining an order relation ≤ such that x ≤ y if and only if x → y = 1 (called the natural order on H), with respect to this order, 1 is the greatest element of H. If H has a smallest element 0, we say that H is bounded, in this case for x ∈ H we have x^* = x → 0.

Proposition 2.1. [3] If H is a bounded Hilbert algebra and x, y ∈ H, then

0^* = 1, 1^* = 0;

Definition 2.2. [5] An algebra (H, → , ∨ , 1) of type (2, 2, 0) is a Hilbert algebra with supremum or sH-Hilbert algebra if

(H, → , 1) is a Hilbert algebra;

A subset D of Hilbert algebra H is called a deductive system (or an implicative filter or simply filter) of H if: (i) 1 ∈ D, (ii) If x ∈ D and x → y ∈ D, then y ∈ D, for all x, y ∈ H.

Let D be a deductive system of a Hilbert algebra. If x ≤ y and x ∈ D, then y ∈ D. We denote Ds (H)= {D: D is a deductive system of H}. If H is bounded, then the deductive system D is proper if and only if 0 ∉ D.

We note that, in some papers the deductive system is called implicative filter too. We use implicative filter in this paper. An implicative filter F of a bounded Hilbert algebra H is a Boolean filter of the first kind if x^** → x ∈ F, for all x ∈ H. Also an implicative filter F of H is called a Boolean filter of the second kind if x ∨ x^* ∈ F, for all x ∈ H. An implicative filter F of a Hilbert algebra with supremum H is a prime filter of the first kind if x ∨ y ∈ F implies x ∈ F or y ∈ F. Also, an implicative filter F of a Hilbert algebra with supremum H is a prime filter of the second kind if x → y ∈ F or y → x ∈ F, for all x, y ∈ H. Also, an implicative filter F of a Hilbert algebra with supremum H is a prime filter of the third kind if (x → y) ∨ (y → x) ∈ F, for all x, y ∈ H. Let F be an implicative filter of a bounded Hilbert algebra H. Define: x ≡  _F y if and only if x → y ∈ F and y → x ∈ F. Then ≡ _F is a congruence relation on H. The set of all congruence classes is denoted by H/F, i.e, H/F : = {[x] : x ∈ H}, where [x] = {y ∈ H:x ≡  _F y}. Define → on H/F as follows: [x] → [y] = [x → y], and 1 = 1/F = F.

Therefore (H/F, → , [1] , [0]) is a bounded Hilbert algebra with respect to F and the order relation on H/F is given by [x] ≤ [y] if and only if x → y ∈ F. Clearly, [x] = [1] if and only if x ∈ F.

Theorem 2.3. [3] For a bounded Hilbert algebra H, the following assertions are equivalent:

x^** = x, for every x ∈ H;

Corollary 2.4 [3] A bounded Hilbert algebra H is a Boolean algebra (relative to natural ordering) if and only if for every x, y ∈ H we have (x → y) → x = x.

Corollary 2.5. [3] For a bounded Hilbert algebra H, the following assertions are equivalent:

H is Boolean algebra (relative to naturalordering);

Definition 2.6. [20] A non-empty subset F of Hilbert algebra H is called

A positive implicative filter of H, if 1 ∈ F and x → ((y → z) → y) ∈ F and x ∈ F imply y ∈ F, for every x, y, z ∈ H.

Theorem 2.7. [20] In a bounded Hilbert algebra H, positive implicative filter, fantastic filter and Boolean filter of the first kind are equivalent.

Definition 2.8. [1] An implicative filter M of a Hilbert algebra H is called maximal if it is not properly contained in any other proper implicative filter of H.

For any Hilbert algebra H, Max (H) denotes the set of all maximal implicative filters of H.

Theorem 2.9. [1] An implicative filter M of a bounded Hilbert algebra H is maximal if and only if for all x ∈ H, if x ∉ M, then x^* ∈ M.

Theorem 2.10. [20] Let F be an implicative filter of bounded Hilbert algebra H. F is a maximal filter if and only if F is a prime filter of the first kind and a Boolean filter of the second kind.

Definition 2.11. [1] Hilbert algebra H is semisimple if the intersection of all maximal implicative filters of H is {1}.

Definition 2.12. [10] The intersection of all maximal implicative filters of Hilbert algebra H is called the radical of H and it is denoted by Rad (H).

Definition 2.13. [1] Let H be a Hilbert algebra. We say that an element x ∈ H is regular if for every y ∈ H we have (x → y) → x = x. We denote by R (H) the set of all regular elements of H. We say that an element x ∈ H is dense if for every r ∈ R (H) we have x → r = r. We denote the set of all dense elements of H by D (H).

Proposition 2.14. [1] If H is a bounded Hilbert algebra, then

D (H) = {x ∈ H : x^* = 0};

Theorem 2.15. [19] The following statements hold:

R (H) = {x^* : x ∈ H} = {x^** : x ∈ H};

A Hilbert algebra is said to be local if and only if it has exactly one maximal filter [6].

Proposition 2.16. [19] H is a local Hilbert algebra if and only if D (H) = H \ {0}.

Definition 2.17. [17] An implicative filter F of H is called an implication filter, if satisfy:

((x → y) → x) → x ∈ F, for all x, y ∈ H such that y ≠ 0.

(If y = 0, then ((x → 0) → x) → x ∈ F, so (x^* → x) → x = x^** → x ∈ F. Thus F is a Boolean filter of the first kind of H.)

Definition 2.18. [17] A bounded Hilbert algebra H is called an implication Hilbert algebra if it satisfies in the following condition:

(I) (x → y) → x = x, for all x, y ∈ H such that y ≠ 0.

(If y = 0, then (x → 0) → x = x^** = x. Thus H is a Boolean algebra.)

3 Stonean implicative filter

In this section, H is a bounded Hilbert algebra with supremum, unless otherwise is stated.

Definition 3.1. An implicative filter F of H is called a Stonean implicative filter if x^* ∨ x^** ∈ F, for all x ∈ H.

The following example shows that the notions of implicative filters and Stonean implicative filters are not the same.

Example 3.2.

Let H = {0, a, b, c, 1}, with 0 < a, b < c < 1, but a, b are incomparable. We define → 0 a b c 1 0 1 1 1 1 1 a b 1 b 1 1 b a a 1 1 1 c 0 a b 1 1 1 0 a b c 1 (H, → , 0, 1) is a Hilbert algebra with supremum, but it is not a local Hilbert algebra. The implicative filter F = {c, 1} is a Stonean implicative filter, since x^* ∨ x^** ∈ F, for all x ∈ H. Also, implicative filters {a, c, 1} and {b, c, 1} are Stonean implicative filters, but G = {1} is not a Stonean implicative filter, since a^* ∨ a^** = b ∨ a = c ∉ G.

Theorem 3.3. (Extension theorem for Stonean implicative filter) Let F be a Stonean implicative filter of H and G be an implicative filter of H where F ⊆ G. Then G is a Stonean implicative filter of H.

Corollary 3.4. The implicative filter {1} of H is a Stonean implicative filter if and only if every implicative filter of H is a Stonean implicative filter.

Theorem 3.5.

If M is a maximal filter of H, then M is a Stonean implicative filter.

Proof.

Let M be a maximal filter of H and x ∈ H. If x ∈ M, then x^** ∈ M, since x ≤ x^**. Therefore x^* ∨ x^** ∈ M. Thus M is a Stonean implicative filter. But if x ∉ M, then x^* ∈ M. Thus x^* ∨ x^** ∈ M, for all x ∈ H. Hence M is a Stonean implicative filter.

In the following example, we show that the converses of (i) and (ii) in the above theorem are not true.

Example 3.6. Let H = {0, a, b, 1}. Define the following operation → 0 a b 1 0 1 1 1 1 a b 1 b 1 b a a 1 1 1 0 a b 1

Then (H, → , 0, 1) is a Hilbert algebra with supremum, but it is not local, because (a → b) ^* ≠ (b → a) ^*. The implicative filters {1} , {a, 1} , {b, 1} and {a, b, 1} are Stonean. We see that F = {1} is not a maximal filter of H. Also, H {0} = {a, b, 1} and D (H) = {1} are Stonean implicative filters but H is not a local Hilbert algebra.

Proposition 3.7. If F is a Boolean filter of the first kind of a local Hilbert algebra of H, then F is a Stonean implicative filter H.

The following example shows that the converse of the above proposition is not true.

Example 3.8. → 0 a b c 1 0 1 1 1 1 1 a 0 1 b b 1 b 0 a 1 a 1 c 0 1 1 1 1 1 0 a b c 1

Then (H, → , 0, 1) is a local Hilbert algebra with supremum. The implicative filter F = {1} is a Stonean implicative filter, since x^* ∨ x^** ∈ F, for all x ∈ H, but it is not a Boolean filter of the first kind of H.

Proposition 3.9. Every Boolean filter of the second kind of a bounded Hilbert algebra H is also a Stonean implicative filter of H.

Proof. Let F be a Boolean filter of the second kind of a bounded Hilbert algebra H and x ∈ H. We have x ≤ x^**, thus x ∨ x^* ≤ x^* ∨ x^**. Therefore x^* ∨ x^** ∈ F, for all x ∈ H. Hence F is a Stonean implicative filter of H. □

In the following example, we show that the converse of Proposition 3.9 is not true.

Example 3.10. In Example 3.8 (ii), F = {1} is a Stonean implicative filter of H, but is not a Boolean filter of the second kind of bounded Hilbert algebra H.

The converse of Proposition 3.9 is true in particular condition.

Proposition 3.11. Let F be a Stonean filter of H and (x → y) → x = x, for all x, y ∈ H. Then F is a Boolean filter of the second kind.

4 Stonean Hilbert algebra

In this section, H is a Hilbert algebra with supremum, unless otherwise is stated.

Definition 4.1. H is called a Stonean Hilbert algebra, if x^* ∨ x^** = 1, for all x ∈ H.

Example 4.2. In Example 3.8 (ii), H = {0, a, b, c, 1} is a Stonean local Hilbert algebra. Also, in Example 3.6, H = {0, a, b, 1} is a Stonean Hilbert algebra but it is not a local Hilbert algebra. Moreover, in Example 3.2 (i), H = {0, a, b, c, 1}, is neither a Stonean Hilbert algebra nor a local Hilbert algebra. In Example 3.2 (iii), H = {0, a, b, c, d, e, f, g, 1} is a Stonean Hilbert algebra.

Theorem 4.3. H is a Stonean Hilbert algebra if and only if every implicative filter F of H is a Stonean implicative filter of H.

Proof. Let F be an implicative filter of the Stonean Hilbert algebra H. Then x^* ∨ x^** = 1, for all x ∈ H. So, x^* ∨ x^** ∈ F, for all x ∈ H. Therefore F is a Stonean implicative filter of H.

Conversely, let every implicative filter F of H be a Stonean implicative filter of H. Then F = {1} is a Stonean implicative filter of H. Therefore x^* ∨ x^** = 1, for all x ∈ H. Hence H is a Stonean Hilbert algebra.

□

Corollary 4.4. H is a Stonean Hilbert algebra if and only if {1} is a Stonean implicative filter.

Theorem 4.5 If H is a local Hilbert algebra, then H is a Stonean Hilbert algebra.

Proof. Let H be a local Hilbert algebra. Then D (H) = H \ {0}, by Proposition 2.16. Thus x^* ∨ x^** = 1, for all x ∈ H. Therefore, H is a Stonean Hilbert algebra. □

In Example 4.2, we showed that the converse of above theorem is not correct.

Corollary 4.6. If H is a local Hilbert algebra, then

H/F is a Stonean Hilbert algebra, for every implicative filter F of H.

In the following theorem we show the relationship between F and H/F.

Theorem 4.7. Let F be an implicative filter of H. Then F is a Stonean implicative filter if and only if H/F is a Stonean Hilbert algebra.

Proof. F is an implicative filter of H if and only if x^* ∨ x^** ∈ F, for all x ∈ H iff [x] ^* ∨ [x] ^** = [1], for all x ∈ H if and only if H/F is a Stonean Hilbert algebra.

Proposition 4.8. Every Boolean algebra H is a Stonean Hilbert algebra.

In the following we show that the converse of above proposition is not correct.

Example 4.9. In Examples 3.8 (ii), H is a Stonean Hilbert algebra, but it is not a Boolean algebra.

Corollary 4.10. If F is a Boolean (maximal) filter of H, then H/F is a Stonean Hilbert algebra.

Corollary 4.11. In a bounded Hilbert algebra H, every positive implicative filter (fantastic filter) is a Stonean implicative filter.

Example 4.12. In Example 3.8 (ii), H = {0, a, b, c, 1} is a Hilbert algebra with supremum. The implicative filter F = {1} is a Stonean implicative filter, but it is not a positive implicative filter (fantastic filter) of H.

Theorem 4.13. Let (x → y) ^* = (y → x) ^*, for all 0 ≠ x, y ∈ H. Then H is a Stonean Hilbert algebra.

Proof. Let (x → y) ^* = (y → x) ^*, for all 0 ≠ x, y ∈ H. For y = 1, we have x^* = 0, for all 0 ≠ x ∈ H. Thus x^* ∨ x^** = 1, for all x ∈ H. Therefore H is a Stonean Hilbert algebra. □

Corollary 4.14.

Let x^** → y^** = y^** → x^**, for all x, y ∈ H \ {0, 1}. Then H is a Stonean Hilbert algebra.

Proof.

(x → y) ^* = (x → y) ^*** = (x^** → y^**) ^* = (y^** → x^**) ^* = (y → x) ^*** = (y → x) ^*, for all 0 ≠ x, y ∈ H.

In the following example we show that the converse of above corollary and Theorem 4.13 is not correct.

Example 4.15. In Example 3.6, H = {0, a, b, 1} is a Stonean Hilbert algebra. But, we have

(a → b) ^* ≠ (b → a) ^*. Thus, the converse of Theorem 4.13 is not correct.

Theorem 4.16. H is a Stonean Hilbert algebra if and only if R (H) is a Stonean Hilbert algebra.

Proof. Let H be a Stonean Hilbert algebra. Then x^* ∨ x^** = 1, for all x ∈ H. Thus x^** ∨ x^*** = x^** ∨ x^* = 1, for all x^* ∈ R (H). Hence R (H) is a Stonean Hilbert algebra.

Conversely, let R (H) be a Stonean Hilbert algebra. Thus x^** ∨ x^*** = 1, for all x^* ∈ R (H). Therefore x^** ∨ x^* = 1, for all x ∈ H. Hence H is a Stonean Hilbert algebra. □

In the following example, we show that the relation between an implication filter (implication algebra) and a Stonean implicative filter (Stonean Hilbert algebra).

Example 4.17.

In Example 3.2 (ii), F = {1} is an implication filter, thus H is an implication algebra. But, F = {1} is not a Stonean implicative filter, also H is not a Stonean Hilbert algebra.

In [16] Hilbert algebra is called “positive implication algebra”.

Theorem 4.18. In a bounded Hilbert algebra H, an implication algebra and a Boolean algebra areequivalent.

In the following example, we show that the bounded condition in above theorem is essential.

Example 4.19. Let H = {a, b, c, d, 1}. We define → a b c d 1 a 1 1 1 d 1 b c 1 c d 1 c b b 1 d 1 d a b c 1 1 1 a b c d 1

Then (H, → , 1) is an implication algebra, since H is a Hilbert algebra such that (x → y) → x = x, for all x, y ∈ H. But it is not a Boolean algebra, since H is not bounded.

In ([9, 15]) A Hertz algebra is an algebra (A, → , ∧ , 1) of type (2, 2, 0) which satisfies the following axioms: (i) x → x = 1; (ii) (x → y) ∧ y = y; (iii) x ∧ (x → y) = x ∧ y; (iv) x → (y ∧ z) = (x → z) ∧ (x → b). In [8] it is proved for a Hilbert algebra H, H is a Hertz algebra if and only if H is a Hilbert algebra with infimum which verifies property (P), which (P): x ← (y ← (x ^ y))=1. .

5 Conclusion and future research

In this paper, we studied the Stonean implicative filter in a Hilbert algebra and investigated some of its properties. The notion of Stonean Hilbert algebra is introduced and we have proved theorems which determine some properties of this notion in Hilbert algebra. The results of this paper will be devoted to study of Hilbert algebra, intuitionistic^,s logic which are different extensions of basic logic.

In the first diagram (Fig. 1) we summarize the results of this paper and the previous results in this field and give the relationships between all types of filters of Hilbert algebra. Also, in the second diagram (Fig. 2) we give the relationships between Hilbert algebra and other algebraic structures. By A → B ( A a B or A [ a ] B ) , we means that A implies B (respectively, A implies B with the condition “a”).

Acknowledgments

The authors are extremely grateful to the referees for their valuable comments and helpful suggestions which helped to improve the presentation of this paper.