Abstract
The concepts of Hilbert implication algebra and generalized Hilbert implication algebra are introduced. The comparison theorem of Hilbert implication algebra and generalized Hilbert implication algebra is proved. In addition, the idea of groupoid and commutative Hilbert implication algebras is investigated. Ideals and filters in Hilbert implication algebras are also discussed. In general, different theorems which show different properties are proved.
1. Introduction
Hilbert algebras are important in investigating certain algebraic logic, since they can be considered as a fragment of any propositional logic containing logical connective implications. The concept of a −closed set and a − homomorphism in lattice implication algebras and some of their properties is elaborated by Roh et al. in [1].
Commutative Hilbert algebra was initiated by Halas in [2]. The concepts of preimplication algebra and implication algebra based on orthosemilattice which generalize the concepts of implication algebra was discussed by Chajda in [3], and the notion of generalized Hilbert algebra with some other properties was investigated by Borzooei and Shohani in [4].
The notion of B-Almost distributive fuzzy lattice in terms of its principal ideal fuzzy lattice was introduced by Assaye et al. in [5]. The concepts of Pseudo - Supplemented almost distributive fuzzy Lattice with some other propertis is introduced by Gerima Tefera in [6].
In this paper, we introduced the concepts of Hilbert implication algebra and generalized Hilbert implication algebras are investigated. Furthermore, groupoids in an implication algebra, commutative properties in a Hilbert implication algebra with some other properties are introduced.
Throughout this paper, represents Hilbert implication algebra unless otherwise mentioned.
2. Preliminaries
Definition 1 (see [7]). An algebra of type is called implication algebra if the following condition holds:(1), (2), (3), (4), , ,
Definition 2 (see [7]). Let be an implication algebra. Then a nonempty subset of an implication algebra is called a subalgebra of if , , then .
3. Main Results
Definition 3. An algebra of type is said to be Hilbert implication algebra if it satisfies the following for all , :
:
:
: If , then
Definition 4. Let be a Hilbert implication algebra and “≤” be a partially ordered relation defined by implies that for , .
Example 1. Let with define “” by the following table:
.
Hence, holds. and .
Which imply .
Hence, holds. and . Implies a .
Hence, holds. Hence, is Hilbert implication algebra with respect to “≤”.
Definition 5. is a groupoid if it satisfies the following axioms for all.
:
: (contraction)
: (quasicommutative)
: (exchange rule)
Lemma 6. A groupoid is a Hilbert implication algebra.
Proof. Let be a groupoid for , , .
by .
Hence, . Thus, holds.
by , since , since , and . Hence, .
Suppose , such that . Assume . Then, . But . Hence, . Again, assume . Then, . But . Hence, . Therefore, . Thus, a groupoid is a Hilbert implication algebra.☐
Definition 7. Let be a Hilbert implication algebra. Then, for any , we have .
Lemma 8. Let be a Hilbert implication algebra. Then, the following hold for all :(1)(2)(3)If , then .
If , then .
if and only if .
Proof. (1), since by Definition 4.(2)Let , and 1 is largest element in . Then, if and if . But 1 is largest element in we have . Hence, .(3)Let . Then, . But , for all Hence, for all .(4)Let and 1 is the largest element in . Suppose . But . So that we have . Imply that (5)Let . Then . By Definition 4, we have . But Hence, .
Conversely suppose , for . .
So that we have , since .
Let , and let .
Now, .
Imply that by hypothesis.
Thus, .
Conversely, let , and . Now .
But .
Therefore, .☐
Example 2. Let with . Then, defined by the following table is not Hilbert implication algebra
Since Hence, is not satisfied.
Lemma 9. Let be implication Hilbert algebra, and . Then, is a poset.
Proof. (1)Let and . Imply that . Hence, “≤” is reflexive(2)Let , and , and . So that we have and Imply . Hence, by Definition 3. Thus, “≤” is antisymmetric.(3)Let with , and . Then, , and Now imply . Imply that by Definition 4.
Therefore, is a poset.☐
Example 3. Let with . Then, given by the following table is a poset.
Since(1), , . Hence, reflexive property holds.(2), and . . Imply that a . Hence, antisymmetric property holds(3), and . Imply that . Hence, the transitive property holds. Thus, is a poset
Theorem 10. Let be Hilbert implication algebra of type . Then, .
Proof. Let . Then, , since . Now . Hence, .☐
Corollary 11. If is a Hilbert implication algebra, then implies that for any .
Lemma 12. Let be a Hilbert implication algebra. Then, for any , the following hold: (1) implies . (2) implies . (3) .
Proof. (1)Let , and . Then by Definition 4, . So that we have if . Implies taking equality(2)Let , and . Now , since , and , since . Hence, (3)Let , and let 1 be the largest element. , since , and . Therefore, for all ☐
Theorem 13. If is a Hilbert implication algebra, then
for any .
Definition 14. A Hilbert implication algebra is said to be commutative if
for any .
Example 4. The table in Example 3 is a commutative Hilbert implication algebra.
Proposition 15. If is a commutative Hilbert implication algebra. Then(1)(2)
Proof. (1)Let . Then, , since , and (2)Let . Then, , since imply , and imply . Hence, for any .☐
Definition 16. Let H be a Hilbert implication algebra and a nonempty subset of is called an ideal of if it satisfies for all :(a)(b)If and . Then
Definition 17. Let be a Hilbert implication algebra and a nonempty subset of is called a filter of if it satisfies the following for all : (1) . (2) If and , then .
Theorem 18. If is a commutative Hilbert implication algebra, and
, then .
Proof. Suppose be a commutative Hilbert implication algebra and , for any . Now , since by , since by definition of Hilbert implication algebra. Hence, .☐
Definition 19. An algebra is said to be a generalized Hilbert implication algebra with a binary operation “” and constant 1 which satisfies the following axioms for all :
:
:
:
: , for all
Proposition 20. Every Hilbert implication algebra is a generalized Hilbert implication algebra. But the converse does not hold.
Proof. Suppose be a Hibert implication algebra. Then, some axiom of generalized Hilbert implication algebra is satisfied by Lemma 8 and .
What is remain to prove , for any .
By of definition of Hilbert implication algebra, we have
. Then, we have the following conditions:(1)(2)(3)So that case one is straightforward. For case two it holds for equality. But case three does not lead to true conclusion.
Hence, . Thus, the forward condition holds.
To show the converse does not holds, consider the following example for and a binary operation defined by the table below:
For all , we have(1), (2), and (3), and . Hence, (4), and . Hence, Therefore, is a generalized Hilbert implication algebra. Since . Hence, is not a Hilbert implication algebra.☐
Lemma 21. In any generalized Hilbert implication algebra , the following holds for all :(1)(2)(3)(4)(5)(6)(7)
Proof. Let be a generalized Hilbert implication algebra (Tables 1–4).(1)Hence, .(2), since . Thus, (3) by definition of generalized Hilbert implication algebra. . Which imply that (4) by definition of generalized Hilbert implication algebra. . Hence, (5). So that we have (6), since . by definition of generalized Hilbert implication algebra. . Therefore, (7). It follows that ☐
4. Conclusion
In this paper, the idea of Hilbert implication algebra and Generalized Hilbert implication algebra is introduced. Some of the properties of Hilbert implication algebra are investigated. Congruence relations on Hilbert implication algebra and some other properties will be considered as future research work.
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Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
The author of this paper would like to thank the referees for their genuine comment for the improvement of the manuscript.