Compatible Deductive Systems of Pulexes

Abstract

The notion of (compatible) deductive system of a pulex is defined and some properties of deductive systems are investigated. We also define a congruence relation on a pulex and show that there is a bijective correspondence between the compatible deductive systems and the congruence relations. We define the quotient algebra induced by a compatible deductive system and study its properties.

1. Introduction

Imai and Iséki [1] introduced the concept of BCK-algebra as a generalization of notions of set difference operation and propositional calculus. The notion of pseudo-BCK algebra was introduced by Georgescu and Iorgulescu [2] as generalization of BCK-algebras not assuming commutativity.

Hájek [3] introduced the concept of BL-algebras as the general semantics of basic fuzzy logic (BL-logic). Iorgulescu studied BCK-algebras and their relation to BL-algebras [4, 5]. BL-algebra has been generalized in different ways [6–8]. Hájek in [9] introduced flea-algebra as a generalization of BL-algebra. He proved that the implication reduct of a flea-algebra is a pseudo-BCK algebra with three additional conditions. The pseudo-BCK algebra with these conditions is called pulex.

In this section some preliminary definitions and theorems are stated. In Section 2, we introduce the notions of deductive system, lattice filter, and subalgebra of pulexes and obtain some properties which are not true in a pseudo-BCK lattice in general. In Section 3, we define concepts of compatible deductive system and congruence relation on a pulex and show that there is a correspondence between the set of all compatible deductive systems of a pulex and the set of all congruence relations on a pulex which is not true in a pseudo-BCK lattice. After that we prove that the quotient algebra defined by a congruence relation is a pulex and we obtain some related results.

Definition 1 (see [10]). A pseudo-BCK algebra is a structure , where is a poset with the greatest element and , are binary operations on such that, for all , , , we have(1), ,(2), ,(3) iff iff .

Theorem 2 (see [11]). An algebra of type is a pseudo-BCK algebra if and only if it satisfies the following:(1) and ;(2), ;(3), ;(4) and implies , the same for .

Definition 3 (see [11]). If the partial order of a pseudo-BCK algebra is a lattice order, with the lattice operations and , then is said to be a pseudo-BCK lattice and it will be denoted by .

Example 4. Let such that , , and are incomparable with . The operations and are given as follows: Then is a pseudo-BCK lattice.

Definition 5 (see [9]). A pulex is a structure such that(1) is a pseudo-BCK lattice;(2), the same for ;(3);(4), the same for .

Remark 6. We see that the pseudo-BCK lattice of Example 4 is not a pulex because .

Definition 7 (see [9]). A flea-algebra is a structure where(1) is a lattice with the greatest element ;(2) is a binary associative operation with as a both-side unit;(3) iff iff (residuation);(4), (prelinearity).

Proposition 8 (see [9]). Let be a flea-algebra. Then is a pulex.

Proposition 9 (see [9, 10]). In any pulex, the following rules hold:(1),(2),(3),(4) iff ,(5) implies that and ,(6) implies that and ,(7),(8) iff ,(9) and ,(10) and ,(11) and ,(12) and .

Proposition 10. In every pulex , the following relations hold for all :

Proof. Since , then by Proposition 9 part (6). We have by Definition 5 part (4). Since , we get that by Proposition 9 part (5). Therefore .
Similarly, we can show .

Proposition 11. Let be a pulex. Then the following relations hold:(1) and ,(2) and , for all , , .

Proof. Since , by Proposition 9 part (6), Hence . On the other hand, we have By Proposition 9 parts , (6), Therefore . By Proposition 9 parts , ( 6),
By part , . Since , we have by Proposition 9 part (5). Thus .

2. Deductive Systems

In this section, we introduce and study deductive systems of a pulex. From logical point of view, deductive systems correspond to the sets of formulas which are closed under the inference rule modus ponens. In what follows, we denote a pulex by .

Definition 12. Let be a subset of a pulex . We say that is a deductive system if(1), and(2)if , then for all .

Example 13. Let be the pulex with the universe such that , , and are incomparable. The operations and are given by the tables below: and are deductive systems of .

Definition 14. Let be a pulex, and let be a nonempty subset of . We say that is a lattice filter of if(1) for all , and(2)if , , and , then .

Deductive systems of a pseudo-BCK lattice may not be a lattice filter. For example, consider deductive system in Example 4. Since , and , then is not a lattice filter. In the following, we study the relationship between deductive systems and lattice filters of a pulex.

Theorem 15. Every deductive system of a pulex is a lattice filter of .

Proof. Let be a deductive system of a pulex . Since , then .(i)Let and such that . So we have . Since is a deductive system, then we have .(ii)Let . By part (9) of Proposition 9, and then by part (i). Since is a pulex, then . is a deductive system, so and then .

Remark 16. The converse of Theorem 15 may not be true. Consider Example 13. is a lattice filter of , but it is not a deductive system because and , but . Hence the converse of the above theorem is not true in a pseudo-BCK lattice too.

Theorem 17. Let be a pulex. Then with is a deductive system of if and only if it satisfies the following condition:

Proof. See Lemma   in [5].

It is easy to verify that the intersection of deductive systems of a pulex is a deductive system of .

Definition 18. Let be a subset of a pulex . The smallest deductive system of containing (i.e., the intersection of all deductive systems of containing ) is called the deductive system generated by and will be denoted by . If , then is written as and is called the principal deductive system generated by .
For convenience, we will write and where indicating the number of occurrence of .

Theorem 19 (see [5]). Let be a pulex, and let be a nonempty subset of and . Then(1) for some ;(2) for some ;(3) implies ;(4).

The set of all deductive systems of a pulex is an algebraic lattice whose compact elements are precisely the finitely generated deductive systems .

Definition 20. A subset of a pulex is called subalgebra of if the constant is in and itself forms a pulex under operations of .

Proposition 21. A nonempty subset of a pulex is a subalgebra of if and only if is closed under operations , , on .

Deductive system of a pseudo-BCK lattice may not be a subalgebra. Consider deductive system of in Example 4. We can show that it is not a subalgebra of . In the following theorem, we prove that this is true in each pulexes.

Proposition 22. Let be a deductive system of a pulex . Then is a subalgebra of .

Proof. Let be a deductive system of a pulex . Then is nonempty subset of . By Theorem 15, is closed under . Suppose that . By Proposition 9 part (9), we have and . By Theorem 15, we get that ; that is, is closed under . Similarly, we can show that is closed under . Hence is a subalgebra of by Proposition 21.

Remark 23. Consider subalgebra of in Example 13. Then is not a lattice filter of . By Theorem 15, is not a deductive system of . Therefore the converse of the above proposition may not be true in general.

3. Compatible Deductive Systems

In this section, we introduce the notions of compatible deductive systems and congruence relations of a pulex and study relationship between them.

Definition 24. Let be a deductive system of a pulex such that Then is called a compatible deductive system of .

Example 25. Let be the pulex with the universe such that . The operations and are given by the tables below: is a compatible deductive system of , but the deductive system of is not compatible because and .

Definition 26. Let be a compatible deductive system of a pulex . Define the relation by for all .

Theorem 27. is an equivalence relation on .

Proof. Clearly, is reflexive and symmetric. Let , . Then , , , and . By Proposition 9 part (12), By Theorem 15, we have and . Since is a deductive system and , , then and ; that is, . Hence is an equivalence relation on .

Definition 28. An equivalence relation on a pulex is called a congruence relation on if , then

Theorem 29. Let be a compatible deductive system of a pulex . Then is a congruence relation on and is called the congruence relation induced by .

Proof. Suppose that . By Proposition 9 part (11), By Theorem 15, we have and and we have . From we obtain . Since is a transitive relation, we have . Similarly, we can show that , , and .
By Proposition 10, Hence by Theorem 15. Thus . Similarly, we can show that . Therefore .
By Proposition 11 part , Hence , by Theorem 15. Thus . Similarly, we can show that . Therefore .

Theorem 30. Let be a congruence relation on a pulex . Then is a compatible deductive system of . is called the compatible deductive system induced by .

Proof. Since is reflexive, . Suppose that , . Then . Since is a congruence relation, then . By transivity, . Hence and is a deductive system of .
We will show that is compatible. Suppose that , then . Since is a congruence relation, then . Hence . By Proposition 9 part (12) . Since is a deductive system, . Similarly, we can show that if , then .

Theorem 31. Let be a pulex. There is a one to one correspondence between the set of all congruence relations on and the set of all compatible deductive systems of .

Proof. We will show that and , Suppose that . Then . Since is a congruence relation, Thus . We have and . Hence .
Conversely, let . Then Thus , ; that is, .

Remark 32. In a pseudo-BCK lattice there is no one-to-one correspondence between congruence relations and compatible deductive systems. For example, is a compatible deductive system in Example 4, but is not a congruence relation on . Suppose that is a congruence relation on . Then implies . Thus ; that is, which is a contradiction.

Definition 33. Let be a compatible deductive system of a pulex , and let be the equivalence relation induced by . The set of all equivalence classes is denoted by . On this set define is called the quotient algebra of with respect to the compatible deductive system .

Theorem 34. Let be a compatible deductive system of a pulex . Then the quotient algebra is a pulex and is called the quotient pulex.

Proof. Since is a congruence relation, all the above relations are well defined. It is easy to show that is a lattice with the greatest element . Hence we have a relation on defined by for all , . By Proposition 11 part , and . It follows that From (24) and (25), we have for all . Now, it is easy to prove that is a pulex.

Theorem 35. Let be a compatible deductive system of pulex , and let be the corresponding quotient algebra. Then the map defined by for all is a homomorphism and its where .

Proof. The proof is straightforward.

Theorem 36. Let be a compatible deductive system of a pulex . There is a bijection from the set of all deductive systems of containing to the set of of all deductive systems of .

Proof. It is easy to prove that(i)if is a deductive system of such that , then is a compatible deductive of the subalgebra of and the quotient pulex is a compatible deductive system of ;(ii)if is a deductive system of , then is a deductive system of and moreover and .Now, define by . By (i) is well define and by (ii) is onto. We will show that is one to one. Suppose that such that . Without loss of generality, we assume that . Then there exists . If , then . So there exists such that ; that is, and . Since , then . Thus which is a contradiction. Hence and is one to one.

4. Conclusion

We introduced the notions of compatible deductive systems and congruence relations on a pulex and obtained some related results. Then we studied the relation between them. We used the compatible deductive systems to define the quotient pulex.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.