1. Introduction
The fuzzy implication is an important operator in fuzzy logic. It plays a very important role in the theoretical establishment and application of fuzzy set theory. It is widely used in various fields, such as approximate reasoning, image processing, fuzzy control and word calculation. In different literature, we can see different definitions of fuzzy implications [
1,
2,
3,
4]. In this paper, we use a widely accepted definition, which is defined in literature [
1] (see Definition 1). In order to further study the fuzzy implication, lots of scholars have given many properties of fuzzy implication, such as exchange principle (EP), ordering principle (OP), etc. (see Definition 2). However, the above properties have strong constraints, therefore, some scholars put forward the pseudo-exchange principle (PEP) with weaker restriction condition [
2]. (PEP) condition is as follows:
It is necessary to construct a new implication operator based on the above properties. For example, Min et al. [
5] constructed a new fuzzy implication from the perspective of additive generator and semigroup. Another construction idea is to construct new fuzzy implication with the help of semigroup order and theory implication. For example, in non-classical mathematical logic, logical propositions are represented by symbols, and the relations between propositions are described by an axiomatic system, the conjunction
and the implication
are related by an adjunction
where ≤ stands for the logical entailment. To explore a more general fuzzy aggregation operator and corresponding implication operator, more general fuzzy logic formal system has become one of the research hotspots in recent years.
For example, algebraic structures with one implication operator, BCK/BCI algebras, BI-algebras, etc.; algebraic structures with double implication operators, quantum B-algebras, pseudo-BCK/BCI algebras [
6,
7,
8,
9], etc.; algebraic structures with binary operators and implication operators, residuated lattices, non-associative residuated lattices, etc. [
10,
11,
12]. The most important structure is a basic implication algebra (BI-algebra for short) proposed in literature [
13]. BI-algebra is the algebraic abstraction of associative/non-associative fuzzy logic, commutative/non-commutative fuzzy logic. This algebra can be used to describe many other algebraic structures, such as BCK/BCI-algebras, quantum B-algebras, pseudo-BCK/BCI-algebras, etc.
In propositional logic system, t-norm is used as the explanation of fuzzy aggregation operator ⊗, meanwhile, many scholars expand their research on t-norm from the aspects of commutativity, associativity, unit element and so on. The researches obtained the concepts of pseudo-t-norm, uninorm, semi-uninorm and so forth [
14]. At the same time, the algebraic structures corresponding to the above binary operators have been proposed successively, such as commutative residuated lattice, non-commutative residuated lattice, non-associative residuated lattice, residuated ordered groupoid, pseudo-BCK/BCI-algebra, etc [
8,
9,
11,
15,
16]. In the same way, in reference [
14], Professor Liu Huawen proposed the concept of a semi-uninorm, which is an extension of a uninorm to remove commutativity and associativity; the conditions of two kinds of implications induced by a semi-uninorm are studied in detail.
Filter theory plays an important role in computer science and non-classical logic. From a logical point of view, various filters correspond to various sets of provable formulas. The properties of several kinds of fuzzy filters on the residual lattice are studied [
10,
11,
12], which makes the structure of filters on residual lattice more clear. Therefore, on the basis of many theoretical researches on filters, the interrelation and hierarchical structure of various filter concepts are systematically analyzed, and construction is particularly important.
According to the above introduction, the research on implication operator, aggregation operator and their relationship have been very deep. In this paper, in order to describe the characteristics of residual implication in depth, we add some new characteristics to BI-algebra. First, we add the (PEP) condition to BI-algebra, so we get strong BI-algebra (SBI-algebra for short). However, basic implication algebra and strong BI-algebra describe algebraic structures with an implication operator. In order to describe the corresponding algebraic structure of noncommutative logic, we extend strong BI-algebra to pseudo-SBI-algebra, which has two kinds of implications. We establish the filter theory and quotient structure of pseudo-SBI-algebra. From the above introduction, it can be seen that binary operation ⊗ is also an important part of fuzzy logic. So, combined pseudo-SBI-algebra with ⊗, we get a new algebra structure, residuated pseudo-SBI-algebra. For the sake of uniform the filter structures of residuated lattices, non-associative residuated lattices and other algebras, we establish the filter and quotient structures of residuated pseudo-SBI-algebras.
2. Preliminaries
Definition 1 ([
1])
. A function is called a fuzzy implication if it satisfies, for all , the following conditions: Definition 2 ([
1,
2])
. A fuzzy implication I is said to satisfy the left neutrality property, if the exchange principle, if the identity principle, if the ordering property, if the pseudo-exchange principle, if Definition 3 ([
13])
. Let be a partially ordered set, which has a binary operation →. Then X is a basic implication algebra(BI-algebra for short) if and only if: for all if then
if then
A BI-algebra X is called normal if and only if:
let if and only if
Definition 4 ([
13])
. Let be a BI-algebra, F is a nonempty set of X. Then F is called a filter of X if it satisfies: if then
if then
if then
if then
Let be a normal BI-algebra, the filter F of X is called regular, when it satisfies:
when and then there exists
Definition 5 ([
17,
18])
. Let X be a non-empty set endows with operation Then If X satisfies:
Then is a BCC-algebra.
If X satisfies and
Then is a BCK-algebra.
Any BCK-algebra is a BCC-algebra.
Definition 6 ([
19])
. Let P be a complete lattice, “&”
be a binary operation on P. Then is called a prequantale if it satisfies: Definition 7 ([
14])
. Let L be a complete lattice, U be a binary operation on L. Then L is called a semi-uninorm if and only if:there exists a neutral element i.e., for all ;
is non-decreasing in each variable.
Definition 8 ([
16])
. A residuated partially-ordered groupoid is a partially ordered set endows with operation ⊗, which satisfies: Definition 9 ([
10,
15,
20,
21])
. Let be a lattice, which has the bottom element 0 and the top element 1. Then for all If L satisfies:
is a monoid, which has the unit 1.
Then is a residuated lattice.
If L satisfies:
is a groupoid which is commutative and has the unit 1.
Then is a non-associative residuated lattice.
If L satisfies , then L is a commutative residuated lattice.
If L satisfies and:
is a groupoid which is non-commutative and has the unit 1.
Then is a residuated lattice-ordered groupoid.
Proposition 1 ([
22])
. Let be a residuated lattice, then, if and only if if and only if
if then
if then
if then
Definition 10 ([
5,
22,
23,
24])
. Let L be a residuated lattice. F is a nonempty subset of X, then F is called a filter if it satisfies:
Definition 11 ([
11])
. Let L be a non-associative residuated lattice, F be a nonempty subset of X. Then F is called a filter if and only if:for any if then
for any let for any then
Definition 12 ([
7,
25])
. A quantum B-algebra is a partially ordered set which satisfies: if then
if and only if
A quantum B-algebra is called commutative if it satisfies:
A quantumB-algebra is called unital if
there exists s.t.
Proposition 2 ([
21])
. An algebra structure is a quantum B-algebra if and only if: for all if then
if and only if
Definition 13 ([
8,
9])
. An algebra structure with a binary operation ≤ is a pseudo-BCK-algebra if and only if verifying the axioms:;
Proposition 3 ([
9,
26])
. Let be a pseudo-BCK-algebra, then, if and only if
if then
if then
if then
if then
Proposition 4 ([
7])
. Let A be a pseudo-BCK-algebra, then A is a quantum B-algebra. 3. Strong BI-Algebras and Left/Right Residuated BI-Algebras
Condition PEP was proposed in literature [
2], which is defined on interval
In this paper, we get rid of the limit of interval, and extend this condition to a partial order set.
BI-algebra is an algebraic framework of general implication operators, which has a wide range of adaptability. However, there are not many characteristics it can extract. If we want to describe the characteristics of residual implication in depth, it is not enough to use BI-algebra alone. We need to add new characteristics to BI-algebra to describe the properties of residual implication. In this section, we use the PEP property of fuzzy implication [
2] for reference, and add PEP condition on the basis of BI-algebra to form strong BI-algebra.
At the same time, in BI-algebra, only implicative operation is involved, not multiplication operation ⊗. In the study of fuzzy logic and related algebra structures, multiplication operation embodies the properties of t-norm, non-associative t-norm, semi-uninorm, quantale and other structures. Therefore, we should not only study the implication operation, but also study the algebraic structure with binary operations like t-norm, non-associative t-norm, semi uninorm and so on. Therefore, in this section, on the basis of BI-algebra, we add multiplication operation ⊗, and get left/right residuated BI-algebras.
Definition 14. A strong BI-algebra(SBI-algebra for short) is a basic implication algebra which satisfies:
Next, we will give equivalent conditions of the above definition.
Proposition 5. Let be a partially ordered set which has a binary operation →. Then is a strong BI-algebra if and only if:
if then
if and only if
Proof. For any assume that . Since , by (2) we have . Thus, Using (2) again, we can get that Thus, the proposition holds. □
Example 1. Then is a strong BI-algebra (SBI-algebra), and is a BI-algebra but it is not strong, since
In paper [
3], Mesiar and Mesiarová give a condition ([
3], Theorem 5) for t-norm to get corresponding residual implication. Through research, if the given t-norm is left continuous, then its corresponding implication satisfies (EP) and (OP) conditions, and the corresponding one also satisfies (PEP) conditions, i.e., the implication is strong BI-algebra. Specific examples are as follows.
Example 2 ([
3])
. Let be given byUsing Theorem 5 in [3], the implication is given by Then is a resiudal implication, because of the left continuity of T, is a strong BI-algebra, and the implication satisfies the (EP), (OP) and (PEP) condition.
Example 3. Let define the operation on X as following: Then is a SBI-algebra, and the implication operator → satisfies the exchange principle, and it does not satisfy the (OP) condition, since
Example 4. Let define the operation on X as following: Then is a BI-algebra, but it is not a strong BI-algebra (SBI-algebra), since Moreover, the implication operator → does not satisfy the ordering property, since
Example 5. Let define the operation on X as following: Then is a strong BI-algebra (SBI-algebra), and the implication operator → satisfies the (OP) condition and the pseudo-exchange principle (PEP).
Proposition 6. Let X be a SBI-algebra. Then,
;
Proof. For any from using Definition 14, we can get
For any by (1), we can get that and Using Definition 3(2), Thus, □
Professor Liu Hua-Wen has proposed fuzzy boundary weak implications in [
4], in this paper, we get the fuzzy weak implication by removing its boundary property.
Definition 15. Fuzzy weak implication (fuzzy w-implication for short) is a function which satisfies:
is non-increasing in its first variable;
is non-decreasing in its second variable.
Clearly, the fuzzy weak implication is the background of strong BI-algebra, now we give the condition that makes this implication to a strong BI-algebra.
Proposition 7. Let → be a fuzzy weak implication with ordering exchange principle (PEP). Then is a strong BI-algebra.
Proof. From the non-increasing and non-decreasing properties of implication operator → in Definition 15, we know that is a BI-algebra, and with (PEP) condition, i.e., if and only if . Thus, is a strong BI-algebra. □
Above all, we have discussed the background of strong BI-algebra, below, we give the relationships between strong BI-algebra and some algebraic structures.
Proposition 8. Every commutative quantum B-algebra is a strong BI-algebra. Conversly, a strong BI-algebra X is a commutative quantum B-algebra, if X satisfies
Let X be a strong BI-algebra, if X satisfies the conditions of BCC-algebra, then X is a BCK-algebra.
Proof. Using Definition 12 and Proposition 5, we know that every commutative quantum B-algebra is a strong BI-algebra.
Let be a strong BI-algebra, , if X satisfies , then from Proposition 2, X is a commutative quantum B-algebra.
According to Definition 5, holds. At the same time, using Proposition 5 and , holds. □
The following
Figure 1 summarizes the relationships between strong BI-algebra and some algebraic structures.
In the above, we have defined strong BI-algebra, an algebraic structure with only an implication operator and ordered relation. In the existing research, it is also very important to express the operators with structure properties such as t-norm, semi-uninorm and quantale. In this paper, ⊗ is used to represent this kind of operators. Next, according to the relationship between implication and multiplication ⊗, we propose some abstract algebraic structures with two kinds of operations.
Definition 16. Let be a partially ordered set, where “” is a binary relation on X.
If X satisfies:
is a BI-algebra;
⇒ and
Then is called a BI-groupoid.
If X satisfies and
if and only if
Then is called a left residuated BI-algebra.
If X satisfies and
if and only if .
Then is called a right residuated BI-algebra.
If is both left and right residuated BI-algebra, then is called residuated SBI-algebra.
We can use
Figure 2 to describe the relationships among these definitions.
Example 6. Let define semi-uninorm ⊗ and its correspondent residual implication → as follows: In this example, ⊗ is non-commutative, → is non-increasing in first variable and non-decreasing in second variabe. However, → does not satisfy the adjoint property, since .
Example 7. Let define t-norm ⊗ and → on X as following: Then the operation → is a residual implication. Because binary operation ⊗ is continuous, ⊗ and → satisfy adjoint property. It is verified that is a residuated SBI-algebra.
Next, we will introduce the example of finite order.
Example 8. Let where Define the operations as Table 1, Table 2 and Table 3: Then, is a left residuated BI-algebra, is a right residuated BI-algebra. In this example, the binary operation ⊗ is non-commutative, → and ⇝ do not satisfy (PEP) condition, since but but
The definitions and properties of residuated lattice, non-associative residuated lattice and pseudo-BL-algebra are introduced in the literatures [
10,
15,
21,
27]. Here we give the relationships between these algebras and left/right residuated BI-algebra.
Proposition 9. Let be a pseudo-BL-algebra, then is a left residuated BI-algebra, is a right residuated BI-algebra;
Let be a residuated lattice, then is a left residuated BI-algebra, is a right residuated BI-algebra;
Let be a non-associative residuated lattice, then is residuated SBI-algebra.
Proof. We can get some properties of pseudo-BL-algebra
from literature [
28] as follows:
if then and
if then and
if and only if if and only if
Using and , we know that and are BI-algebras.
Then use condition , we know that is a left residuated BI-algebra, and is a right residuated BI-algebra;
Similar to the proof of (1), the conclusion in Proposition 9(2) is tenable by using the related properties in Proposition 1.
By the propositions of non-associative residuated lattice in literature [
29], the conclusion is established. □
Next, we give some propositions of left/right residuated BI-algebras and residuated SBI-algebras.
Proposition 10. Let be a left residuated BI-algebra, then,
if then and
Proof. using the reflexivity of ≤, we have Applying Definition 16(2), there exists
can be obtained form the reflexivity of ≤. Applying Definition 16(2), it is clear that
assume , from (1), we have using , Applying to Definition 16(2), there exists
assume , by Definition 16(2), we have that From (1), there exists Thus, Then using Definition 16(2), □
Proposition 11. Let be a right residuated BI-algebra, then,
if then and
Proposition 12. Let be a residuated SBI-algebra, then for any
there exists and
there exists and
if then there exists and
The proofs of Propositions 11 and 12 are similar to the proof of Proposition 10, so the proofs are omitted here.
Proposition 13. Let be a residuated SBI-algebra, then the binary operation ⊗ is commutative, i.e.,
Proof. Let
be a residuated SBI-algebra, then the following equations hold
Using the reflexivity, according to Equation (12), we have then from Equation (13), Conversly, we can get Above all, using antisymmetry, we have □