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Symmetry 2018, 10(11), 573; https://doi.org/10.3390/sym10110573

Article
Q-Filters of Quantum B-Algebras and Basic Implication Algebras
1
Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, China
2
Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China
3
Department of Mathematics, Shahid Beheshti University, Tehran 1983963113, Iran
4
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea
*
Author to whom correspondence should be addressed.
Received: 26 September 2018 / Accepted: 26 October 2018 / Published: 1 November 2018

Abstract

:
The concept of quantum B-algebra was introduced by Rump and Yang, that is, unified algebraic semantics for various noncommutative fuzzy logics, quantum logics, and implication logics. In this paper, a new notion of q-filter in quantum B-algebra is proposed, and quotient structures are constructed by q-filters (in contrast, although the notion of filter in quantum B-algebra has been defined before this paper, but corresponding quotient structures cannot be constructed according to the usual methods). Moreover, a new, more general, implication algebra is proposed, which is called basic implication algebra and can be regarded as a unified frame of general fuzzy logics, including nonassociative fuzzy logics (in contrast, quantum B-algebra is not applied to nonassociative fuzzy logics). The filter theory of basic implication algebras is also established.
Keywords:
fuzzy implication; quantum B-algebra; q-filter; quotient algebra; basic implication algebra

1. Introduction

For classical logic and nonclassical logics (multivalued logic, quantum logic, t-norm-based fuzzy logic [1,2,3,4,5,6]), logical implication operators play an important role. In the study of fuzzy logics, fuzzy implications are also the focus of research, and a large number of literatures involve this topic [7,8,9,10,11,12,13,14,15,16]. Moreover, some algebraic systems focusing on implication operators are also hot topics. Especially with the in-depth study of noncommutative fuzzy logics in recent years, some related implication algebraic systems have attracted the attention of scholars, such as pseudo-basic-logic (BL) algebras, pseudo- monoidal t-norm-based logic (MTL) algebras, and pseudo- B, C, K axiom (BCK)/ B, C, I axiom (BCI) algebras [17,18,19,20,21,22,23] (see also References [5,6,7]).
For formalizing the implication fragment of the logic of quantales, Rump and Yang proposed the notion of quantum B-algebras [24,25], which provide a unified semantic for a wide class of nonclassical logics. Specifically, quantum B-algebras encompass many implication algebras, like pseudo-BCK/BCI algebras, (commutative and noncommutative) residuated lattices, pseudo- MV/BL/MTL algebras, and generalized pseudo-effect algebras. New research articles on quantum B-algebras can be found in References [26,27,28]. Note that all hoops and pseudo-hoops are special quantum B-algebras, and they are closely related to L-algebras [29].
Although the definition of a filter in a quantum B-algebra is given in Reference [30], quotient algebraic structures are not established by using filters. In fact, filters in special subclasses of quantum B-algebras are mainly discussed in Reference [30], and these subclasses require a unital element. In this paper, by introducing the concept of a q-filter in quantum B-algebras, we establish the quotient structures using q-filters in a natural way. At the same time, although quantum B-algebra has generality, it cannot include the implication structure of non-associative fuzzy logics [31,32], so we propose a wider concept, that is, basic implication algebra that can include a wider range of implication operations, establish filter theory, and obtain quotient algebra.

2. Preliminaries

Definition 1.
Let (X, ≤) be partially ordered set endowed with two binary operations → and ↝ [24,25]. Then, (X, →, ↝, ≤) is called a quantum B-algebra if it satisfies: ∀x, y, z∈X,
(1) y→z ≤ (x→y)→(x→z);
(2) y↝z ≤ (x↝y) ↝ (x↝z);
(3) y ≤ z ⇒ x→y ≤ x→z;
(4) x ≤ y→z ⟺ y ≤ x↝z.
If u∈X exists, such that u→x = u↝x = x for all x in X, then u is called a unit element of X. Obviously, the unit element is unique. When a unit element exists in X, we call X unital.
Proposition 1.
An algebra structure (X, →, ↝, ≤) endowed with a partially order ≤ and two binary operations → and ↝ is a quantum B-algebra if and only if it satisfies [4]: ∀x, y, z∈X,
(1) x→(y↝z) = y↝(x→z);
(2) y ≤ z ⇒ x→y ≤ x→z;
(3) x ≤ y→z ⟺ y ≤ x↝z.
Proposition 2.
Let (X, →, ↝, ≤) be a quantum B-algebra [24,25,26]. Then, (∀ x, y, z∈X)
(1) y ≤ z ⇒ x↝y ≤ x↝z;
(2) y ≤ z ⇒ z↝x ≤ y↝x;
(3) y ≤ z ⇒ z→x ≤ y→x;
(4) x ≤ (x↝y)→y, x ≤ (x→y)↝y;
(5) x→y = ((x→y)↝y)→y, x↝y = ((x↝y)→y)↝y;
(6) x→y ≤ (y→z)↝(x→z);
(7) x↝y ≤ (y↝z)→(x↝z);
(8) assume that u is the unit of X, then u ≤ x↝y ⟺ x ≤ y ⟺ u ≤ x→y;
(9) if 0∈X exists, such that 0 ≤ x for all x in X, then 0 = 0↝0 = 0→0 is the greatest element (denote by 1), and x→1 = x↝1=1 for all x∈X;
(10) if X is a lattice, then (x ∨ y)→z = (x→z)∧(y→z), (x∨y)↝z = (x↝z) ∨(y↝z).
Definition 2.
Let (X, ≤) be partially ordered set and Y⊆ X [24]. If x ≥ y∈Y implies x∈Y, then Y is called to be an upper set of X. The smallest upper set containing a given x∈X is denoted by ↑x. For quantum B-algebra X, the set of upper sets is denoted by U(X). For A, B∈U(X), define
A·B = {x∈X|b∈B: b→x∈A}.
We can verify that A·B = {x∈X|∃a∈A: a↝x∈B} = {x∈X|∃a∈A, b∈B: a≤b→x} = {x∈X|∃a∈A, b∈B: b ≤ a↝x}.
Definition 3.
Let A be an empty set, ≤ be a binary relation on A [17,18], → and ↝ be binary operations on A, and 1 be an element of A. Then, structure (A, →, ↝, ≤, 1) is called a pseudo-BCI algebra if it satisfies the following axioms: ∀ x, y, z∈A,
(1) 
x→y ≤ (y→z)↝(x→z), x↝y ≤ (y↝z)→(x↝z),
(2) 
x ≤ (x↝y)→y, x ≤ (x→y)↝y;
(3) 
x ≤ x;
(4) 
x ≤ y, y ≤ x ⇒ x = y;
(5) 
x ≤ y ⟺ x→y = 1 ⟺ x↝y = 1.
If pseudo-BCI algebra A satisfies: x→1 = 1 (or x↝1 = 1) for all x∈A, then A is called a pseudo-BCK algebra.
Proposition 3.
Let (A, →, ↝, ≤, 1) be a pseudo-BCI algebra [18,19,20]. We have (∀x, y, z∈A)
(1) if 1 ≤ x, then x = 1;
(2) if x ≤ y, then y→z ≤ x→z and y↝z ≤ x↝z;
(3) if x ≤ y and y ≤ z, then x ≤ z;
(4) x→(y↝z) = y↝(x→z);
(5) x ≤ y→z ⟺ y ≤ x↝z;
(6) x→y ≤ (z→x)→(z→y); x↝y ≤ (z↝x)↝(z↝y);
(7) if x ≤ y, then z→x ≤ z→y and z↝x ≤ z↝y;
(8) 1→x = 1↝x =x;
(9) y→x = ((y→x)↝x)→x, y↝x = ((y↝x)→x)↝ x;
(10) x→y ≤ (y→x) ↝1, x↝y ≤ (y↝x)→1;
(11) (x→y)→1 = (x→1)↝(y→1), (x↝y)↝1 = (x↝1)→(y↝1);
(12) x→1 = x↝1.
Proposition 4.
Let (A, →, ↝, ≤, 1) be a pseudo-BCK algebra [17], then (∀x, y A): x ≤ y→x, x ≤ y↝x.
Definition 4.
Let X be a unital quantum B-algebra [24]. If there exists x∈X, such that x→u = x↝u = u, then we call that x integral. The subset of integral element in X is denoted by I(X).
Proposition 5.
Let X be a quantum B-algebra [24]. Then, the following assertions are equivalent:
(1) X is a pseudo-BCK algebra;
(2) X is unital, and every element of X is integral;
(3) X has the greatest element, which is a unit element.
Proposition 6.
Every pseudo-BCI algebra is a unital quantum B-algebra [25]. And, a quantum B-algebra is a pseudo-BCI algebra if and only if its unit element u is maximal.
Definition 5.
Let (A, →, ↝, ≤, 1) be a pseudo-BCI algebra [20,21]. When the following identities are satisfied, we call X an antigrouped pseudo-BCI algebra:
∀x∈A, (x→1)→1 = x or (x↝1)↝1 = x.
Proposition 7.
Let (A, →, ↝, ≤, 1) be a pseudo-BCI algebra [20]. Then, A is antigrouped if and only if the following conditions are satisfied:
(G1) for all x, y, z∈A, (x→y)→(x→z) = y→z, and
(G2) for all x, y, z∈A, (x↝y)↝(x↝z) = y↝z.
Definition 6.
Let (A, →, ↝, ≤, 1) be a pseudo-BCI algebra and F ⊆ X [19,20]. When the following conditions are satisfied, we call F a pseudo-BCI filter (briefly, filter) of X:
(F1) 1∈F;
(F2) x∈F, x→y∈F ⟹ y∈F;
(F3) x∈F, x↝y∈F ⟹ y∈F.
Definition 7.
Let (A, →, ↝, ≤, 1) be a pseudo-BCI algebra and F be a filter of X [20,21]. When the following condition is satisfied, we call F an antigrouped filter of X:
(GF) ∀x∈X, (x→1)→1∈F or (x↝1)↝1∈F ⟹ x∈F.
Definition 8.
A subset F of pseudo-BCI algebra X is called a p-filter of X if the following conditions are satisfied [20,21]:
(P1) 1∈F,
(P2) (x→y)↝(x→z)∈F and y∈F imply z∈F,
(P3) (x↝y)→(x↝z)∈F and y∈F imply z∈F.

3. Q-Filters in Quantum B-Algebra

In Reference [30], the notion of filter in quantum B-algebra is proposed. If X is a quantum B-algebra and F is a nonempty set of X, then F is called the filter of X if FU(X) and F·FF. That is, F is a filter of X, if and only if: (1) F is a nonempty upper subset of X; (2) (zX, yF, yzF) ⇒ zF. We denote the set of all filters of X by F(X).
In this section, we discuss a new concept of q-filter in quantum B-algebra; by using q-filters, we construct the quotient algebras.
Definition 9.
A nonempty subset F of quantum B-algebra X is called a q-filter of X if it satisfies:
(1) 
F is an upper set of X, that is, F∈U(X);
(2) 
for all x∈F, x→x∈F and x↝x∈F;
(3) 
x∈F, y∈X, x→y∈F ⟹ y∈F.
(4) 
A q-filter of X is normal if x→y∈F⟺ x↝y∈F.
Proposition 8.
Let F be a q-filter of quantum B-algebra X. Then,
(1) x∈F, y∈X, x↝y∈F ⟹ y∈F.
(2) x∈F and y∈X ⟹ (x↝y)→y∈F and (x→y)↝y∈F.
(3) if X is unital, then Condition (2) in Definition 9 can be replaced by u∈F, where u is the unit element of X.
Proof. 
(1) Assume that xF, yX, and xyF. Then, by Proposition 2 (4), x ≤ (xy)→y. Applying Definition 9 (1) and (3), we get that yF.
(2) Using Proposition 2 (4) and Definition 9 (1), we can get (2).
(3) If X is unital with unit u, then uu = u. Moreover, applying Proposition 2 (8), uxx and uxx from xx, for all xX. Therefore, for unital quantum B-algebra X, Condition (2) in Definition 8 can be replaced by condition “uF. □
By Definition 6, and Propositions 6 and 8, we get the following result (the proof is omitted).
Proposition 9.
Let (A, →, ↝, ≤, 1) be a pseudo-BCI algebra. Then, an empty subset of A is a q-filter of A (as a quantum B-algebra) if and only if it is a filter of A (according to Definition 6).
Example 1.
Let X = {a, b, c, d, e, f}. Define operations and ↝ on X as per the following Cayley Table 1 and Table 2; the order on X is defined as follows: b ≤ a ≤ f; e ≤ d ≤ c. Then, X is a quantum B-algebra (we can verify it with the Matlab software (The MathWorks Inc., Natick, MA, USA)), but it is not a pseudo-BCI algebra. Let F1 = {f}, F2 = {a, b, f}; then, F1 is a filter but not a q-filter of X, and F2 is a normal q-filter of X.
Theorem 1.
Let X be a quantum B-algebra and F a normal q-filter of X. Define the binary ≈F on X as follows:
x ≈F y ⟺ x→y∈F and y→x∈F, where x, y∈X.
Then,
(1) ≈F is an equivalent relation on X;
(2) ≈F is a congruence relation on X, that is, x≈F y ⟹ (z→x) ≈F (z→y), (x→z) ≈F (y→z), (z↝x) ≈F (z↝y), (x↝z) ≈F (y↝z), for all z∈X.
Proof. 
(1) For any xX, by Definition 9 (2), xxF, it follows that xF x.
For all x, yX, if xF y, we can easily verify that yF x.
Assume that xF y, yF z. Then, xyF, yxF, yzF, and zyF, since
yz ≤ (xy)→(xz) by Definition 1 (1).
From this and Definition 9, we have xzF. Similarly, we can get zxF. Thus, xF z.
Therefore, ≈F is an equivalent relation on X.
(2) If xF y, then xyF, yxF. Since
xy ≤ (zx)→(zy), by Definition 1 (1).
yx ≤ (zy)→(zx), by Definition 1 (1).
Using Definition 9 (1), (zx)→(zy)∈F, (zy)→(zx)∈F. It follows that (zx) ≈F (zy).
Moreover, since
xy ≤ (yz)↝(xz), by Proposition 2 (6).
yx ≤ (xz)↝(yz), by Proposition 2 (6).
Then, form xyF and yxF; using Definition 9 (1), we have (yz)↝(xz)∈F, (xz)↝(yz)∈F. Since F is normal, by Definition 9 we get (yz)→(xz)F, (xz)→(yz)∈F. Thus, (xz) ≈F (yz).
Similarly, we can get that xF y ⟹ (zx) ≈F (zy) and (xz) ≈F (yz). □
Definition 10.
A quantum B-algebra X is considered to be perfect, if it satisfies:
(1) for any normal q-filter F of X, x, y in X,(there exists an∈X, such that [x→y]F = [a→a]F ) ⟺ (there exists b∈X, such that [x↝y]F = [b↝b]F ).
(2) for any normal q-filter F of X, (X/≈F →, ↝, ≤) is a quantum B-algebra, where quotient operations → and ↝ are defined in a canonical way, and ≤ is defined as follows:
[x]F ≤ [y]F ⟺ (there exists a∈X such that [x]F→[y]F = [a→a]F)
⟺ (there exists b∈X such that [x]F↝[y]F = [b↝b]F).
Theorem 2.
Let (A, →, ↝, ≤, 1) be a pseudo-BCI algebra, then A is a perfect quantum B-algebra.
Proof. 
By Proposition 6, we know that A is a quantum B-algebra.
(1) For any normal q-filter F of A, x, yA, if there exists aA, such that [xy]F = [aa]F, then
[xy]F = [aa]F = [1]F.
It follows that (xy)→1∈F, 1→(xy) = xyF. Applying Proposition 3 (11) and (12), we have
(x→1)↝(y→1) = (xy)→1∈F.
Since F is normal, from (x→1)↝(y→1)∈F and xyF we get that
(x→1)→(y→1)∈F and xyF.
Applying Proposition 3 (11) and (12) again, (xy)→1 = (x→1)→(y→1). Thus,
(xy)→1 = (x→1)→(y→1)∈F and 1→(xy) = xyF.
This means that [xy]F = [1]F = [1↝1]F. Similarly, we can prove that the inverse is true. That is, Definition 10 (1) holds for A.
(2) For any normal q-filter F of pseudo-BCI algebra A, binary ≤ on A/F is defined as the following:
[x]F ≤ [y]F ⟺ [x]F→[y]F = [1]F.
We verify that ≤ is a partial binary on A/F.
Obviously, [x]F ≤ [x]F for any xA.
If [x]F ≤ [y]F and [y]F ≤ [x]F, then [x]F→[y]F = [xy]F = [1]F, [y]F→[x]F = [yx]F = [1]F. By the definition of equivalent class, xy = 1→(xy) ∈F, yx = 1→(yx)∈F. It follows that xF y; thus, [x]F = [y]F.
If [x]F ≤ [y]F and [y]F ≤ [z]F, then [x]F→[y]F = [xy]F = [1]F, [y]F→[z]F = [yz]F = [1]F. Thus,
xy = 1→(xy)∈F, (xy)→1∈F;
yz = 1→(yz)∈F, (yz)→1∈F.
Applying Definition 3 and Proposition 3,
yz ≤ (xy)→(xz),
(xy)→1 = (x→1)↝(y→1) ≤ ([(y→1)↝(z→1)]→[(x→1)↝(z→1)]) = [(yz)→1]→[(xz)→1].
By Definition 9,
1→(xz) = xzF, (xz)→1∈F.
This means that (xz) ≈F 1, [xz]F = [1]F. That is, [x]F→[z]F =[xz]F = [1]F, [x]F ≤ [z]F.
Therefore, applying Theorem 1, we know that (A/F,, [1]F) is a quantum B-algebra and pseudo-BCI algebra. That is, Definition 10 (2) holds for A.
Hence, we know that A is a perfect quantum B-algebra.
The following examples show that there are some perfect quantum B-algebras that may not be a pseudo-BCI algebra.
Example 2.
Let X = {a, b, c, d, e, 1}. Define operations and ↝ on X as per the following Cayley Table 3 and Table 4, the order on X is defined as the following: b ≤ a ≤ 1; e ≤ d ≤ c. Then, X is a pseudo-BCI algebra (we can verify it with Matlab). Denote F1 = {1}, F2 = {a, b, 1}, F3 = X, then Fi (i = 1, 2, 3) are all normal q-filters of X, and quotient algebras (X/≈Fi →, ↝, [1]Fi) are pseudo-BCI algebras. Thus, X is a perfect quantum B-algebra.
Example 3.
Let X = {a, b, c, d, e, f}. Define operations and ↝ on X as per the following Cayley Table 5 and Table 6, the order on X is defined as follows: b ≤ a ≤ f; e ≤ d ≤ c. Then, X is a quantum B-algebra (we can verify it with Matlab), but it is not a pseudo-BCI algebra, since e↝e≠ e→e. Denote F = {a, b, f}, then F, X are all normal q-filters of X, quotient algebras (X/≈F →, ↝, ≤), (X/≈X →, ↝, ≤) are quantum B-algebras, and X is a perfect quantum B-algebra.

4. Basic Implication Algebras and Filters

Definition 11.
Let (A, ∨, ∧, ⨂, →, 0, 1) be a type-(2, 2, 2, 2, 0, 0) algebra [32]. A is called a nonassociative residuated lattice, if it satisfies:
(A1) (A, ∨, ∧, 0, 1) is a bounded lattice;
(A2) (A,, 1) is a commutative groupoid with unit element 1;
(A3) ∀x, y, z∈A, xy ≤ z ⟺ x ≤ y→z.
Proposition 10.
Let (A, ∨, ∧, ⨂, →, 0, 1) be a nonassociative residuated lattice [32]. Then, (∀ x, y, z∈A)
(1) x ≤ y ⟺ x→y = 1;
(2) x ≤ y ⇒ x⨂z ≤ y⨂z;
(3) x ≤ y ⇒ y→z ≤ x→z;
(4) x ≤ y ⇒ z→x ≤ z→y;
(5) x⨂(y∨z) = (x⨂y)∨(x⨂z);
(6) x→(y∧z) = (x→y)∧(x→z);
(7) (y∨z)→x = (y→x)∧(z→x);
(8) (x→y)⨂x ≤ x, y;
(9) (x→y)→y ≥ x, y.
Example 4.
Let A = [0, 1], operationon A is defined as follows:
xy = 0.5xy + 0.5max{0, x + y − 1}, x, yA.
Then,is a nonassociative t-norm on A (see Example 1 in Reference [32]). Operation → is defined as follows:
x→y = max{z∈[0, 1]|zx ≤ y}, x, y∈A.
Then, (A, max, min, ⨂, →, 0, 1) is a nonassoiative residuated lattice (see Theorem 5 in Reference [32]). Assume that x = 0.5, y = 0.2, z = 0.1, then
y z = 0.2 0.1 = max { a [ 0 , 1 | a 0.2     0.1 } = 5 6 .
x y = 0.55 0.2 = max { a [ 0 , 1 | a 0.55     0.2 } = 17 31 .
x z = 0.55 0.1 = max { a [ 0 , 1 | a 0.55     0.1 } = 4 11 .
( x y )     ( x z ) =   17 31 4 11 = max { a [ 0 , 1 | z 17 31     4 11 } = 67 88 .  
Therefore,
y z ( x y ) ( x z ) .
Example 4 shows that Condition (1) in Definition 1 is not true for general non-associative residuated lattices, that is, quantum B-algebras are not common basic of non-associative fuzzy logics. So, we discuss more general implication algebras in this section.
Definition 12.
A basic implication algebra is a partially ordered set (X, ≤) with binary operation →, such that the following are satisfied for x, y, and z in X:
(1) x ≤ y ⇒ z→x ≤ z→y;
(2) x ≤ y ⇒ y→z ≤ x→z.
A basic implication algebra is considered to be normal, if it satisfies:
(3) for any x, y∈X, x→x = y→y;
(4) for any x, y∈X, x ≤ y ⟺ x→y = e, where e = x→x = y→y.
We can verify that the following results are true (the proofs are omitted).
Proposition 11.
Let (X, →, ≤) be a basic implication algebra. Then, for all x, y, z∈X,
(1) x ≤ y ⇒ y→x ≤ x→x ≤ x→y;
(2) x ≤ y ⇒ y→x ≤ y→y ≤ x→y;
(3) x ≤ y and u ≤ v ⇒ y→u ≤ x→v;
(4) x ≤ y and u ≤ v ⇒ v→x ≤ u→y.
Proposition 12.
Let (X, →, ≤, e) be a normal basic implication algebra. Then for all x, y, z∈X,
(1) x→x = e;
(2) x→y = y→x = e ⇒ x = y;
(3) x ≤ y ⇒ y→x ≤ e;
(4) if e is unit (that is, for all x in X, e→x = x), then e is a maximal element (that is, e ≤ x ⇒ e = x).
Proposition 13.
(1) If (X, →, ↝, ≤) is a a quantum B-algebra, then (X, →, ≤) and (X, ↝, ≤) are basic implication algebras; (2) If (A, →, ↝, ≤, 1) is a pseudo-BCI algebra, then (A, →, ≤, 1) and (A, ↝, ≤, 1) are normal basic implication algebras with unit 1; (3) If (A, ∨, ∧, ⨂, →, 0, 1) is a non-associative residuated lattice, then (A, →, ≤, 1) is a normal basic implication algebra.
The following example shows that element e may not be a unit.
Example 5.
Let X = {a, b, c, d, 1}. Define abcd1 and operation on X as per the following Cayley Table 7. Then, X is a normal basic implication algebra in which element 1 is not a unit. (X, →, ≤) is not a commutative quantum B-algebra, since
c = 1 c b   =   ( c d ) ( 1 d ) .
The following example shows that element e may be not maximal.
Example 6.
Let X = {a, b, c, d, 1}. Define abcd, abc1 and operation on X as per the following Cayley Table 8. Then, X is a normal basic implication algebra, and element 1 is not maximal and is not a unit.
Definition 13.
A nonempty subset F of basic implication algebra (X, →, ≤) is called a filter of X if it satisfies:
(1) F is an upper set of X, that is, x∈F and x ≤ y∈X ⟹ y∈F;
(2) for all x∈F, x→x∈F;
(3) x∈F, y∈X, x→y∈F ⟹ y∈F;
(4) x∈X, y→z∈F ⟹ (x→y)→(x→z)∈F;
(5) x∈X, y→z∈F ⟹ (z→x)→(y→x)∈F.
For normal basic implication algebra (X, →, ≤, e), a filter F of X is considered to be regular, if it satisfies:
(6) x∈X, (x→y)→e∈F and (y→z)→e∈F ⟹ (x→z)→e∈F.
Proposition 14.
Let (X, →, ≤, e) be a normal basic implication algebra and F ⊆ X. Then, F is a filter of X if and only if it satisfies:
(1) e∈F;
(2) x∈F, y∈X, x→y∈F ⟹ y∈F;
(3) x∈X, y→z∈F ⟹ (x→y)→(x→z)∈F;
(4) x∈X, y→z∈F ⟹ (z→x)→(y→x)∈F.
Obviously, if e is the maximal element of normal basic implication algebra (X, →, ≤, e), then any filter of X is regular.
Theorem 3.
Let X be a basic implication algebra and F a filter of X. Define binary ≈F on X as follows:
x≈F y ⟺ x→y∈F and y→x∈F, where x, y∈X.
Then
(1) ≈F is a equivalent relation on X;
(2) ≈F is a congruence relation on X, that is, x≈F y ⟹ (z→x) ≈F (z→y), (x→z) ≈F (y→z), for all z∈X.
Proof 
(1) ∀xX, from Definition 13 (2), xxF, thus xF x. Moreover, ∀x, yX, if xF y, then yF x.
If xF y and yF z. Then xyF, yxF, yzF, and zyF. Applying Definition 13 (4) and (5), we have
(xy)→(xz)∈F, (zy)→(zx)∈F.
From this and Definition 13 (3), we have xzF, zxF. Thus, xF z.
Hence, ≈F is a equivalent relation on X.
(2) Assume xF y. By the definition of bianary relation ≈F, we have xyF, yxF. Using Definition 13 (4),
(zx)→(zy)∈F, (zy)→(zx)∈F.
This means that (zx) ≈F (zy). Moreover, using Definition 13 (5), we have
(yz)→(xz)∈F, (xz)→(yz)∈F.
Hence, (xz) ≈F (yz). □
Theorem 4.
Let (X, →, ≤, e) be a normal basic implication algebra and F a regular filter of X. Define quotient operation → and binary relation ≤ on X/≈F as follows:
[x]F→[y]F = [x]F→[y]F, ∀x, y∈X;
[x]F≤ [y]F ⟺ [x]F →[y]F = [e]F, ∀x, y∈X.
Then, (X/≈F, →, ≤, [e]F) is a normal basic implication algebra, and (X, →, ≤, e) ∼ (X/≈F, →, ≤, [e]F).
Proof. 
Firstly, we prove that binary relation ≤ on X/≈F is a partial order.
(1) ∀xX, obviously, [x]F ≤ [x]F.
(2) Assume that [x]F ≤ [y]F and [y]F ≤ [x]F, then
[x]F→[y]F = [xy]F = [e]F, [y]F→[x]F = [yx]F = [e]F.
It follows that e→(xy)∈F, e→(yx)∈F. Applying Proposition 14 (1) and (2), we get that (xy)∈F and (yx)∈F. This means that [x]F = [y]F.
(3) Assume that [x]F ≤ [y]F and [y]F ≤ [z]F, then
[x]F→[y]F = [xy]F = [e]F, [y]F→[z]F = [yz]F = [e]F.
Using the definition of equivalent relation F, we have
e→(xy)∈F, (xy)→eF; e→(yz)∈F, (yz)→eF.
From e→(xy)∈F and e→(yz)∈F, applying Proposition 14 (1) and (2), (xy)∈F and (yz)∈F. By Proposition 14 (4), (xy)→(xz)∈F. It follows that (xz)∈F. Hence, (xx)→(xz)∈F, by Proposition 14 (4). Therefore,
e→(xz) = (xx)→(xz)∈F.
Moreover, from (xy)→eF and (yz)→eF, applying regularity of F and Definition 13 (6), we get that (xz)→eF.
Combining the above e→(xz)∈F and (xz)→eF, we have xz ≈F e, that is, [xz]F = [e]F. This means that [x]F ≤ [z]F. It follows that the binary relation ≤ on X/≈F is a partially order.
Therefore, applying Theorem 3, we know that (X/≈F,, [e]F) is a normal basic implication algebra, and (X, →, ≤, e) (X/≈F,, [e]F) in the homomorphism mapping f: XX/≈F; f(x) = [x]F. □
Example 7.
Let X = {a, b, c, d, 1}. Define operations on X as per the following Cayley Table 9, and the order binary on X is defined as follows: a ≤ b ≤ c ≤ 1, b ≤ d ≤ 1. Then (X, →, ≤, 1) is a normal basic implication algebra (it is not a quantum B-algebra). Denote F = {1}, then F is regular filters of X, and the quotient algebras (X, →, ≤, 1) is isomorphic to (X/≈F, →, [1]F).
Example 8.
Denote X = {a, b, c, d, 1}. Define operations on X as per the following Cayley Table 10, and the order binary on X is defined as follows: a ≤ b ≤ c ≤ 1, b ≤ d ≤ 1. Then (X, →, ≤, 1) is a normal basic implication algebra (it is not a quantum B-algebra). Let F = {1, d}, then F is a regular filters of X, and the quotient algebras (X/≈F, →, [1]F) is presented as the following Table 11, where X/≈F = {{a}, {b, c}, [1]F = {1, d}}. Moreover, (X, →, ≤, 1) ∼ (X/≈F →, [1]F).

5. Conclusions

In this paper, we introduced the notion of a q-filter in quantum B-algebras and investigated quotient structures; by using q-filters as a corollary, we obtained quotient pseudo-BCI algebras by their filters. Moreover, we pointed out that the concept of quantum B-algebra does not apply to non-associative fuzzy logics. From this fact, we proposed the new concept of basic implication algebra, and established the corresponding filter theory and quotient algebra. In the future, we will study in depth the structural characteristics of basic implication algebras and the relationship between other algebraic structures and uncertainty theories (see References [33,34,35,36]). Moreover, we will consider the applications of q-filters for Gentzel’s sequel calculus.

Author Contributions

X.Z. initiated the research and wrote the draft. B.R.A., Y.B.J., and X.Z. completed the final version.

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 61573240).

Conflicts of Interest

The authors declare no conflict of interest.

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Table 1. Cayley table of operation .
Table 1. Cayley table of operation .
abcdef
afacccf
bfacccf
cccfabc
dccffac
eccffac
fabcdef
Table 2. Cayley table of operation ↝.
Table 2. Cayley table of operation ↝.
abcdef
afaccdf
bffcccf
cccfaac
dccfaac
eccffac
fabcdef
Table 3. Cayley table of operation .
Table 3. Cayley table of operation .
abcde1
a1accc1
b11ccc1
ccc1abc
dcc11ac
ecc111c
1abcde1
Table 4. Cayley table of operation ↝.
Table 4. Cayley table of operation ↝.
abcde1
a1accd1
b11ccc1
ccc1aac
dcc11ac
ecc111c
1abcde1
Table 5. Cayley table of operation .
Table 5. Cayley table of operation .
abcdef
afacccf
bffcccf
cccfabc
dccffac
eccfffc
fabcdef
Table 6. Cayley table of operation ↝.
Table 6. Cayley table of operation ↝.
abcdef
afaccdf
bffcccf
cccfaac
dccffac
eccffac
fabcdef
Table 7. Cayley table of operation .
Table 7. Cayley table of operation .
abcd1
a11111
bd1111
cdd111
dbcd11
1bbcb1
Table 8. Cayley table of operation .
Table 8. Cayley table of operation .
abcd1
a11111
bc1111
ccc111
daca1c
1abbc1
Table 9. Cayley table of operation .
Table 9. Cayley table of operation .
abcd1
a11111
bd1111
cbd1d1
dacc11
1abcd1
Table 10. Cayley table of operation .
Table 10. Cayley table of operation .
abcd1
a11111
bd1111
cbd1d1
dabc11
1abcd1
Table 11. Quotient algebra (X/≈F,, [1]F).
Table 11. Quotient algebra (X/≈F,, [1]F).
{a}{b,c}[1]F
{a}[1]F[1]F[1]F
{b,c}{b,c}[1]F[1]F
[1]F{a}{b,c}[1]F

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