On BRK-Algebras

Abstract

The notion of BRK-algebra is introduced which is a generalization of BCK/BCI/BCH/Q/QS/BM-algebras. The concepts of 𝐺-part, 𝑝-radical, medial of a BRK-algebra are introduced and studied their properties. We proved that the variety of all medial BRK-algebras is congruence permutable and showed that every associative BRK-algebra is a group.

1. Introduction

In 1996, Imai and Iséki [1] introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. These algebras have been extensively studied since their introduction. In 1983, Hu and Li [2] introduced the notion of a BCH-algebra which is a generalization of the notion of BCK and BCI-algebras and studied a few properties of these algebras. In 2001, Neggers et al. [3] introduced a new notion, called a Q-algebra and generalized some theorems discussed in BCI/BCK-algebras. In 2002, Neggers and Kim [4] introduced a new notion, called a B-algebra, and obtained several results. In 2007, Walendziak [5] introduced a new notion, called a BF-algebra, which is a generalization of B-algebra. In [6], C. B. Kim and H. S. Kim introduced BG-algebra as a generalization of B-algebra. We introduce a new notion, called a BRK-algebra, which is a generalization of BCK/BCI/BCH/Q/QS/BM-algebras. The concept of 𝐺-part, 𝑝-radical, and medial of a BRK-algebra are introduced and studied their properties.

2. Preliminaries

First, we recall certain definitions from [2–5, 7, 8] that are required in the paper.

Definition 2.1. A BCI-algebra is an algebra (𝑋,∗,0) of type (2, 0) satisfying the following conditions: (B₁)(𝑥∗𝑦)∗(𝑥∗𝑧)≤(𝑧∗𝑦), (B₂)𝑥∗(𝑥∗𝑦)≤𝑦,(B₃)𝑥≤𝑥,(B₄)𝑥≤𝑦and𝑦≤𝑥imply𝑥=𝑦,(B₅)𝑥≤0implies𝑥=0,where𝑥≤𝑦isdefinedby𝑥∗𝑦=0,forall𝑥,𝑦,𝑧∈𝑋.

If (B₅) is replaced by (B₆): 0≤𝑥, then the algebra is called a BCK-algebra. It is known that every BCK-algebra is a BCI-algebra but not conversely.

Definition 2.2. A BCH-algebra is an algebra (𝑋,∗,0) of type (2,0) satisfying (B₃), (B₄), and (B₇): (𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦.

It is shown that every BCI-algebra is a BCH-algebra but not conversely.

Definition 2.3. A Q-algebra is an algebra (𝑋,∗,0) of type (2,0) satisfying (B₃), (B₇),  and  (B₈): 𝑥∗0=𝑥.

A Q-algebra is said to be a QS-algebra if it satisfies the additional relation:(B₉)(𝑥∗𝑦)∗(𝑥∗𝑧)=𝑧∗𝑦,

for any 𝑥,𝑦,𝑧∈𝑋. It is shown that every BCH-algebra is a Q-algebra but not conversely.

Definition 2.4. A B-algebra is an algebra (𝑋,∗,0) of type (2,0) satisfying (B₃), (B₈),  and  (B₁₀): (𝑥∗𝑦)∗𝑧=𝑥∗(𝑧∗(0∗𝑦)).

A B-algebra is said to be 0-commutative if 𝑎∗(0∗𝑏)=𝑏∗(0∗𝑎) for any 𝑎,𝑏∈𝑋. In [3], it is shown that Q-algebras and B-algebras are different notions.

Definition 2.5. A BF-algebra is an algebra (𝑋,∗,0) of type (2,0) satisfying (B₃), (B₈), and (B₁₁): 0∗(𝑥∗𝑦)=(𝑦∗𝑥).

It is shown that every B-algebra is BF-algebra but not conversely.

Definition 2.6. A BM-algebra is an algebra (𝑋,∗,0) of type (2,0) satisfying (B₈) and  (B₁₂): (𝑥∗𝑦)∗(𝑥∗𝑧)=𝑧∗𝑦.

Definition 2.7. A BH-algebra is an algebra (𝑋,∗,0) of type (2,0) satisfying (B₃), (B₄), and (B₈).

Definition 2.8. A BG-algebra is an algebra (𝑋,∗,0) of type (2,0) satisfying (B₃), (B₈), and (BG): (𝑥∗𝑦)∗(0∗𝑦)=𝑥.

3. BRK-Algebras

In this section, we define the notion of BRK-algebra and observe that the axioms in the definition are independent.

Definition 3.1. A BRK-algebra is a nonempty set 𝐴 with a constant 0 and a binary operation ∗ satisfying axioms:(B₈)𝑥∗0=𝑥, (B₁₃)(𝑥∗𝑦)∗𝑥=0∗𝑦forany𝑥,𝑦∈𝐴.

For brevity, we also call 𝐴 a BRK-algebra. In 𝐴, we can define a binary relation ”≤” by 𝑥≤𝑦 if and only if 𝑥∗𝑦=0.

Example 3.2. Let 𝐴∶=ℝ−{−𝑛},0≠𝑛∈ℤ⁺ where ℝ is the set of all real numbers and ℤ⁺ is the set of all positive integers. If we define a binary operation ∗ on 𝐴 by 𝑥∗𝑦=𝑛(𝑥−𝑦)𝑛+𝑦,(3.1) then (𝐴,∗,0) is an BRK-algebra.

Example 3.3. Let 𝐴={0,1,2} in which ∗ is defined by ∗012002211002200.(3.2) Then (𝐴,∗,0) is a BRK-algebra.

We know that every BCK-algebra is a BCI-algebra and every BCI-algebra is a BCH-algebra and every BCH-algebra is a Q-algebra. We can observe that every Q-algebra is a BRK-algebra but converse needs not be true.

Example 3.4. Let 𝐴={0,1,2,3} in which ∗ is defined by ∗012300101110102210133230.(3.3) Then (𝐴,∗,0) is a BRK-algebra, which is not a BCK/BCI/BCH/Q-algebra.

We know that every QS-algebra is a BM-algebra and we can observe that every BM-algebra is a BRK-algebra but converses need not be true.

Example 3.5. Let 𝐴={0,1,2,3} in which ∗ is defined by ∗012300220110022200233110.(3.4) Then (𝐴,∗,0) is a BRK-algebra, which is not a QS/BM-algebra.

It is easy to see that B/BG/BF/BH-algebra and BRK-algebras are different notions. For example, Example 3.3 is a BRK-algebra which is not a BH-algebra and Example 3.4 is an BRK-algebra which is not B/BG/BF-algebra. Consider the following example. Let 𝐴={0,1,2,3,4,5} be a set with the following table:∗012345002134511024532210534334502144531025534210.(3.5) Then (𝐴,∗,0) is a B/BF/BG/BH-algebra which is not an BRK-algebra.

We observe that the two axioms (B₈) and (B₁₃) are independent. Let 𝐴={0,1,2} be a set with the following left table:∗012001211122212∗012001011012010.(3.6) Then the axiom (B₈) holds but not (B₁₃), since (1∗2)∗1=2∗1=1≠2=0∗2. Similarly, the set 𝐴={0,1,2} with the above right table satisfies the axiom (B₁₃) but not (B₈), since 2∗0=0≠2.

Proposition 3.6. If (𝐴,∗,0) is a BRK-algebra, then, for any 𝑥,𝑦∈𝐴, the following conditions hold: (1)𝑥∗𝑥=0,(2)𝑥∗𝑦=0⇒0∗𝑥=0∗𝑦.

Proof. Let (𝐴,∗,0) be a BRK-algebra and 𝑥,𝑦∈𝐴. Then(1)𝑥∗𝑥=(𝑥∗0)∗𝑥=0∗0=0 (by B₈ and B₁₃),(2)𝑥∗𝑦=0⇒(𝑥∗𝑦)∗𝑥=0∗𝑥⇒0∗𝑦=0∗𝑥.

Proposition 3.7. Every BRK-algebra A satisfies the following property: 0∗(𝑥∗𝑦)=(0∗𝑥)∗(0∗𝑦),(3.7) for any 𝑥,𝑦∈𝐴.

Proof. Let 𝑥,𝑦∈𝐴. Then 0∗(𝑥∗𝑦)==([((0(𝑥∗∗𝑦𝑦))∗∗(𝑥𝑥)∗∗𝑦()𝑥)∗∗(𝑦0)]∗∗𝑦()0∗𝑦b)yBb₁₃yB₁₃=(0∗𝑥)∗(0∗𝑦).(3.8)

Theorem 3.8. Every BRK-algebra 𝐴 satisfying 𝑥∗(𝑥∗𝑦)=𝑥∗𝑦 for all 𝑥,𝑦∈𝐴 is a trivial algebra.

Proof. Putting 𝑥=𝑦 in the equation 𝑥∗(𝑥∗𝑦)=𝑥∗𝑦, we obtain 𝑥∗0=0⇒𝑥=0. Hence, 𝐴 is a trivial algebra.

Theorem 3.9. Every BRK-algebra 𝐴 satisfying (𝑥∗𝑦)∗(𝑥∗𝑧)=𝑧∗𝑦 for all 𝑥,𝑦,𝑧∈𝐴 is a BCI-algebra.

Proof. Let (𝐴,∗,0) be a BRK-algebra and (𝑥∗𝑦)∗(𝑥∗𝑧)=𝑧∗𝑦 for all 𝑥,𝑦,𝑧∈𝐴. Then(1)(𝑥∗𝑦)∗(𝑥∗𝑧)∗(𝑧∗𝑦)=(𝑧∗𝑦)∗(𝑧∗𝑦)=0, (2)(𝑥∗(𝑥∗𝑦))∗𝑦=((𝑥∗0)∗(𝑥∗𝑦))∗𝑦=(𝑦∗0)∗𝑦=𝑦∗𝑦=0, (3)𝑥∗𝑥=0,(4)Let 𝑥∗𝑦=0=𝑦∗𝑥. Then 𝑥=𝑥∗0=𝑥∗(𝑥∗𝑦)=(𝑥∗0)∗(𝑥∗𝑦)=𝑦∗0=𝑦,(5)𝑥∗0=0⇒𝑥=0.

Theorem 3.10. Every 0-commutative B-algebra is a BRK-algebra.

Proof. Let 𝐴 be a 0-commutative B-algebra. Then 𝑥∗(𝑥∗𝑦)=𝑦 for all 𝑥,𝑦∈𝐴. Hence, (𝑥∗𝑦)∗𝑥=𝑥∗(𝑥∗(0∗𝑦))=0∗𝑦.

The following theorem can be proved easily.

Theorem 3.11. Let (A,∗,0) be a BRK-algebra. Then, for any x,y∈A, the following conditions hold.(1)If (𝑥∗𝑦)∗(0∗(0∗𝑦))=(𝑥∗𝑦)∗𝑦, then 0∗(0∗(0∗𝑦))=0∗𝑦.(2)If (𝑥∗𝑦)∗(0∗𝑦)=(𝑥∗𝑦)∗𝑦, then 0∗(0∗𝑦)=0∗𝑦.(3)If 𝑥∗(𝑦∗𝑥)=𝑥∗(0∗(𝑥∗𝑦)), then 0∗(𝑦∗𝑥)=0∗(0∗(𝑥∗𝑦)).

4. 𝐺-Part of BRK-Algebras

In this section, we define 𝐺-part, 𝑝-radical and medial of a BRK-algebra. We give a necessary and sufficient condition for a BRK-algebra to become a medial BRK-algebra and investigate the properties of 𝐺-part in BRK-algebras.

Definition 4.1. A nonempty subset 𝐼 of a BRK-algebra 𝐴 is called a subalgebra of 𝐴 if 𝑥∗𝑦∈𝐼 whenever 𝑥,𝑦∈𝐼.

Definition 4.2. A nonempty subset 𝐼 of a BRK-algebra 𝐴 is called an ideal of 𝐴 if for any 𝑥,𝑦∈𝐴:(i)0∈𝐼, (ii)𝑥∗𝑦∈𝐼 and 𝑦∈𝐼 imply 𝑥∈𝐼.

Obviously, {0} and 𝐴 are ideals of 𝐴. We call {0} and 𝐴 the zero ideal and the trivial ideal of 𝐴, respectively. An ideal 𝐼 is said to be proper if 𝐼≠𝐴.

Definition 4.3. An ideal 𝐼 of a BRK-algebra 𝐴 is called a closed ideal of 𝐴 if 0∗𝑥∈𝐼 for all 𝑥∈𝐼.

Example 4.4. Let 𝐴={0,1,2} in which ∗ is defined by ∗012002211002200.(4.1) Then (𝐴,∗,0) is a BRK-algebra and the set 𝐼={0,2} is a subalgebra, an ideal, and a closed ideal of 𝐴.

Definition 4.5. Let 𝐴 be a BRK-algebra. For any subset 𝑆 of 𝐴, we define 𝐺(𝑆)={𝑥∈𝑆∣0∗𝑥=𝑥}.(4.2) In particular, if 𝑆=𝐴, then we say that 𝐺(𝐴) is the 𝐺-part of a BRK-algebra.

For any BRK-algebra 𝐴, the set:𝐵(𝐴)={𝑥∈𝐴∣0∗𝑥=0}(4.3) is called a 𝑝-radical of 𝐴. Clearly, 𝐵(𝐴) is a subalgebra and an ideal of 𝐴.

A BRK-algebra 𝐴 is said to be 𝑝-semisimple if 𝐵(𝐴)={0}.

The following property is obvious:𝐺(𝐴)∩𝐵(𝐴)={0}.(4.4)

Lemma 4.6. If (𝐴,∗,0) is a BRK-algebra and 𝑎∗𝑏=𝑎∗𝑐 for 𝑎,𝑏,𝑐∈𝐴, then 0∗𝑏=0∗𝑐.

Proof. Let (𝐴,∗,0) be a BRK-algebra and 𝑎,𝑏,𝑐∈𝐴. Then by (B₁₃), 𝑎∗𝑏=𝑎∗𝑐⇒(𝑎∗𝑏)∗𝑎=(𝑎∗𝑐)∗𝑎⇒0∗𝑏=0∗𝑐.

Theorem 4.7. Let (𝐴,∗,0) be a BRK-algebra. Then a left cancellation law holds in 𝐺(𝐴).

Proof. Let 𝑎,𝑏,𝑐∈𝐺(𝐴) with 𝑎∗𝑏=𝑎∗𝑐. Then, by Lemma 4.6,  0∗𝑏=0∗𝑐. Since 𝑏,𝑐∈𝐺(𝐴), we obtain 𝑏=𝑐.

Proposition 4.8. Let (𝐴,∗,0) be a BRK-algebra. If 𝑥∈𝐺(𝐴), then 0∗𝑥∈𝐺(𝐴).

Proof. Let 𝑥∈𝐺(𝐴). Then 0∗𝑥=𝑥 and hence 0∗(0∗𝑥)=0∗𝑥. Therefore, 0∗𝑥∈𝐺(𝐴).

Converse of the above proposition needs not be true. From Example 4.4, we can see that 0∗1=2∈{0,2}=𝐺(𝐴) but 1∉𝐺(𝐴).

Theorem 4.9. If 𝑥,𝑦∈𝐺(𝐴), then 𝑥∗𝑦∈𝐺(𝐴).

Proof. Let 𝑥,𝑦∈𝐺(𝐴). Then 0∗𝑥=𝑥 and 0∗𝑦=𝑦. Hence, 0∗(𝑥∗𝑦)=(0∗𝑥)∗(0∗𝑦)=𝑥∗𝑦. Therefore, 𝑥∗𝑦∈𝐺(𝐴).

Proposition 4.10. If (𝐴,∗,0) is a BRK-algebra and 𝑥,𝑦∈𝐴, then 𝑦∈𝐵(𝐴)⟺(𝑥∗𝑦)∗𝑥=0.(4.5)

Proof. Let (𝐴,∗,0) be a BRK-algebra and 𝑥,𝑦∈𝐴. Then, by (B₁₃),  𝑦∈𝐵(𝐴)⇔0∗𝑦=0⇔(𝑥∗𝑦)∗𝑥=0.

Theorem 4.11. If 𝑆 is a subalgebra of a BRK- algebra (𝐴,∗,0), then 𝐺(𝐴)∩𝑆=𝐺(𝑆).

Proof. Clearly, 𝐺(𝐴)∩𝑆⊆𝐺(𝑆). If 𝑥∈𝐺(𝑆), then 0∗𝑥=𝑥 and 𝑥∈𝑆⊆𝐴. Hence, 𝑥∈𝐺(𝐴). Therefore, 𝑥∈𝐺(𝐴)∩𝑆. Thus, 𝐺(𝐴)∩𝑆=𝐺(𝑆).

Theorem 4.12. Let (𝐴,∗,0) be a BRK- algebra. If 𝐺(𝐴)=𝐴, then 𝐴 is 𝑝-semisimple.

Proof. Assume that 𝐺(𝐴)=𝐴. Then {0}=𝐺(𝐴)∩𝐵(𝐴)=𝐴∩𝐵(𝐴)=𝐵(𝐴). Hence, 𝐴 is 𝑝-semisimple.

Theorem 4.13. Every closed ideal of a BRK- algebra is a subalgebra.

Proof. Let 𝐼 be a closed ideal of a BRK-algebra (𝐴,∗,0) and 𝑥,𝑦∈𝐼. Then 0∗𝑦∈𝐼. By (B₁₃), (𝑥∗𝑦)∗𝑥=0∗𝑦∈𝐼. Since 𝐼 is an ideal and 𝑥∈𝐼, we have 𝑥∗𝑦∈𝐼. So 𝐼 is a subalgebra of 𝐴.

Note that the converse of the above theorem is not true. In Example 3.4, the set {0,1,2} is a subalgebra but not a closed ideal.

Theorem 4.14. Let 𝐼 be a subset of a BRK-algebra 𝐴. Then 𝐼 is a closed ideal of 𝐴 if and only if it satisfies (i) 0∈𝐼 (ii) 𝑥∗𝑧∈𝐼,𝑦∗𝑧∈𝐼 and 𝑧∈𝐼 imply 𝑥∗𝑦∈𝐼, for all 𝑥,𝑦,𝑧∈𝐴.

Proof. Let 𝐼 be a closed ideal of 𝐴. Clearly, 0∈𝐼. Assume that 𝑥∗𝑧,𝑦∗𝑧,𝑧∈𝐼. Since 𝐼 is an ideal, we have 𝑥,𝑦∈𝐼 which implies that 𝑥∗𝑦∈𝐼 because 𝐼 is a closed ideal and hence a subalgebra of 𝐴. Conversely, assume that 𝐼 satisfies (i) and (ii). Let 𝑥∗𝑦,𝑦∈𝐼. Since 0∗0,𝑦∗0,0∈𝐼, by (ii) we have 0∗𝑦∈𝐼. From (ii), again it follows that 𝑥=𝑥∗0∈𝐼 so that 𝐼 is an ideal of 𝐴. Now suppose that 𝑥∈𝐼. Since 0∗0,𝑥∗0,0∈𝐼, we obtain 0∗𝑥∈𝐼 by (ii). This completes the proof.

Definition 4.15. A BRK-algebra (𝐴,∗,0) is said to be positive implicative if ((𝑥∗𝑦)∗𝑦)∗(0∗𝑦)=𝑥∗𝑦(4.6) for all 𝑥,𝑦∈𝐴.

The BRK-algebra in Example 3.3 is positive implicative.

Definition 4.16. Let (𝐴,∗,0) be a BRK-algebra. For a fixed 𝑎∈𝐴. The map 𝑅_𝑎∶𝐴→𝐴 given by 𝑅_𝑎(𝑦)=𝑦∗𝑎 for all 𝑦∈𝐴 is called right translation of 𝐴. Similarly the map 𝐿_𝑎∶𝐴→𝐴 given by 𝐿_𝑎(𝑦)=𝑎∗𝑦 for all 𝑦∈𝐴 is called a left translation of 𝐴.

Definition 4.17. Let (𝐴,∗,0) be a BRK-algebra. For a fixed 𝑎∈𝐴. The map 𝑇_𝑎∶𝐴→𝐴 given by 𝑇_𝑎(𝑦)=(𝑦∗𝑎)∗(0∗𝑎) for all 𝑦∈𝐴 is called a weak right translation of 𝐴. Similarly, the map 𝑀_𝑎∶𝐴→𝐴 given by 𝑀_𝑎(𝑦)=(𝑎∗𝑦)∗(0∗𝑦) for all 𝑦∈𝐴 is called a weak left translation of 𝐴.

Theorem 4.18. A BRK-algebra (𝐴,∗,0) is positive implicative if and only if 𝑅_𝑧=𝑇_𝑧∘𝑅_𝑧 for all 𝑧∈𝐴.

Proof. Let 𝐴 be a BRK-algebra and 𝑅_𝑧=𝑇_𝑧∘𝑅_𝑧 for 𝑧∈𝐴. Then 𝑦∗𝑧=𝑅_𝑧(𝑦)==𝑇𝑇_𝑧𝑧∘𝑅_𝑧(𝑦)=𝑇_𝑧𝑅_𝑧(𝑦)(𝑦∗𝑧)=((𝑦∗𝑧)∗𝑧)∗(0∗𝑧),∀𝑦,𝑧∈𝐴.(4.7) Hence, 𝐴 is positive implicative BRK-algebra. Conversely, assume that 𝐴 is positive implicative BRK-algebra. Let 𝑥,𝑦∈𝐴. Then 𝑅_𝑥(𝑦)=𝑦∗𝑥==(𝑇(_𝑥𝑦∗𝑥)∗𝑥)∗(0∗𝑥)=𝑅_𝑥(𝑦)∗𝑥∗(0∗𝑥)𝑅_𝑥(𝑦)=𝑇_𝑥∘𝑅_𝑥(𝑦).(4.8) Hence, 𝑅_𝑥=𝑇_𝑥∘𝑅_𝑥.

Definition 4.19. A BRK-algebra (𝐴,∗,0) satisfying (𝑥∗𝑦)∗(𝑧∗𝑢)=(𝑥∗𝑧)∗(𝑦∗𝑢)(4.9) for any 𝑥,𝑦,𝑧 and 𝑢∈𝐴, is called a medial BRK-algebra.

Example 4.20. Let 𝐴∶=ℝ−{−𝑛},0≠𝑛∈ℤ⁺ where ℝ is the set of all real numbers and ℤ⁺ is the set of all positive integers. If we define a binary operation ∗ on 𝐴 by 𝑥∗𝑦=𝑛(𝑥−𝑦)𝑛+𝑦,(4.10) then (𝐴,∗,0) is a medial BRK-algebra.

Theorem 4.21. If 𝐴 is a medial BRK-algebra, then, for any 𝑥,𝑦,𝑧∈𝐴, the following hold:(i)𝑥∗(𝑦∗𝑧)=(𝑥∗𝑦)∗(0∗𝑧),(ii)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦.

Proof. Let 𝐴 be a medial BRK-algebra and 𝑥,𝑦,𝑧∈𝐴. Then(i)(𝑥∗𝑦)∗(0∗𝑧)=(𝑥∗0)∗(𝑦∗𝑧)=𝑥∗(𝑦∗𝑧),(ii)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑦)∗(𝑧∗0)=(𝑥∗𝑧)∗(𝑦∗0)=(𝑥∗𝑧)∗𝑦.

By the above theorem, the following corollary follows.

Corollary 4.22. Every medial BRK-algebra is a Q-algebra.

Theorem 4.23. Let 𝐴 be a medial BRK-algebra. Then the right cancellation law holds in 𝐺(𝐴).

Proof. Let 𝑎,𝑏,𝑥∈𝐺(𝐴) with 𝑎∗𝑥=𝑏∗𝑥. Then, for any 𝑦∈𝐺(𝐴),𝑥∗𝑦=(0∗𝑥)∗𝑦=(0∗𝑦)∗𝑥=𝑦∗𝑥. Therefore, 𝑎=0∗𝑎=(𝑥∗𝑎)∗𝑥=(𝑎∗𝑥)∗𝑥=(𝑏∗𝑥)∗𝑥=(𝑥∗𝑏)∗𝑥=0∗𝑏=𝑏.(4.11)

Now, we give a necessary and sufficient condition for a BRK-algebra to become a medial BRK-algebra.

Theorem 4.24. A BRK-algebra 𝐴 is medial if and only if it satisfies:(i)𝑥∗𝑦=0∗(𝑦∗𝑥)forall𝑥,𝑦∈𝐴, (ii)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦forall𝑥,𝑦,𝑧∈𝐴.

Proof. Suppose (𝐴,∗,0) is medial and 𝑥,𝑦,𝑧∈𝐴. Then(i)0∗(𝑦∗𝑥)=(𝑥∗𝑥)∗(𝑦∗𝑥)=(𝑥∗𝑦)∗(𝑥∗𝑥)=(𝑥∗𝑦)∗0=𝑥∗𝑦, (ii)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑦)∗(𝑧∗0)=(𝑥∗𝑧)∗(𝑦∗0)=(𝑥∗𝑧)∗𝑦. Conversely, assume that the conditions hold. Then (𝑥∗𝑦)∗(𝑧∗𝑢)============0((((((((0(00(((00𝑢𝑢𝑥𝑥𝑥𝑥∗∗∗∗∗∗∗∗∗∗∗∗((((𝑦((((((𝑧𝑧𝑧𝑧((𝑧𝑧𝑦𝑧𝑧))0𝑧)))∗∗∗∗∗∗∗∗∗∗∗∗(𝑥(𝑥(𝑢(𝑧𝑥𝑥𝑥𝑢𝑧𝑦(𝑦)))0)))))∗∗∗∗∗∗∗∗∗∗∗∗𝑥(𝑢𝑦𝑦(𝑥(𝑦((𝑢𝑢𝑢))))𝑥00))))∗∗)))∗∗∗∗∗∗∗(((𝑦𝑦bbb𝑧𝑢𝑢𝑢))𝑦𝑦(yyy))))))0))P∗((∗(((((((iir)ibbbbbbb𝑢o))𝑦yyyyyyy)p)o(((((((s((iiiiiiiii))iii)bbt)))))))iyy)))o&nP(ir3)o().ip7)o)asnitdion(i)3).7)(4.12) Therefore, 𝐴 is medial.

Corollary 4.25. A BRK-algebra 𝐴 is medial if and only if it is a medial QS-algebra.

The following theorem can be proved easily.

Theorem 4.26. An algebra (𝐴,∗,0) of type (2,0) is a medial BRK-algebra if and only if it satisfies: (i)𝑥∗(𝑦∗𝑧)=𝑧∗(𝑦∗𝑥),(ii)𝑥∗0=𝑥,(iii)𝑥∗𝑥=0.

Corollary 4.27. If 𝐴 is a medial BRK-algebra, then 𝑥∗(𝑥∗𝑦)=𝑦 for all 𝑥,𝑦∈𝐴.

Corollary 4.28. The class of all of medial BRK-algebras forms a variety, written 𝜈(𝑀𝑅).

Proposition 4.29. A variety 𝜈 is congruence-permutable if and only if there is a term 𝑝(𝑥,𝑦,𝑧) such that 𝜈⊨𝑝(𝑥,𝑥,𝑦)≈𝑦,𝜈⊨𝑝(𝑥,𝑦,𝑦)≈𝑥.(4.13)

Corollary 4.30. The variety 𝜈(𝑀𝑅) is congruence permutable.

Proof. Let 𝑝(𝑥,𝑦,𝑧)=𝑥∗(𝑦∗𝑧). Then by Corollary 4.25 and (B₈), we have 𝑝(𝑥,𝑥,𝑦)=𝑦 and 𝑝(𝑥,𝑦,𝑦)=𝑥, and so the variety 𝜈(𝑀𝑅) is congruence permutable.

The following example shows that a BRK-algebra may not satisfy the associative law.

Example 4.31. Let 𝐴={0,1,2} be a set with the following table: ∗012002211002200.(4.14) Then (𝐴,∗,0) is a BRK-algebra, but associativity does not hold since (1∗2)∗1=0∗1=2≠1=1∗0=1∗(2∗1).

Theorem 4.32. If 𝐴 is an associative BRK-algebra, then, for any 𝑥∈𝐵(𝐴),  𝑥=0.

Proof. Let 𝑥∈𝐵(𝐴). Then 0=0∗𝑥=(𝑥∗𝑥)∗𝑥=𝑥∗(𝑥∗𝑥)=𝑥∗0=𝑥.

Theorem 4.33. If 𝐴 is an associative BRK-algebra, then 𝐺(𝐴)=𝐴.

Proof. Let 𝐴 be an associative BRK-algebra. Clearly, 𝐺(𝐴)⊆𝐴. Let 𝑥∈𝐴. Then 0∗𝑥=(𝑥∗𝑥)∗𝑥=𝑥∗(𝑥∗𝑥)=𝑥∗0=𝑥. Hence, 𝑥∈𝐺(𝐴). Therefore, 𝐺(𝐴)=𝐴.

Now, we prove that every associative BRK-algebra is a group.

Theorem 4.34. Every BRK-algebra (𝐴,∗,0) satisfying the associative law is a group under the operation ”∗”.

Proof. Putting 𝑥=𝑦=𝑧 in the associative law (𝑥∗𝑦)∗𝑧=𝑥∗(𝑦∗𝑧) and using (B₃) and (B₈), we obtain 0∗𝑥=𝑥∗0=𝑥. This means that 0 is the zero element of 𝐴. By (B₃), every element 𝑥 of 𝐴 has as its inverse the element 𝑥 itself. Therefore, (𝐴,∗) is a group.

5. Conclusion and Future Research

In this paper, we have introduced the concept of BRK-algebra and studied their properties. In addition, we have defined 𝐺-part, 𝑝-radical, and medial of BRK-algebra and proved that the variety of medial algebras is congruence permutable. Finally, we proved that every associative BRK-algebra is a group.

In our future work, we introduce the concept of fuzzy BRK-algebra, interval-valued fuzzy BRK-algebra, intuitionistic fuzzy structure of BRK-algebra, intuitionistic fuzzy ideals of BRK-algebra, and intuitionistic (T,S)-normed fuzzy subalgebras of BRK-algebras, intuitionistic 𝐿-fuzzy ideals of BRK-algebra.

I hope this work would serve as a foundation for further studies on the structure of BRK-algebras.