Int. J. Anal. Appl. (2022), 20:31 A Brief Note On QS/BP/BOI-Algebras Aiyared Iampan1,∗, Pongpun Julatha2 and Nareupanat Lekkoksung3 1Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 2Department of Mathematics, Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand 3Division of Mathematics, Faculty of Engineering, Rajamangala University of Technology Isan, Khon Kaen Campus, Khon Kaen 40000, Thailand ∗Corresponding author: aiyared.ia@up.ac.th Abstract. In this paper, the three types of the logical algebra of interest are QS-algebras, BOI-algebras, and BP-algebras, which are continuously studied by algebraists. Since the terms for each of the above logical algebras appear to be independent of each other, it may be understood that they are different. Therefore, the main purpose of this brief note is to show that QS-algebras, BOI-algebras, and BP- algebras satisfying (BM) as one. 1. Introduction and Preliminaries Several algebras with one binary and one nullary operations were proposed to develop an algebraic replica of implication reduction in classical or non-classical propositional logics. In 1966, Imai and Iséki [4,5] introduced BCK-algebras and BCI-algebras, which were important algebras and also inspired the creation of new types of algebra. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. In 1999, Ahn and Kim [2] introduced the concept of QS-algebras (see Definition 1.1) and thereafter in 2013 Ahn and Han [1] introduced the concept of BP-algebras (see Definition 1.2). In 2019, El-Gendy [3] introduced the concept of BOI-algebras (see Definition 1.3). The three types of the logical algebra of interest in this paper are QS-algebras, BOI-algebras, and BP-algebras, which are continuously studied by algebraists. Since the terms for each of the above Received: May 16, 2022. 2010 Mathematics Subject Classification. 06F35, 03G25, 03B05. Key words and phrases. QS-algebra; BOI-algebra; BP-algebra. https://doi.org/10.28924/2291-8639-20-2022-31 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-31 2 Int. J. Anal. Appl. (2022), 20:31 logical algebras appear to be independent of each other, it may be understood that they are different. Therefore, the main purpose of this brief note is to show that QS-algebras, BOI-algebras, and BP- algebras satisfying (BM) as one. Before we begin the study, let’s review the definitions of QS-algebras, BP-algebras, and BOI-algebras according to Definitions 1.1, 1.2, and 1.3, respectively. Definition 1.1. [2] A QS-algebra is a non-empty set X with a constant 0 and a binary operation ∗ satisfying axioms: (∀x ∈X)(x ∗x =0), (QS1) (∀x ∈X)(x ∗0= x), (QS2) (∀x,y,z ∈X)((x ∗y)∗z =(x ∗z)∗y), (QS3) (∀x,y,z ∈X)((z ∗x)∗ (z ∗y)= y ∗x). (BM) Example 1.1. [2] Let X = {0,1,2} with the following Cayley table: ∗ 0 1 2 0 0 0 0 1 1 0 0 2 2 0 0 Then X is a QS-algebra. Ahn and Kim proved the following proposition (see [2]). Proposition 1.1. If X =(X;∗,0) is a QS-algebra, then (∀x,y ∈X)((x ∗ (x ∗y))∗y =0), (1.1) (∀x,y,z ∈X)(((x ∗z)∗ (y ∗z))∗ (x ∗y)=0), (1.2) (∀x ∈X)(0∗ (0∗ (0∗x))=0∗x). (1.3) Definition 1.2. [1] A BP-algebra is a non-empty set X with a constant 0 and a binary operation ∗ satisfying axioms: (∀x ∈X)(x ∗x =0), (QS1) (∀x,y ∈X)(x ∗ (x ∗y)= y), (BP1) (∀x,y,z ∈X)((x ∗z)∗ (y ∗z)= x ∗y). (BP2) Int. J. Anal. Appl. (2022), 20:31 3 Example 1.2. [1] Let X = {0,1,2,3} with the following Cayley table: ∗ 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 Then X is a BP-algebra. Ahn and Han proved the following proposition (see [1]). Proposition 1.2. If X =(X;∗,0) is a BP-algebra, then (∀x ∈X)(0∗ (0∗x)= x), (1.4) (∀x,y ∈X)(0∗ (x ∗y)= y ∗x), (1.5) (∀x ∈X)(x ∗0= x). (QS2) Definition 1.3. [3] A BOI-algebra is a non-empty set X with a constant 0 and a binary operation ∗ satisfying axioms: (∀x ∈X)(x ∗x =0), (QS1) (∀x,y ∈X)(x ∗ (x ∗y)= y), (BP1) (∀x,y,z ∈X)((x ∗y)∗z =(x ∗z)∗y). (QS3) Example 1.3. [3] Let X = {0,1,2} with the following Cayley table: ∗ 0 1 2 0 0 2 1 1 1 0 2 2 2 1 0 Then X is a BOI-algebra. El-Gendy proved the following proposition (see [3]). 4 Int. J. Anal. Appl. (2022), 20:31 Proposition 1.3. If X =(X;∗,0) is a BOI-algebra, then (∀x ∈X)(x ∗0= x), (QS2) (∀x,y ∈X)(0∗ (x ∗y)= y ∗x), (1.6) (∀x,y,z ∈X)((z ∗x)∗ (z ∗y)= y ∗x), (BM) (∀x,y,z ∈X)((x ∗z)∗ (y ∗z)= x ∗y), (BP2) (∀x,y ∈X)((x ∗y)∗ (0∗y)= x), (1.7) (∀x,y ∈X)(x ∗ (x ∗ (x ∗y))= x ∗y), (1.8) (∀x,y ∈X)((x ∗y)∗x =0∗y), (1.9) (∀x,y,z ∈X)((x ∗y)∗z =(x ∗y)∗ (0∗z)). (1.10) 2. Results In this section, we will link that QS-algebras and BOI-algebras are the same, and BOI-algebras and BP-algebras satisfying (BM) are the same. Theorem 2.1. Every QS-algebra is a BOI-algebra. Proof. It only needs to show (BP1). Replacing x by 0 and z by x in (BM) and by (QS2), we obtain (BP1). � Theorem 2.2. Every BOI-algebra is a QS-algebra. Proof. It only needs to show (QS2) and (BM). It is immediately obtained by (QS2) and (BM). � From Theorems 2.1 and 2.2, we get the following theorem. Theorem 2.3. QS-algebras and BOI-algebras are the same. Theorem 2.4. Every BOI-algebra is a BP-algebra. Proof. It only needs to show (BP2). It is immediately obtained by (BP2). � Theorem 2.5. Every BP-algebra satisfying (BM) is a BOI-algebra. Proof. It only needs to show (QS3). Let X = (X;∗,0) be a BP-algebra satisfying (BM). Let x,y,z ∈X. Then (x ∗y)∗z =(x ∗y)∗ (x ∗ (x ∗z)) (by (BP1)) =(x ∗z)∗y, (by (BM)) so (QS3) is satisfied. Hence, X is a BOI-algebra. � From Theorems 2.4 and 2.5, we get the following theorem. Int. J. Anal. Appl. (2022), 20:31 5 Theorem 2.6. BOI-algebras and BP-algebras satisfying (BM) are the same. From Theorems 2.3 and 2.6, we get the following corollary. Corollary 2.1. QS-algebras, BOI-algebras, and BP-algebras satisfying (BM) are the same. We can summarize the diagram as follows. Acknowledgment The authors wish to express their sincere thanks to the referees for the valuable suggestions which lead to an improvement of this paper. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] S.S. Ahn, J.S. Han, On BP-Algebras, Hacettepe J. Math. Stat. 42 (2013), 551–557. [2] S.S. Ahn, H.S. Kim, On QS-Algebras, J. Chungcheong Math. Soc. 12 (1999), 33–41. [3] O.R. El-Gendy, On BOI-Algebras, Int. J. Math. Comput. Appl. Res. 9 (2019), 13–28. [4] Y. Imai, K. Iséki, On Axiom Systems of Propositional Calculi. XIV, Proc. Japan Acad. 42 (1966), 19–22. https: //doi.org/10.3792/pja/1195522169. [5] K. Iséki, An Algebra Related With a Propositional Calculus, Proc. Japan Acad. 42 (1966), 26–29. https://doi. org/10.3792/pja/1195522171. https://doi.org/10.3792/pja/1195522169 https://doi.org/10.3792/pja/1195522169 https://doi.org/10.3792/pja/1195522171 https://doi.org/10.3792/pja/1195522171 1. Introduction and Preliminaries 2. Results Acknowledgment References