Int. J. Anal. Appl. (2023), 21:43 Tri-Endomorphisms on BCH-Algebras Patchara Muangkarn1, Cholatis Suanoom1, Jirayu Phuto2, Aiyared Iampan3,∗ 1Science and Applied Science Center, Program of Mathematics, Kamphaeng Phet Rajabhat University, Kamphaeng Phet 62000, Thailand 2Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand 3Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand ∗Corresponding author: aiyared.ia@up.ac.th Abstract. In this paper, we use the concept of endomorphisms and bi-endomorphisms as a model to create tri-endomorphisms on of BCH-algebras. We introduce the concepts of left tri-endomorphisms, central tri-endomorphisms, right tri-endomorphisms, and complete tri-endomorphisms of BCH-algebras and provide some properties. In addition, we obtain the properties between those tri-endomorphisms and some subsets of BCH-algebras. 1. Introduction The algebraic structures of BCK-algebras and BCI-algebras were studied by Iséki and his colleague [4,5]. In 1983, Hu and Li [3] generalized a new class of algebras from BCI-algebras, namely, a BCH- algebra. Next, Bandru and Rafi [1] introduced a new algebra, called a G-algebra. BCH-algebras are also being studied extensively later, [2,3]. In this paper, we use the concept of endomorphisms and bi-endomorphisms as a model to create tri-endomorphisms. We introduce the concepts of left tri-endomorphisms, central tri-endomorphisms, right tri-endomorphisms, and complete tri-endomorphisms of BCH-algebras and provide some proper- ties. Before studying, we will review the definitions and well-known results. Received: Jul. 31, 2022. 2020 Mathematics Subject Classification. 06F35, 03G25. Key words and phrases. BCH-algebra; left tri-endomorphism; central tri-endomorphism; right tri-endomorphism; com- plete tri-endomorphism. https://doi.org/10.28924/2291-8639-21-2023-43 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-43 2 Int. J. Anal. Appl. (2023), 21:43 Definition 1.1. [3] A BCH-algebra is a non-empty set X with an element 0 and a binary operation ∗ satisfying the following conditions: (BCH1) (∀x ∈X)(x ∗x =0), (BCH2) (∀x,y ∈X)(x ∗y =0,y ∗x =0⇒ x = y), (BCH3) (∀x,y,z ∈X)((x ∗y)∗z =(x ∗z)∗y). In a BCH-algebra X =(X,∗,0), the binary relation ≤ on X is defined as follows: (∀x,y ∈X)(x ≤ y ⇔ x ∗y =0). Example 1.1. Let X = {0,a,b,c} with the following Cayley table as follows: ∗ 0 a b c 0 0 0 0 0 a a 0 c c b b 0 0 b c c 0 0 0 Then X =(X,∗,0) is a BCH-algebra. Proposition 1.1. [2,3] Let X =(X,∗,0) be a BCH-algebra. Then the following hold: for all x,y ∈X, (BCH4) (∀x,y ∈X)(x ∗ (x ∗y)≤ y), (BCH5) (∀x ∈X)(x ∗0=0⇒ x =0), (BCH6) (∀x,y ∈X)(0∗ (x ∗y)= (0∗x)∗ (0∗y)), (BCH7) (∀x ∈X)(x ∗0= x), (BCH8) (∀x,y ∈X)((x ∗y)∗x =0∗y), (BCH9) (∀x,y ∈X)(x ≤ y ⇒ 0∗x =0∗y). For a BCH-algebra X = (X,∗,0), some interesting subsets of X play a significant rule in the investigation of its properties described below. Definition 1.2. A non-empty subset Y of a BCH-algebra X =(X,∗,0) is called a subalgebra of X if x ∗y ∈ Y for all x,y ∈ Y . A non-empty subset I of a BCH-algebra X =(X,∗,0) is called an ideal of X if (1) 0∈ I, (2) (∀x,y ∈X)(x ∗y ∈ I,x ∈ I ⇒ y ∈ I). 2. Main results In this section, we introduce the concepts of left tri-endomorphisms, central tri-endomorphisms, right tri-endomorphisms, and complete tri-endomorphisms of BCH-algebras as follows. Definition 2.1. Let X =(X,∗,0) be a BCH-algebra. A mapping f :X3 →X is called Int. J. Anal. Appl. (2023), 21:43 3 (1) a left tri-endomorphism on X if (∀w,x,y,z ∈X)(f (x ∗w,y,z)= f (x,y,z)∗ f (w,y,z)), (2) a central tri-endomorphism on X if (∀w,x,y,z ∈X)(f (x,y ∗w,z)= f (x,y,z)∗ f (x,w,z)), (3) a right tri-endomorphism on X if (∀w,x,y,z ∈X)(f (x,y,z ∗w)= f (x,y,z)∗ f (x,y,w)), (4) a complete tri-endomorphism on X if (∀a,b,c,x,y,z ∈X)(f (x ∗a,y ∗b,z ∗c)= f (x,y,z)∗ f (a,b,c)). Example 2.1. In Example 1.1, we define fl :X3 →X by fl(x,y,z)=  x if y = z =0, 0 otherwise. Then fl is a left tri-endomorphism on X. Proposition 2.1. Let X =(X,∗,0) be a BCH-algebra and fl be a left tri-endomorphism on X. Then (1) (∀y,z ∈X)(fl(0,y,z)=0), (2) (∀w,x,y,z ∈X)(x ≤w ⇒ fl(x,y,z)≤ fl(w,y,z)). Proof. (1) Let y,z ∈X. Then, by BCH1, we have fl(0,y,z)= fl(0∗0,y,z)= fl(0,y,z)∗fl(0,y,z)= 0. (2) Let w,x,y,z ∈X be such that x ≤w. Then, by (1), we have 0= fl(0,y,z)= fl(x∗w,y,z)= fl(x,y,z)∗ fl(w,y,z). Hence, fl(x,y,z)≤ fl(w,y,z). � Similarly, the properties of central and right tri-endomorphisms are easily obtained. Proposition 2.2. Let X = (X,∗,0) be a BCH-algebra and fc be a central tri-endomorphism on X. Then (1) (∀x,z ∈X)(fc(x,0,z)=0), (2) (∀w,x,y,z ∈X)(y ≤w ⇒ fc(x,y,z)≤ fc(x,w,z)). Proposition 2.3. Let X =(X,∗,0) be a BCH-algebra and fr be a right tri-endomorphism on X. Then (1) (∀x,y ∈X)(fr(x,y,0)=0), (2) (∀w,x,y,z ∈X)(z ≤w ⇒ fr(x,y,z)≤ fr(x,y,w)). Theorem 2.1. Let X = (X,∗,0) be a BCH-algebra and f be a complete tri-endomorphism on X. Then (1) f (0,0,0)=0, (2) if S is a subalgebra of X, then f (S3) is also a subalgebra of X, (3) if S is an ideal of X and f is bijective, then f (S3) is also an ideal of X, (4) if f is a left tri-endomorphism on X, then f (x,y,z)∗ f (x,0,0)=0 for any x,y,z ∈X, (5) if f is a central tri-endomorphism on X, then f (x,y,z)∗ f (0,y,0)=0 for any x,y,z ∈X, (6) if f is a right tri-endomorphism on X, then f (x,y,z)∗ f (0,0,z)=0 for any x,y,z ∈X, 4 Int. J. Anal. Appl. (2023), 21:43 (7) if f is a left and right (central and right, left and central) tri-endomorphism on X, then f (x,y,z)=0 for any x,y,z ∈X, i.e., f is the zero map. Proof. (1) By BCH1, we have f (0,0,0)= f (0∗0,0∗0,0∗0)= f (0,0,0)∗ f (0,0,0)=0. (2) Suppose that S is a subalgebra of X. Let a,b ∈ f (S3). Then there exist (x1,y1,z1),(x2,y2,z2) ∈ S3 such that a = f (x1,y1,z1) and b = f (x2,y2,z2). Thus a ∗ b = f (x1,y1,z1) ∗ f (x2,y2,z2) = f (x1 ∗ x2,y1 ∗ y2,z1 ∗ z2) ∈ f (S3). Hence, f (S3) is a subalgebra of X. (3) Suppose that S is an ideal of X and f is bijective. Since 0 ∈ S and by (1), we have 0 = f (0,0,0)∈ f (S3). Assume that x∗y ∈ f (S3) and x ∈ f (S3). There exist (x1,y1,z1),(x2,y2,z2)∈S3 such that x ∗ y = f (x1,y1,z1) and x = f (x2,y2,z2). Since f is surjective, there exists (a,b,c)∈X3 such that y = f (a,b,c). Thus f (S3)3 f (x1,y1,z1)= x ∗y = f (x2,y2,z2)∗ f (a,b,c)= f (x2∗a,y2∗ b,z2 ∗ c). Since f is injective, we have x2 ∗ a,y2 ∗ b,z2 ∗ c ∈ S. Since S is an ideal of X, we get a,b,c ∈S. Thus y = f (a,b,c)∈ f (S3). Hence, f (S3) is an ideal of X. (4)-(6) It is obvious from Propositions 2.1-2.3. (7) Suppose that f is a left and right tri-endomorphism onX. Letx,y,z ∈X. Then, by Propositions 2.1 and 2.3, BCH1, BCH7 0= f (0,y,z)= f (x∗x,y∗0,z∗0)= f (x,y,z)∗f (x,0,0)= f (x,y,z)∗0= f (x,y,z). Hence, f is the zero map on X. � Let Tl(X) (resp., Tc(X), Tr(X) and T(X)) be the set of all left tri-endomorphisms (resp., right, central and complete tri-endomorphisms) on a BCH-algebra X =(X,∗,0). We define an operation ? on Tl(X) by (∀x,y,z ∈ X)((f ? g)(x,y,z) = f (x,y,z)∗g(x,y,z)). Let f ∈ Tl(X) and x,y,z ∈ X. Then (f ?f )(x,y,z)= f (x,y,z)∗f (x,y,z)=0. This means that f ?f =0X, where 0X :X3 →X is a function that maps all members to 0. Let f ,g ∈Tl(X) be such that f ?g =0X and g?f =0X. Then for all x,y,z ∈X, 0= (f ?g)(x,y,z)= f (x,y,z)∗g(x,y,z) and 0= (g ? f )(x,y,z)= g(x,y,z)∗ f (x,y,z). Since g(x,y,z), f (x,y,z) ∈ X, we have f (x,y,z) = g(x,y,z) for all x,y,z ∈ X. Hence, f = g. Let f ,g,h ∈Tl(X) and x,y,z ∈X. Then ((f ?g)?h)(x,y,z)= (f ?g)(x,y,z)∗h(x,y,z)=( f (x,y,z)∗g(x,y,z) ) ∗h(x,y,z)= ( f (x,y,z)∗h(x,y,z) ) ∗g(x,y,z)= (f ?h)(x,y,z)∗g(x,y,z)= ((f ?h)?g)(x,y,z). Hence, (f ?g)?h =(f ?h)?g. Theorem 2.2. (Tl(X),?,0X),(Tc(X),?,0X),(Tr(X),?,0X), and (T(X),?,0X) are BCH-algebras. Let X = (X,∗,0) be a BCH-algebra. We define the binary operation � on X3 as follows: (∀(a,b,c),(x,y,z) ∈ X3)((a,b,c) � (x,y,z) = (a ∗ x,b ∗ y,c ∗ z)). Then X3 = (X,�,(0,0,0)) is a BCH-algebra. Theorem 2.3. Let X =(X,∗,0) be a BCH-algebra and S1,S2,S3 be subsets of X. Then (1) S1 ×S2 ×S3 is a subalgebra of X3 if and only if S1,S2 and S3 are subsets of X, (2) S1 ×S2 ×S3 is an ideal of X3 if and only if S1,S2 and S3 are ideals of X. Int. J. Anal. Appl. (2023), 21:43 5 Proof. (1) Suppose that S1 × S2 × S3 is a subalgebra of X3. Firstly, we will show that S1 is a subalgebra of X. Let a,b ∈ S1. Let x ∈ S2 and u ∈ S3. Then (a,x,u),(b,x,u) ∈ S1 ×S2 ×S3. Thus (a∗b,0,0)= (a∗b,x∗x,u∗u)= (a,x,u)�(b,x,u)∈S1×S2×S3, that is, a∗b ∈S1. Hence, S1 is a subalgebra of X. On the other hand, we can show that S2 and S3 are subalgebras of X. Conversely, let (x,y,z),(a,b,c) ∈ S1 ×S2 ×S3. Then x ∗a ∈ S1,y ∗b ∈ S2, and z ∗c ∈ S3, so (x,y,z)� (a,b,c)= (x ∗a,y ∗b,z ∗c)∈S1 ×S2 ×S3. Hence, S1 ×S2 ×S3 is a subalgebra of X3. (2) Suppose that S1×S2×S3 is an ideal of X3. Since (0,0,0)∈S1×S2×S3, we have 0∈Si for all i =1,2,3. Assume that a∗x ∈S1 and a∈S1. Let b ∈S2 and c ∈S3. Then (a,b,c)∈S1×S2×S3 and (x,b,c)∈X3. Thus (a,b,c)�(x,b,c)= (a∗x,b∗b,c ∗c)= (a∗x,0,0)∈S1×S2×S3. Since S1 ×S2 ×S3 is an ideal of X3, we have (x,b,c) ∈ S1 ×S2 ×S3, that is, x ∈ S1. Hence, S1 is an ideal of X. Similarly, we can show that S2 and S3 are ideals of X. Conversely, suppose that S1, S2 and S3 are ideals of X. Since 0 ∈ Si for all i = 1,2,3, we have (0,0,0)∈S1×S2×S3. Assume that (a,b,c)∗(x,y,z)∈S1×S2×S3 and (a,b,c)∈S1×S2×S3. We get (a ∗ x,b ∗ y,c ∗ z) ∈ S1 ×S2 ×S3. Since a ∗ x,a ∈ S1, we have x ∈ S1. Moreover, we can obtain that y ∈ S2 and z ∈ S3. This implies that (x,y,z) ∈ S1 ×S2 ×S3. Hence, S1 ×S2 ×S3 is an ideal of X3. � Acknowledgment: This research project was supported by the Thailand Science Research and Inno- vation Fund and the University of Phayao (Grant No. FF66-UoE017). Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] R.K. Bandru, N. Rafi, On G-algebras, Sci. Magna, 8 (2012), 1-7. [2] W.A. Dudek, J. Thomys, On decompositions of BCH-algebras, Math. Japon. 35 (1990), 1131-1138. https: //cir.nii.ac.jp/crid/1572261549626076288. [3] Q.P. Hu, X. Li, On BCH-algebras, Math. Seminar Notes, 11 (1983), 313-320. [4] K. Iséki, An algebra related with a propositional calculus, Proc. Japan Acad. Ser. A Math. Sci. 42 (1966), 26-29. https://doi.org/10.3792/pja/1195522171. [5] K. Iséki, On BCI-algebras, Math. Seminar Notes, 8 (1980), 125-130. https://cir.nii.ac.jp/crid/1572261549626076288 https://cir.nii.ac.jp/crid/1572261549626076288 https://doi.org/10.3792/pja/1195522171 1. Introduction 2. Main results References