EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 997-1004 ISSN 1307-5543 – ejpam.com Published by New York Business Global On direct product of d-Algebras Maliwan Phattarachaleekul Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham 44150, Thailand Abstract. The main aim of this work is to introduce and study the notions of ideal direct product d-algebras, d-ideal direct product d-algebras, sub-direct product d-algebras, edge direct product and positive implicative direct product d-algebras and investigate their characterizations. 2020 Mathematics Subject Classifications: 06F35 Key Words and Phrases: d-algebras, direct product d-algebras, ideal direct product d-algebras, d-ideal direct product d-algebras, ideal, edge direct product d-algebras, positive implicative, direct product d-algebra 1. Introduction The concept of d-algebras was first introduced by J. Neggers and H. S. Kim ([9]). A d-algebraX = (X, ∗, 0) is an algebra of type (2, 0), that is, a nonempty setX together with a binary operation ∗ and a constant 0 satisfying some axioms In [1], they introduced and investigated several relations between d-algebras and BCK-algebras and showed that the class of oriented digraphs corresponds in a simple way to the class of edge d-algebras and that arbitrary d-algebras also determine unique edge d-algebras in a natural manner. In 1999, J. Neggers, Y. B. Jun and H. S. Kim ([8]), introduced the notions of a d-subalgebra, d-ideal, and a d∗-ideal in d-algebras, and investigated relations among them. Furthermore, they are able to define the ideal of a quotient d-algebra and to prove a fundamental theorem of d-morphisms for d-algebras as a consequence. S. S. Ahn and K. S. So ([1], defined left-regular maps on d-algebras.These mappings show behaviors reminiscent or homomorphisms on d-algebras. In particular, they have introduced the kernels, annihilators, co-annihilators and some of their properties for these mappings, especially in the setting of positive implicative d-algebras. The study of multipliers have been made by various researchers in the context of C*-algebras, rings and semigroups in ([6]). In 2012, M. A. Chaudhry and F. Ali ([3]) introduced the concept of a multiplier on d-algebra and obtain some properties of multipliers of d-algebras. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4738 Email address: maliwan.t@msu.ac.th (M. Phattarachaleekul) https://www.ejpam.com 997 © 2023 EJPAM All rights reserved. M. Phattarachaleekul / Eur. J. Pure Appl. Math, 16 (2) (2023), 997-1004 998 The concept of the direct product, was first defined in groups and obtained the prop- erties that a direct product of groups is also a group. In 1999, J. Neggers and H. S. Kim ([9] ) introduced the concept of a direct product of d-algebras, they investigate several re- lations between projection mappings and d-morphisms on a direct sum of edge d-algebras, In 2020, A. Setiani, S. Gemawati and L. Deswita ([10]) introduced the notions of a direct product of BP-algebra and some of related properties are investigated. Also, the notion of direct product of 0-commutative BP-algebra and BP-homomorphisms were studied. In 2022, C. Chanmanee, R. Chinram, R. Prasertpong, P. Julatha, and A. Iampan ([2]) gave the concept an external direct produc and a weak direct product of B-algebras and they provided several fundamental theorems of (anti-)B-homomorphisms in view of the external direct product B-algebras. In this paper, we introduce the concept of an ideal direct product d-algebra, a d-ideal direct product d-algebra, sub-direct product d-algebra, an edge direct product and a positive implicative direct product d-algebra. 2. Preliminaries First, we will review some essential notations and definitions of d-algebras and ordinary senses that are needed for this study in this section. Definition 1. [9] A d-algebras is a non-empty set X with a constant 0 and a binary operation ∗ satisfying the following axioms: (i) x ∗ x = 0, (ii) 0 ∗ x = 0, (iii) x ∗ y = 0 and y ∗ x = 0 imply x = y for all x, y ∈ X. A nonempty subset S of a d-algebra X is said to be a sub-algebra of X if x ∗ y ∈ S for all x, y ∈ S. Definition 2. [1] A d-algebras (X, ∗, 0) is said to be a positive implicative if (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z) for all x, y, z ∈ X. Example 1. [1] Let X = {0, a, b, c} be a set with a binary operation ∗ on X defined by the following table: ∗ 0 a b c 0 0 0 0 0 a a 0 a 0 b b b 0 0 c c c c 0 M. Phattarachaleekul / Eur. J. Pure Appl. Math, 16 (2) (2023), 997-1004 999 Then (X, ∗, 0) is a positive implicative d-algebra. Example 2. [5] Let X = {0, a, b, c} be a set with a binary operation ∗ on X defined by the following table: ∗ 0 a b c 0 0 0 0 0 a a 0 0 b b b b 0 0 c c c c 0 Then (X, ∗, 0) is a d-algebra but not positive implicative because (a∗b)∗c = 0∗c = 0 ̸= b = b ∗ 0 = (a ∗ c) ∗ (b ∗ c). The set S1 = {b, c} is not a sub-algebra of X whereas S2 = {0, a, b} is a sub-algebra of X. Definition 3. [7] Let (X, ∗, 0) be a d-algebra and x ∈ X . Define x ∗X :={x ∗ a |a ∈ X}. We say that X is edge if x ∗X = {x, 0} for all x ∈ X. Example 3. [7] Let X = {0, 1, 2, 3} be a set with the binary operation ∗ on X defined by the following table: ∗ 0 1 2 3 0 0 0 0 0 1 1 0 0 1 2 2 2 0 0 3 3 3 3 0 Then (X, ∗, 0) is an edge d-algebra. Example 4. [4] Let X = {0, 1, 2, 3} be a set with the following table: ∗ 0 1 2 3 0 0 0 0 0 1 1 0 1 0 2 2 2 0 0 3 3 3 1 0 Since 3 ∗X = {3, 1, 0} ≠ {3, 0}, then (X, ∗, 0) is not an edge d-algebra. M. Phattarachaleekul / Eur. J. Pure Appl. Math, 16 (2) (2023), 997-1004 1000 Theorem 1. [7] Let (X, ∗, 0) be an edge d-algebra. Then the following conditions are satisfiesd : (i) x ∗ 0 = x, (ii) (x ∗ y) ∗ z = (x ∗ z) ∗ y, (iii) x ∗ (x ∗ y) = y , for any x, y, z ∈ X. Definition 4. [3] Let (X, ∗, 0) be a d-algebra and I a subset of X, then I is called an ideal of X if it satisfies the following conditions: (i) 0 ∈ I , (ii) If x ∗ y ∈ I and y ∈ I imply x ∈ I. Definition 5. [3] Let (X, ∗, 0) be a d-algebra and I a nonempty subset of X, then I is called a d-ideal of X if it satisfies the following conditions : (i) If x ∗ y ∈ I and y ∈ I imply x ∈ I, (ii) If x ∈ I and y ∈ X imply x ∗ y ∈ I. Clealy, If I is a d-ideal of a d-algebra X, then x ∗ x = 0 ∈ I for any x ∈ I and then I is an ideal of X, but the converse need not be true as the following example: Example 5. [9] Let X = {0, a, b, c} be a set with binary operation ∗ on X defined by the following table: ∗ 0 a b c 0 0 0 0 0 a a 0 0 b b b b 0 0 c c c c 0 Then (X, ∗, 0) is a d-algebra and I := {0, a} is an ideal of X, but not a d- ideal of X, since a ∗ c = b /∈ I. Theorem 2. [9] Let I be a d-ideal of a d-algebra X. If x ∈ I and y ∈ X such that y ∗ x = 0, then y ∈ I. 3. Direct product d-Algebras J. Neggers and H. S. Kim ([9] ) introduced the concept of a direct product of d-algebras as follows. Let {(Xi, ∗, 0) | i ∈ I} be a non-empty family of d-algebras and ∏ i∈I Xi= {(xi)i∈I | xi ∈ Xi}. Then (0i)i∈I where 0i ∈ Xi. serves as 0 of ∏ i∈I Xi. Define a binary M. Phattarachaleekul / Eur. J. Pure Appl. Math, 16 (2) (2023), 997-1004 1001 operation ⊙ on ∏ i∈I Xi by (xi)i∈I ⊙ (yi)i∈I = (xi ∗ yi)i∈I for all (xi)i∈I , (yi)i∈I ∈ ∏ i∈I Xi. Then ( ∏ i∈I Xi,⊙, (0i)i∈I) is a d-algebra, called a direct product d-algebra. That is a direct product d-algebra ( ∏ i∈I Xi,⊙, (0i)i∈I) is satisfies the following conditions : (i) (xi)i∈I ⊙ (xi)i∈I = (0i)i∈I , (ii) (0i)i∈I)⊙ (xi)i∈I = (0i)i∈I , (iii) (xi)i∈I ⊙ (yi)i∈I = (0i)i∈I and (yi)i∈I ⊙ (xi)i∈I = (0i)i∈I implies (xi)i∈I = (yi)i∈I for all (xi)i∈I , (yi)i∈I ∈ ∏ i∈I Xi. Definition 6. Let ( ∏ i∈I Xi,⊙, (0i)i∈I) be a direct product d-algebra. A non-empty subset∏ i∈I Ni of ∏ i∈I Xi is said to be an ideal direct product d-algebra if it satisfies the following conditions : (I1) (0i)i∈I ∈ ∏ i∈I Ni, (I2) (xi)i∈I ∗ (yi)i∈I ∈ ∏ i∈I Ni and (yi)i∈I ∈ ∏ i∈I Ni implies (xi)i∈I ∈ ∏ i∈I Ni. Definition 7. Let ( ∏ i∈I Xi,⊙, (0i)i∈I) be a direct product d-algebra. A non-empty subset∏ i∈I Ni of ∏ i∈I Xi is said to be a d-ideal direct product d-algebras if it satisfies the following conditions: (D1) (xi)i∈I ⊙ (yi)i∈I ∈ ∏ i∈I Ni and (yi)i∈I ∈ ∏ i∈I Ni implies (xi)i∈I ∈ ∏ i∈I Ni, (D2) (xi)i∈I ∈ ∏ i∈I Ni and (yi)i∈I ∈ ∏ i∈I Xi implies (xi)i∈I ⊙ (yi)i∈I ∈ ∏ i∈I Ni . Example 6. [1], [9] Let X1 = {0, 1, 2, 3} and X2 = {0′ , a, b, c}. Define binary operations ∗ on X1 and ∗′ on X2. defined by the following two tables, respectively. ∗ 0 1 2 3 0 0 0 0 0 1 1 0 1 0 2 2 2 0 0 3 3 3 3 0 ∗′ 0 ′ a b c 0 ′ 0 ′ 0 ′ 0 ′ 0 ′ a a 0 ′ 0 ′ b b b b 0 ′ 0 ′ c c c c 0 ′ By example 4 and example 5, (X1, ∗, 0) and (X2, ∗′, 0 ′ ) are d-algebras. Consider an ideal N1 = {0, 1} of (X1 and ideal N2 = {0′ , a} of X2, we have N1 × N2 = {(0, 0′ ), (0, a), (1, 0 ′ )}, (1, a), } is an ideal direct product d-algebra but not a d-ideal direct product d-algebra, since then (0, a)⊙ (2, c) = (0∗2, a∗′ c) = (0, b) /∈ N1×N2. M. Phattarachaleekul / Eur. J. Pure Appl. Math, 16 (2) (2023), 997-1004 1002 Definition 8. Let ( ∏ i∈I Xi,⊙, (0i)i∈I) be a direct product d-algebra, a nonempty subset∏ i∈I Ni of ∏ i∈I Xi is said to be a sub-direct product of ∏ i∈I Xi if (xi)i∈I ⊙ (yi)i∈I ∈ ∏ i∈I Ni for all (xi)i∈I , (yi)i∈I ∈ ∏ i∈I Ni. Theorem 3. Every d-ideal direct product d-algebra is an ideal direct product d-algebra . Proof. Let ( ∏ i∈I Xi,⊙, (0i)i∈I) be a direct product d-algebra and ∏ i∈I Ni be a d-ideal of∏ i∈I Xi. Since xi ∗ xi = 0i for all i ∈ I implies that (xi)i∈I ⊙ (xi)i∈I = (0i)i∈I ∈ ∏ i∈I Ni for any (xi)i∈I ∈ ∏ i∈I Ni . Thus ∏ i∈I Ni is an ideal of ∏ i∈I Xi. Theorem 4. Every d-ideal a direct product d-algebra is a sub-direct product d-algebra. Proof. It is Clear by definition 7 and 8 Definition 9. Let ( ∏ i∈I Xi,⊙, (0i)i∈I) be a direct product d-algebra and (ai)i∈I ∈ ∏ i∈I Xi. Define the set (ai)i∈I ⊙ ∏ i∈I Xi:= {(ai)i∈I ⊙ (xi)i∈I | (xi)i∈I ∈ ∏ i∈I Xi}. We say that ∏ i∈I Xi is to be an edge direct product of d-algebra if (ai)i∈I ⊙ ∏ i∈I Xi:= {(ai)i∈I , (0i)i∈I}. Example 7. [9],[7] Let X1 = {0, 1, 2, 3} and X2 = {0′ , a, b, c} be the set with a binary operation ∗ and ∗′ respectively that following 2 of tables : ∗ 0 1 2 3 0 0 0 0 0 1 1 0 0 1 2 2 2 0 0 3 3 3 3 0 ∗′ 0 ′ a b c 0 ′ 0 ′ 0 ′ 0 ′ 0 ′ a a 0 ′ 0 ′ b b b b 0 ′ 0 ′ c c c c 0 ′ Then (X1, ∗, 0) and (X2, ∗ ′ , 0) are edge d-algebras. But X1 ×X2 is not an edge direct product d-algebra, because of (2, a)⊙(X1×X2) = {(2, a), (2, 0), (0, a), (0, 0′ )} ≠ {(0, 0′ ), (2, a)}. Theorem 5. Let ( ∏ i∈I Xi,⊙, (0i)i∈I) be an edge direct product d-algebra and ∏ i∈I Ni be an ideal direct product of ∏ i∈I Xi. If (ni)i∈I ∈ ∏ i∈I Ni and (xi)i∈I ∈ ∏ i∈I Xi, then (xi)i∈I ⊙ ((xi)i∈I ⊙ (ni)i∈I) ∈ ∏ i∈I Ni. Proof. Consider ((xi)i∈I⊙((xi)i∈I⊙(ni)i∈I))⊙(ni)i∈I = ((xi)i∈I⊙(ni)i∈I))⊙((xi)i∈I⊙ (ni)i∈I)) = (0i)i∈I , by definition 7 and theorem 1, (xi)i∈I ⊙ ((xi)i∈I ⊙ (ni)i∈I) ∈ ∏ i∈I Ni. Definition 10. A direct product d-algebra ( ∏ i∈I Xi,⊙, (0i)i∈I) is said to be positive im- plicative if ((xi)i∈I ⊙ (yi)i∈I) ⊙ (zi)i∈I = ((xi)i∈I ⊙ (zi)i∈I) ⊙ ((yi)i∈I ⊙ (zi)i∈I) for all (xi)i∈I , (yi)i∈I , (zi)i∈I ∈ ∏ i∈I Xi. REFERENCES 1003 Theorem 6. Let {(Xi, ∗, 0i) | i ∈ I} be a non-empty family of positive implicative d-algebra, then ( ∏ i∈I Xi,⊙, (0i)i∈I) is a positive implicative direct product d-algebra. Proof. Let (xi)i∈I , (yi)i∈I , (zi)i∈I ∈ ∏ i∈I Xi. Then ((xi)i∈I ⊙ (yi)i∈I)⊙ (zi)i∈I = (xi ∗ yi)i∈I ∗ (zi))i∈I = (xi ∗ zi)i∈I ∗ (yi ∗ zi)i∈I = ((xi)i∈I ⊙ (zi)i∈I)⊙ ((yi)i∈I ⊙ (zi)i∈I). Thus ∏ i∈I Xi is a positive implicative direct product d-algebra. Theorem 7. Every ideal of a positive implicative direct product d-algebra is a d-ideal direct product d-algebra. Proof. Let ( ∏ i∈I Xi,⊙, (0i)i∈I) be a positive implicative direct product d-algebra and∏ i∈I Ni is an ideal of ∏ i∈I Xi. By Definition 10, we have (ni)i∈I ⊙ (xi)i∈I)⊙ (ni)i∈I = (ni ∗ xi)i∈I ⊙ (ni)i∈I = ((ni ∗ xi) ∗ ((ni))i∈I = ((ni ∗ ni) ∗ (xi ∗ ni))i∈I = (0i ∗ (xi ∗ ni))i∈I = (0i)i∈I ∈ I. Hence ((ni)i∈I ⊙ (xi)i∈I) ∈ I, implies that ∏ i∈I Ni is a d-ideal direct product of d-algebras. 4. Conclusion In this paper, we give the concept of ideal, d-ideal, sub-direct product and edge in a direct product d-algebra and we prove relationship between ideal direct product and d-ideal direct product of d-algebras. Moreover, we shown that a direct product of edge d-algebras is not an edge direct product d-algebra. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] S. S. Ahn and K. S. So. On kernels and annihilators of left-regular mappings in d-algebras. Honam Mathematical Journal, 30(4):645–658, 2008. REFERENCES 1004 [2] C. Chanmanee, R. Chinram, R. Prasertpong, P. Julatha, and A. Iampan. Direct product of infinite family of b-algebras. 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