EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6493 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Note on Quotient Ternary Semirings and Isomorphism Theorems Montakarn Petapirak1, Arbaz Jehan Khan1, Ronnason Chinram1,∗ 1 Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand Abstract. In this paper, we present a variety of notions of quotient ternary semirings, along with examples and remarks. Moreover, we establish various isomorphism theorems for ternary semirings. 2020 Mathematics Subject Classifications: 16Y60, 16Y99 Key Words and Phrases: Q-ideals, congruences, k-ideals, quotient ternary semirings, isomor- phism theorems 1. Introduction The notion of ideals is fundamental in ring theory, as it plays an important role in defining quotient rings. Therefore ideal theory has been a major area of research in the study of rings. Meanwhile, the concept of congruences is crucial for defining quotient semigroups. Semirings, as algebraic structures, were definitely one of natural choices of generalizations of rings. Many properties that hold for rings can be extended to semirings. Quotient semirings were previously studied in [1–3]. The concept of ternary semirings was first introduced in 2003 [4]. Every semiring can always be turned to a ternary semiring, though a ternary semiring does not always reduce to a semiring. The notion of ternary semirings can, in some sense, be regarded as a generalization of semirings, but it goes beyond a simple generalization, because certain concepts, such as lateral ideals, have no analog in semirings. In 2011, Chaudhari and Ingale [5] introduced a partitioning ideal (shortly, Q-ideal) of a ternary semiring. Q-ideals are useful to develop the quotient structures of ternary semirings. In 2021, Sunitha et al. [6] provided the characterization of full k-ideals in ternary semirings. In 2024, Sanborisoot and Ayutthaya [7] constructed a congruence relation with respect to a full k-ideal on a ternary semiring for the purpose of forming a ternary ring from the quotient ternary semiring. Quotient ternary semirings ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6493 Email addresses: montakarn.p@psu.ac.th (M. Petapirak), arbazjehankhan@gmail.com (A. J. Khan), ronnason.c@psu.ac.th (R. Chinram) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Petapirak, A. J. Khan, R. Chinram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6493 2 of 9 can be defined by congruences and/or ideals. Moreover, additional research papers related to ternary semirings, published during 2023-2025, can be found in [8–11]. In this paper, we aim to present various notions of quotient ternary semirings, exam- ine their properties, and provide a discussion on isomorphism theorems concerning these quotient ternary semirings. 2. Preliminaries In this section, we will recall some basic definitions of ternary semirings. Definition 1 ([4]). A non-empty set R together with a binary operation, called the addition, and the ternary multiplication, denoted by juxtaposition, is said to be a ternary semiring if R is an additive commutative semigroup satisfying the following conditions: (i) (abc)de = a(bcd)e = ab(cde), (ii) (a+ b)cd = acd+ bcd, (iii) a(b+ c)d = abd+ acd, (iv) ab(c+ d) = abc+ abd for all a, b, c, d, e ∈ R. Definition 2 ([4]). Let R be a ternary semiring. If 0 ∈ R such that 0 + x = x and 0xy = x0y = xy0 = 0 for all x, y ∈ R, then 0 is called a zero element. In this case, R is called a ternary semiring with zero. Definition 3 ([4]). An additive semigroup S of a ternary semiring R is called a ternary subsemiring of R if s1s2s3 ∈ S for all s1, s2, s3 ∈ S. Definition 4 ([4]). An additive subsemigroup I of a ternary semiring R is called (1) a left ideal of R if r1r2a ∈ I for all r1, r2 ∈ R and a ∈ I, (2) a right ideal of R if ar1r2 ∈ I for all r1, r2 ∈ R and a ∈ I, (3) a lateral ideal of R if r1ar2 ∈ I for all r1, r2 ∈ R and a ∈ I, (4) an ideal of R if I is a left ideal, a right ideal, and a lateral ideal of R. An ideal I of R is called a proper ideal if I ̸= R. Definition 5. An ideal I of a semiring R is called a k-ideal of R if, for any x, y ∈ R, x ∈ I and x+ y ∈ I, it follows y ∈ I. Definition 6. Let R and T be two ternary semirings and φ be a mapping which maps R into T . Then the mapping φ : R → T is called a homomorphism of R into T if the following conditions hold: M. Petapirak, A. J. Khan, R. Chinram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6493 3 of 9 (i) φ(a+ b) = φ(a) + φ(b), (ii) φ(abc) = φ(a)φ(b)φ(c) for all a, b, c ∈ R. Definition 7. Let R and T be two ternary semirings. The homomorphism φ : R → T is called an isomorphism of R onto T if φ is a bijection. If φ is an isomorphism, we say that R and T are isomorphic and use the notation R ∼= T . 3. Quotient ternary semirings In this section, we will present the various kinds of quotient ternary semirings con- structed by using different concepts. 3.1. Quotient ternary semirings modulo Q-ideals Firstly, we recall some results from [5]. Definition 8 ([5]). An ideal I of a ternary semiring R is called a partitioning ideal (shortly, Q-ideal) if there exists a subset Q of R such that (1) R = ∪{q + I | q ∈ Q}, (2) for any q1, q2 ∈ Q, (q1 + I) ∩ (q2 + I) ̸= ∅ ⇔ q1 = q2. Let I be a Q-ideal of a ternary semiring R. We let R/I(Q) = {q+ I | q ∈ Q} and define the addition ⊕ and ternary multiplication, for q1, q2, q3 ∈ Q, by (q1 + I)⊕ (q2 + I) = q∗ + I and (q1 + I)(q2 + I)(q3 + I) = q′ + I where q∗ ∈ Q is a unique element such that q1 + q2 + I ⊆ q∗ + I and q′ ∈ Q is a unique element such that q1q2q3 + I ⊆ q′ + I. Then R/I(Q) forms a ternary semiring under this addition and ternary multiplication. This ternary semiring will be called a quotient ternary semiring of R by a Q-ideal I [5]. Example 1. We consider a ternary semiring R = Z− 0 under the usual addition and ternary multiplication of integers. Let I = 5Z− 0 = {0,−5,−10,−15, . . .}. It is easy to show that I is a Q-ideal of Z− 0 where Q = {0,−1,−2,−3,−4} and Z− 0 /I(Q) = {0 + I,−1 + I,−2 + I,−3 + I,−4 + I} where 0 + I = {0,−5,−10,−15, . . .}, −1 + I = {−1,−6,−11,−16, . . .}, −2 + I = {−2,−7,−12,−17, . . .}, −3 + I = {−3,−8,−13,−18, . . .}, −4 + I = {−4,−9,−14,−19, . . .}. Later, let J = 2Z− 0 \ {−2} = {0,−4,−6,−8, . . .}. Clearly, J is an ideal. The quotient ternary semiring Z− 0 /J(Q) does not easily hold according to this concept. M. Petapirak, A. J. Khan, R. Chinram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6493 4 of 9 Definition 9 ([5]). Let R and T be two ternary semirings such that T has a zero 0T . An onto homomorphism φ : R → T is called maximal if, for each a ∈ T , there exists a unique qa ∈ φ−1({a}) such that x + ker(φ) ⊆ qa + ker(φ) for each x ∈ φ−1({a}) where ker(φ) = {x ∈ R | φ(x) = 0T }. Example 2. Consider the ternary semiring Z− 0 under the usual addition and ternary mul- tiplication of integers, and the ternary semiring Z3 under the usual addition and ternary multiplication of integers modulo 3. Let φ : Z− 0 → Z3 be defined by φ(n) = n for all n ∈ Z− 0 . Then φ is an onto homomorphism and ker(φ) = 3Z− 0 = {0,−3,−6,−9, . . .}. Let a ∈ Z3 and qa ∈ φ−1({a}). Assume that a = 0 and suppose that qa ̸= 0. So 0 /∈ qa + ker(φ). We have that 0 ∈ φ−1({a}). Then, by definition, we obtain that 0 ∈ 0 + ker(φ) ⊆ qa + ker(φ), a contradiction. We can conclude that if a = 0, then qa must be 0. Similarly, if a = 1, then qa must be −2, and if a = 2, then qa must be −1. Therefore, φ is maximal. In addition, taking Q = {0,−1,−2}, it follows that, Z− 0 /ker(φ)(Q) = {0 + ker(φ),−1 + ker(φ),−2 + ker(φ)} where 0 + ker(φ) = {0,−3,−6,−9, . . .}, −1 + ker(φ) = {−1,−4,−7,−10, . . .}, −2 + ker(φ) = {−2,−5,−8,−11, . . .}. Theorem 1 ([5]). Let R and T be two ternary semirings such that T has a zero 0T . If φ : R → T is a maximal homomorphism, then there exists a subset Q of R such that ker(φ) is a Q-ideal of R and R/ker(φ)(Q) ∼= T . 3.2. Quotient ternary semirings modulo congruences Definition 10. Let R be a ternary semiring. An equivalence relation ρ on R is called a congruence on R if, for all a1, a2, b1, b2, c1, c2 ∈ R, the following conditions hold: (1) If (a1, a2) ∈ ρ and (b1, b2) ∈ ρ, then (a1 + b1, a2 + b2) ∈ ρ, (2) If (a1, a2) ∈ ρ, (b1, b2) ∈ ρ and (c1, c2) ∈ ρ, then (a1b1c1, a2b2c2) ∈ ρ. We denote the congruence class of a ∈ R by a + ρ and let R/ρ = {a + ρ | a ∈ R}. Furthermore, define a binary addition ⊕ and ternary multiplication on R/ρ, for all a+ ρ, b+ ρ, c+ ρ ∈ R/ρ, by (a+ ρ)⊕ (b+ ρ) = (a+ b) + ρ and (a+ ρ)(b+ ρ)(c+ ρ) = abc+ ρ. Then R/ρ is a ternary semiring under this binary addition and ternary multiplication. Proposition 1. Let R and T be ternary semirings and φ : R→ T be a ternary semiring homomorphism. Define a relation K(φ) on R by K(φ) = {(a, b) ∈ R×R | φ(a) = φ(b)}. Then K(φ) is a congruence on R. M. Petapirak, A. J. Khan, R. Chinram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6493 5 of 9 Proof. Straightforward. Example 3. Let R = Z− be a ternary semiring under the usual addition and ternary multiplication of integers, and T = Z4 be a ternary semiring under the usual addition and ternary multiplication of integers modulo 4. Define a function φ : R→ T by φ(a) = a for all a ∈ R. Then φ is a homomorphism. By Proposition 1, K(φ) is a congruence on R. We have that R/K(φ) = {−1 +K(φ),−2 +K(φ),−3 +K(φ),−4 +K(φ)} where −1 +K(φ) = {−1,−5,−9,−13, . . .}, −2 +K(φ) = {−2,−6,−10,−14, . . .}, −3 +K(φ) = {−3,−7,−11,−15, . . .}, −4 +K(φ) = {−4,−8,−12,−16, . . .}. 3.3. Quotient ternary semirings modulo ideals Let I be an ideal of a ternary semiring R. We define a relation ρI on R as ρI = {(x, y) ∈ R×R | x+ a = y + b for some a, b ∈ I}. This means that (x, y) ∈ ρI if and only if there exist a1, a2 ∈ I satisfying x+ a1 = y + a2. Then ρI is a congruence relation on R, and we denote the congruence class of x by a coset x+ I. The collection of all congruence classes is denoted by R/I. Example 4. We consider a ternary semiring Z− 0 under the usual addition and ternary multiplication of integers. (1) Let I = 5Z− 0 = {0,−5,−10,−15, . . .}. It is easy to show that I is a k-ideal of Z− 0 . Then Z− 0 /I = {0 + I,−1 + I,−2 + I,−3 + I,−4 + I} where 0 + I = {0,−5,−10,−15, . . .}, −1 + I = {−1,−6,−11,−16, . . .}, −2 + I = {−2,−7,−12,−17, . . .}, −3 + I = {−3,−8,−13,−18, . . .}, −4 + I = {−4,−9,−14,−19, . . .}. This quotient ternary semiring obtained here is similar to Z− 0 /I(Q) considered in Example 1. Note that every k-ideal of a ternary semiring R is a Q-ideal of R. M. Petapirak, A. J. Khan, R. Chinram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6493 6 of 9 (2) Let J = 2Z− 0 ∖ {−2} = {0,−4,−6,−8, . . .}. It is clear that J is an ideal of Z− 0 but not a k-ideal. We have that Z− 0 /J = {0 + J,−1 + J} where 0 + J = {0,−2,−4,−6, . . .}, −1 + J = {−1,−3,−5,−7, . . .}. It is also worth noting that, as in Example 1, considering a quotient ternary semiring Z− 0 /J(Q) is not straightforward. 4. Main Results In this section, we will consider quotient ternary semirings via congruences and ideals from Subsections 3.2 and 3.3, respectively. Firstly, as shown in Example 4 (2), we see that −2+J = 0+J but −2 /∈ J . Moreover, J ⊆ 0 + J and 0 + J need not be equal to J . The following proposition shows some properties of cosets in a ternary semiring R/I. Proposition 2. Let R be a ternary semiring, I an ideal of R, and a, b ∈ R. Then the following statements hold: (1) If R has a zero and I is a k-ideal of R, then 0 + I = I. (2) If R has a zero and a ∈ I, then a+ I = 0 + I. (3) If I is a k-ideal of R and a ∈ I, then a+ I = b+ I if and only if b ∈ I. (4) If R has a zero and I is a k-ideal of R, then a+ I = 0 + I if and only if a ∈ I. Proof. (1) Assume 0 ∈ R and I is a k-ideal of R. Clearly, I ⊆ 0 + I. Let x ∈ 0 + I. Then (0, x) ∈ ρI . Thus there exist a, b ∈ I such that x + a = 0 + b. Given that I is a k-ideal, x+ a ∈ I and a ∈ I imply that x ∈ I. Hence 0 + I = I. (2) Assume 0 ∈ R and a ∈ I. Since 0 + a = a + 0 = a where a ∈ I, by the definition of a congruence relation ρI , we have (0, a) ∈ ρI . Hence 0 + I = a+ I. (3) Let I be a k-ideal of R and a ∈ I. Assume b ∈ R such that a + I = b + I. Then (a, b) ∈ ρI which implies a+ u = b+ v for some u, v ∈ I. Then b+ v = a+ u ∈ I because a, u ∈ I. Since I is a k-ideal, v ∈ I and b+ v ∈ I imply that b ∈ I. Conversely, we assume b ∈ I. Then a, b ∈ I and a+ b = b+ a give (a, b) ∈ ρI and consequently, a+ I = b+ I. (4) Assume that a+ I = 0+ I. Since 0 ∈ I, by (3), we have that a ∈ I. Conversely, it is clear by (1) and (2). Let R be any ternary semiring and I be an ideal of R. The k-closure of I is defined by Ck(I) = {r ∈ R | r + x ∈ I for some x ∈ I}. Then Ck(I) is the smallest k-ideal of R containing I, as shown in [11]. Theorem 2. Let I be an ideal of a ternary semiring R. Then R/I ∼= R/Ck(I). M. Petapirak, A. J. Khan, R. Chinram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6493 7 of 9 Proof. Define φ : R/I → R/Ck(I) by φ(a+ I) = a+ Ck(I) for all a ∈ R. First, we let a, b ∈ R such that a + I = b + I. So there exist x, y ∈ I such that a + x = b + y. Since x, y ∈ I ⊆ Ck(I), a+x = a+y yields a+Ck(I) = b+Ck(I). Then φ is well-defined. Clearly, φ is an onto homomorphism. Let a+ I, b+ I ∈ R/I such that φ(a+ I) = φ(b+ I). Thus a+Ck(I) = b+Ck(I). So there exist x, y ∈ Ck(I) such that a+x = b+y. Since x, y ∈ Ck(I), there exist s, r ∈ I such that x+ s ∈ I and y + r ∈ I. Thus a+ x+ r + s = b+ y + r + s and x + r + s, y + r + s ∈ I. Hence a + I = b + I. This shows that φ is one-to-one and R/I ∼= R/Ck(I). Let R and T be ternary semirings and φ : R → T be a homomorphism. Recall from Proposition 1 that K(φ) = {(a, b) ∈ R × R | φ(a) = φ(b)} is a congruence on R. Let im(φ) denote the sets of all images of φ, that is, im(φ) = {φ(x) | x ∈ R}. Proposition 3. Let R and T be ternary semirings and φ : R→ T be a ternary semiring homomorphism. Then R/K(φ) ∼= im(φ). Proof. Define ψ : R/K(φ) → im(φ) by ψ(a + K(φ)) = φ(a) for all a ∈ R. Let a +K(φ) = b +K(φ). Hence φ(a) = φ(b). Thus ψ is well-defined. Clearly, ψ is an onto homomorphism. Next, let a + K(φ), b + K(φ) ∈ R/K(φ) be such that ψ(a + K(φ)) = ψ(b+K(φ)). Then φ(a) = φ(b) which implies (a, b) ∈ K(φ). Hence a+K(φ) = b+K(φ). Proposition 4. Let R and T be ternary semirings with zeroes 0R and 0T , respectively, and φ : R → T be a homomorphism such that φ(0R) = 0T . Then ker(φ) = {x ∈ R | φ(x) = 0T } is a k-ideal of R. Proof. Let φ : R → T be a homomorphism where R and T are ternary semirings. Since 0R ∈ ker(φ), we see that ker(φ) ̸= ∅. Let a, b ∈ ker(φ). Therefore φ(a) = 0T and φ(b) = 0T . Since φ is a homomorphism, we have φ(a + b) = φ(a) + φ(b) = 0T . This implies that a+ b ∈ ker(φ). Next, we let r ∈ ker(φ) and a, b ∈ R. We have that φ(rab) = φ(r)φ(a)φ(b) = 0Tφ(a)φ(b) = 0T . Hence rab ∈ ker(φ). Similarly, arb, abr ∈ ker(φ). Then ker(φ) is an ideal of R. Furthermore, let a, b ∈ R be such that a + b ∈ ker(φ) and a ∈ ker(φ). Then 0T = φ(a + b) = φ(a) + φ(b) = 0T + φ(b) = φ(b), so b ∈ ker(φ). Therefore, ker(φ) is a k-ideal of R. Let R and T be ternary semirings with zeroes 0R and 0T , respectively, and φ : R→ T be a homomorphism such that φ(0R) = 0T . A homomorphism φ is called a k-kernel homomorphism if for all s1, s2 ∈ R, φ(s1) = φ(s2) implies s1 + a = s2 + b for some a, b ∈ ker(φ). Next, we provide an analogue of the first isomorphism theorem. Theorem 3. Let R and T be ternary semirings and φ : R→ T be a homomorphism with ker(φ). If φ is a k-kernel homomorphism, then R/ker(φ) ∼= im(φ). Proof. Assume that φ is a k-kernel homomorphism. Let ψ : R/ker(φ) → im(φ) be defined by ψ(s+ker(φ)) = φ(s) for all s ∈ R. Let s1, s2, s3 be any three elements in R. To prove that ψ is well-defined, assume that s1 + ker(φ) = s2 + ker(φ). Then s1 + a = s2 + b for some a, b ∈ ker(φ). So φ(a) = φ(b) = 0T and φ(s1) = φ(s1) + φ(a) = φ(s1 + a) = φ(s2 + b) = φ(s2) + φ(b) = φ(s2). M. Petapirak, A. J. Khan, R. Chinram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6493 8 of 9 In addition, as φ is a homomorphism, we have ψ((s1 + ker(φ)) + (s2 + ker(φ))) = ψ(s1 + s2 + ker(φ)) = φ(s1 + s2) = φ(s1) + φ(s2) = ψ(s1 + ker(φ)) + ψ(s2 + ker(φ)) and ψ((s1 + ker(φ))(s2 + ker(φ))(s3 + ker(φ))) = ψ(s1s2s3 + ker(φ)) = φ(s1s2s3) = φ(s1)φ(s2)φ(s3) = ψ(s1 + ker(φ))ψ(s2 + ker(φ))ψ(s3 + ker(φ)). Hence, ψ is a homomorphism. Clearly, ψ is onto. Next, we will show that ψ is one- to-one. Assume that s1, s2 ∈ R such that ψ(s1 + ker(φ)) = ψ(s2 + ker(φ)). Then φ(s1) = φ(s2). By assumption, we obtain s1+a = s2+b for some a, b ∈ ker(φ). Therefore, s1 + ker(φ) = s2 + ker(φ) and this leads to the result R/ker(φ) ∼= im(φ). Example 5. Let R = Z− be a ternary semiring under the usual addition and ternary multiplication of integers and T = Z4 be a ternary semiring under the usual addition and ternary multiplication of integers modulo 4. Define a function φ : R→ T by φ(a) = a for all a ∈ R. Then φ is a homomorphism. We have that ker(φ) = {−4,−8,−12,−16, . . .} is a k-ideal of R. We have that R/ker(φ) = {−1 + ker(φ),−2 + ker(φ),−3 + ker(φ),−4 + ker(φ)} where −1 + ker(φ) = {−1,−5,−9,−13, . . .}, −2 + ker(φ) = {−2,−6,−10,−14, . . .}, −3 + ker(φ) = {−3,−7,−11,−15, . . .}, −4 + ker(φ) = {−4,−8,−12,−16, . . .}. We end this section with a summary of the relationship between the isomorphisms in the following corollary. Corollary 1. Let R and T be ternary semirings with zeroes 0R and 0T , respectively, and φ : R→ T be a homomorphism such that φ(0R) = 0T . If φ is a k-kernel homomorphism, then R/ker(φ) ∼= R/K(φ). 5. Conclusion In this paper, we present multiple approaches for constructions of quotient ternary semirings, namely quotient ternary semirings modulo congruences and those modulo ideals. M. Petapirak, A. J. Khan, R. Chinram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6493 9 of 9 We investigate some relationships between ideals and quotient structures in ternary semir- ings. 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