EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5859 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Filters of Implicative Negatively Partially Ordered Ternary Semigroups Kansada Nakwan1, Panuwat Luangchaisri1, Thawhat Changphas1,∗ 1 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand Abstract. In this paper, we study a special set in an implicative n.p.o.(negatively partially or- dered) ternary semigroup, and prove that a filter can be represented by the union of such sets. Indeed, let (T, [ ],≤, [ ]∗) be an implicative n.p.o. ternary semigroup. For any a, b ∈ T , we define S(a, b) := {c ∈ T : [aa[bbc]∗]∗ = 1}. We have the following: (1) A non-empty subset F of T is a filter if and only if it satisfies the following conditions: (F3) 1 ∈ F ; (F4) for any a, b, c ∈ T , if [abc]∗ ∈ F and a, b ∈ F , then c ∈ F . (2) If T is commutative and F is a filter of T , then F = ⋃ a,b∈F S(a, b). 2020 Mathematics Subject Classifications: 20M12, 06F99, 06A06, 06A12 Key Words and Phrases: Implicative negatively partially ordered ternary semigroup (INPOTS), filter, left self-distributive 1. Introduction Implicative negatively partially ordered semigroups and filters were introduced and studied in [3] by Chan and Shum. The implicative negatively partially ordered semigroup is a generalization of the implicative semilattice (cf. [2], [8]), it is closed to implications in mathematical logic (cf. [1], [4]). As demonstrated in [8], filters play a crucial role in implicative semilattice theory. Quotient structures of implicative negatively partially ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5859 Email addresses: kansada.n@kkumail.com (K. Nakwan), panulu@kku.ac.th (P. Luangchaisri), thacha@kku.ac.th (T. Changphas) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) K. Nakwan, P. Luangchaisri, T. Changphas / Eur. J. Pure Appl. Math, 18 (2) (2025), 5859 2 of 8 ordered semigroups through filters were constructed in [3]. Additionally, in [6], filters within commutative implicative negatively partially ordered semigroups were examined. In [5], the author introduced a set in an implicative negatively partially ordered semigroup, and gave an equivalent condition of a filter. Moreover, it is obtained that a filter is the union of that special sets. In this paper, we follow these concepts to derive a special set in an implicative nega- tively partially ordered ternary semigroup, and prove that a filter can be represented by the union of such sets. Also, some important results are investigated. 2. Preliminaries We collects results obtained in negatively partially ordered ternary semigroups. Definition 1. [7] A system (T, [ ],≤) is called a NPOTS (negatively partially ordered ternary semigroup) if (1) (T, [ ]) is a ternary semigroup; (2) a partially order ≤ on T is compatible with [ ]; (3) ∀a, b, c ∈ T, [abc] ≤ a, [abc] ≤ b, [abc] ≤ c. Definition 2. [7] A NPOTS (T, [ ],≤) is called an INPOTS (implicative negatively par- tially ordered ternary semigroup) if there is an additional ternary multiplication [ ]∗ on T such that for all a, b, c, u ∈ T , u ≤ [cbc]∗ ⇔ [uab] ≤ c. Here, [ ]∗ is a ternary implication. A multiplicative identity of a ternary semigroup (T, [ ]) is an element 1 of T satisfying the condition [1a1] = [11a] = [a11] = a for any a ∈ T . Example 1. Consider the INPOTS (T, [ ],≤, [ ]∗) defined as follows: [ ] 1 a 0 11 1 0 0 1a 0 0 0 10 0 0 0 [ ] 1 a 0 aa 0 0 0 a1 0 0 0 a0 0 0 0 [ ] 1 a 0 00 0 0 0 01 0 0 0 0a 0 0 0 [ ]∗ 1 a 0 11 1 a a 1a 1 1 1 10 1 1 1 [ ]∗ 1 a 0 aa 1 1 1 a1 1 1 1 a0 1 1 1 [ ]∗ 1 a 0 00 1 1 1 01 1 1 1 0a 1 1 1 K. Nakwan, P. Luangchaisri, T. Changphas / Eur. J. Pure Appl. Math, 18 (2) (2025), 5859 3 of 8 and ≤= {(0, 0), (1, 1), (a, a), (a, 1), (0, a), (0, 1)}. We place x1x2 in the first column and x3 in the first row to express the calculation [x1x2x3] using a multiplication table. Observed that the greatest element 1 is not identity since [1a1] = 0 ̸= a. The following shows that not every NPOTS with identity admits the INPOTS. Example 2. Let us consider a NPOTS (T, [ ],≤) defined as follows: [ ] 1 2 3 4 6 11 1 2 3 4 6 12 2 2 6 4 6 13 3 3 3 6 6 14 4 4 6 4 6 16 6 6 6 6 6 [ ] 1 2 3 4 6 21 2 2 6 4 6 22 2 2 6 4 6 23 6 6 6 6 6 24 4 4 6 4 6 26 6 6 6 6 6 [ ] 1 2 3 4 6 31 3 6 3 6 6 32 6 6 6 6 6 33 3 6 3 6 6 34 6 6 6 6 6 36 6 6 6 6 6 [ ] 1 2 3 4 6 41 4 4 6 4 6 42 4 4 6 4 6 43 6 6 6 6 6 44 4 4 6 4 6 46 6 6 6 6 6 [ ] 1 2 3 4 6 61 6 6 6 6 6 62 6 6 6 6 6 63 6 6 6 6 6 64 6 6 6 6 6 66 6 6 6 6 6 and ≤ = {(1, 1), (2, 2), (3, 3), (4, 4), (6, 6), (3, 1), (2, 1), (4, a), (4, 1), (6, 4), (6, 2), (6, 3), (6, 1)}. The element 1 is the greatest element. Suppose T is an INPOTS with ternary implication [ ]∗. Clearly, [322] = 6 ≤ 4 and [422] = 2 ≤ 2. Then 3 ≤ [224]∗ and 4 ≤ [224]∗, so [224]∗ = 1. As 1 ≤ [224]∗, we have 2 = [122] ≤ 4. This is a contradiction. Hence, T is not an INPOTS. Definition 3. [9] An INPOTS (T, [ ],≤, [ ]∗) is called commutative if [a1a2a3] = [aα(1)aα(2)aα(3)] for any permutation α ∈ S3. K. Nakwan, P. Luangchaisri, T. Changphas / Eur. J. Pure Appl. Math, 18 (2) (2025), 5859 4 of 8 The example shows an infinite commutative INPOTS. Example 3. [7] Let Z+ be a TS such that [abc] = abc for all a, b, c ∈ Z+. Consider ≤= {(a, b) ∈ Z+ × Z+ : b | a}. Here, b | a means b divides a. Then (Z+, [ ],≤) is a commutative NPOTS; 1 is the greatest element. Define [abc]∗ = c gcd(ab, c) for all a, b, c ∈ Z+. Then (Z+, [ ],≤, [ ]∗) is a commutative INPOTS. Theorem 1. [7] Let (T, [ ],≤, [ ]∗) be an INPOTS. Then, for a, b ∈ T , (1) a ≤ [aaa]∗; (2) [aaa]∗ = [bbb]∗; (3) [aaa]∗ is the greatest element of T ; then an INPOTS always contains the greatest element. Let 1 be the greatest element of a NPOTS (T, [ ],≤) if exists. Assume 1 is the multi- plicative identity. Then, for any u, v, w ∈ T , [uvw] = 1 ⇔ u = 1, v = 1, w = 1. Throughout this paper, we assume 1 is both the multiplicative identity and the greatest element of an INPOTS. Theorem 2. [7] Let (T, [ ],≤, [ ]∗) be an INPOTS. Then, for a, b, c, u, v ∈ T , (1) a ≤ 1, [aaa]∗ = 1, a = [11a]∗; (2) a ≤ [bc[abc]]∗; (3) a ≤ [aa[aaa]]∗; (4) a ≤ [bca]∗; (5) if a ≤ b, then [buv]∗ ≤ [auv]∗ and [uva]∗ ≤ [uvb]∗; (6) a ≤ b ⇔ [a1b]∗ = 1 ⇔ [1ab]∗ = 1; (7) [ab[cuv]∗]∗ = [[abc]uv]∗ = [a[bcu]v]∗. 3. Filters of implicative negatively partially ordered ternary semigroups We begin with filters of an INPOTS. Definition 4. [7] Let (T, [ ],≤, [ ]∗) be an INPOTS. Then ∅ ≠ F ⊆ T is called a filter of T if K. Nakwan, P. Luangchaisri, T. Changphas / Eur. J. Pure Appl. Math, 18 (2) (2025), 5859 5 of 8 (F1) [abc] ∈ F for any a, b, c ∈ F ; (F2) for any x, y ∈ T , a ≤ b and a ∈ F imply b ∈ F . Proposition 1. Let (T, [ ],≤, [ ]∗) be an INPOTS. Then ∅ ≠ F ⊆ T is a filter if and only if it holds the conditions: (F3) 1 ∈ F ; (F4) for any a, b, c ∈ T , if [abc]∗ ∈ F and a, b ∈ F , then c ∈ F . Proof. Assume that F is a filter of T . Since 1 is the greatest element of T , 1 ∈ F . It is observed that for any a, b, c ∈ T , from [abc]∗ ≤ [abc]∗, we have [[abc]∗ab] ≤ c. (2.1) Let a, b, c ∈ T be such that [abc]∗ ∈ F and a, b ∈ F . By assumption, we have [[abc]∗ab] ∈ F . Using (2.1), we get [[abc]∗ab] ≤ c. This implies that c ∈ F . Hence, F satisfies (F3) and (F4). Conversely, assume that F satisfies (F3) and (F4). If a, b ∈ T such that a ≤ b and a ∈ F , then by Theorem 2 (6) we have [1ab]∗ = 1 ∈ F . By (F4), b ∈ F . Thus, F satisfies (F2). Let a, b, c ∈ F . By Theorem 2 (2), a ≤ [bc[abc]]∗, and so by (F2) we get [bc[abc]]∗ ∈ F . From (F4), [abc] ∈ F . Hence, F satisfies (F1). Consequently, F is a filter of T . Definition 5. Let (T, [ ],≤, [ ]∗) be an INPOTS. For any a, b ∈ T , define S(a, b) := {c ∈ T : [aa[bbc]∗]∗ = 1}. Observe that 1, b ∈ S(a, b) for any a, b ∈ T . Proposition 2. For a commutative INPOTS (T, [ ],≤, [ ]∗), a ∈ S(a, b) for all a, b ∈ T . Proof. Let a, b ∈ T . By Theorem 2 (7), [aa[bba]∗]∗ = [a[abb]a]∗ = [a[bba]a]∗ = [ab[baa]∗]∗ = [[abb]aa]∗ = [[bba]aa]∗ = [bb[aaa]∗]∗ = [bb1]∗. From Theorem 2 (4), 1 ≤ [bb1]∗ ≤ 1. This implies that [aa[bba]∗]∗ = 1, and so a ∈ S(a, b). K. Nakwan, P. Luangchaisri, T. Changphas / Eur. J. Pure Appl. Math, 18 (2) (2025), 5859 6 of 8 Proposition 3. Let (T, [ ],≤, [ ]∗) be an INPOTS, and b ∈ T . If [buv]∗ = 1 for all u, v ∈ T , then S(a, b) = T = S(b, a) for all a ∈ T . Proof. Assume that [buv]∗ = 1 for all u, v ∈ T and a ∈ T . Clearly, S(a, b) ⊆ T and S(b, a) ⊆ T . By assumption, we have [aa[bba]∗]∗ = [aa1]∗. Since 1 ≤ [aa1]∗ ≤ 1, we have [aa[bba]∗]∗ = 1, and then a ∈ S(a, b). Thus, T ⊆ S(a, b). By assumption, [bb[aaa]∗]∗ = [bb1]∗ = 1. This shows that a ∈ S(b, a), and so T ⊆ S(b, a). Example 4. Let us consider the INPOTS (T, [ ],≤, [ ]∗) defined as follows: [ ] 1 2 3 4 5 7 11 1 2 3 4 5 7 12 2 3 3 5 7 7 13 3 3 3 7 7 7 14 4 5 7 4 5 7 15 5 7 7 5 7 7 17 7 7 7 7 7 7 [ ] 1 2 3 4 5 7 21 2 3 3 5 7 7 22 3 3 3 7 7 7 23 3 3 3 7 7 7 24 5 7 7 5 7 7 25 7 7 7 7 7 7 27 7 7 7 7 7 7 [ ] 1 2 3 4 5 7 31 3 3 3 7 7 7 3a 3 3 3 7 7 7 33 3 3 3 7 7 7 34 7 7 7 7 7 7 35 7 7 7 7 7 7 37 7 7 7 7 7 7 [ ] 1 2 3 4 5 7 41 4 5 7 4 5 7 42 5 7 7 5 7 7 43 7 7 7 7 7 7 44 4 5 7 4 5 7 45 5 7 7 5 7 7 47 7 7 7 7 7 7 [ ] 1 2 3 4 5 7 51 5 7 7 5 7 7 52 7 7 7 7 7 7 53 7 7 7 7 7 7 54 5 7 7 5 7 7 55 7 7 7 7 7 7 57 7 7 7 7 7 7 [ ] 1 2 3 4 5 7 71 7 7 7 7 7 7 72 7 7 7 7 7 7 73 7 7 7 7 7 7 74 7 7 7 7 7 7 75 7 7 7 7 7 7 77 7 7 7 7 7 7 [ ]∗ 1 2 3 4 5 7 11 1 2 3 4 5 7 12 1 1 2 4 4 5 13 1 1 1 4 4 4 14 1 2 3 1 2 3 15 1 1 2 1 1 2 17 1 1 1 1 1 1 [ ]∗ 1 2 3 4 5 7 21 1 1 2 4 4 5 22 1 1 1 4 4 4 23 1 1 1 4 4 4 24 1 1 3 1 1 2 25 1 1 1 1 1 1 27 1 1 1 1 1 1 K. Nakwan, P. Luangchaisri, T. Changphas / Eur. J. Pure Appl. Math, 18 (2) (2025), 5859 7 of 8 [ ]∗ 1 2 3 4 5 7 31 1 1 1 4 4 4 3a 1 1 1 4 4 4 33 1 1 1 4 4 4 34 1 1 1 1 1 1 35 1 1 1 1 1 1 37 1 1 1 1 1 1 [ ]∗ 1 2 3 4 5 7 41 1 2 3 1 2 3 42 1 1 2 1 1 2 43 1 1 1 1 1 1 44 1 2 3 1 2 3 45 1 1 2 1 1 2 47 1 1 1 1 1 1 [ ]∗ 1 2 3 4 5 7 51 1 1 2 1 1 2 52 1 1 1 1 1 1 53 1 1 1 1 1 1 54 1 1 2 1 1 2 55 1 1 1 1 1 1 57 1 1 1 1 1 1 [ ]∗ 1 2 3 4 5 7 71 1 1 1 1 1 1 72 1 1 1 1 1 1 73 1 1 1 1 1 1 74 1 1 1 1 1 1 75 1 1 1 1 1 1 77 1 1 1 1 1 1 and ≤ = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (7, 7), (3, 1), (3, 2), (2, 1), (4, 1), (5, 4), (5, 1), (5, 2)(7, 1), (7, 2), (7, 3), (7, 4), (7, 5)}. By Proposition 3, we have S(a, 7) = S(7, a) = T for all a ∈ T . Furthermore, we have S(1, 1) = {1}, S(1, 2) = S(2, 1) = S(1, 3) = S(3, 1) = S(2, 3) = S(3, 2) = S(2, 2) = S(3, 3) = {1, 2, 3}, S(1, 4) = S(4, 1) = S(4, 4) = {1, 4}, and S(2, 4) = S(4, 2) = S(3, 4) = S(4, 3) = S(1, 5) = S(2, 5) = S(3, 5) = S(4, 5) = S(5, 5) = S(5, 1) = S(5, 2) = S(5, 3) = S(5, 4) = T . We observe that for any a, b ∈ T , S(a, b) is a filter of T . Theorem 3. Let (T, [ ],≤, [ ]∗) be a commutative INPOTS. If F is a filter, then S(a, b) ⊆ F for all a, b ∈ F . Proof. Let F be a filter of T and let a, b ∈ F . If c ∈ S(a, b), then [aa[bbc]∗]∗ = 1 ∈ F , and by (F4) we have c ∈ F . Theorem 4. Let (T, [ ],≤, [ ]∗) be a commutative INPOTS. If F is a filter of T , then F = ⋃ a,b∈F S(a, b). Proof. Let F be a filter of T . By Proposition 2, c ∈ S(c, 1) for any c ∈ F . Then F ⊆ ⋃ c∈F S(c, 1) ⊆ ⋃ a,b∈F S(a, b). For the reverse inclusion, let c′ ∈ ⋃ a,b∈F S(a, b). Then there exist x, y ∈ F such that c′ ∈ S(x, y). By Theorem 3, c′ ∈ F . This shows that ⋃ a,b∈F S(a, b) ⊆ F . K. Nakwan, P. Luangchaisri, T. Changphas / Eur. J. Pure Appl. Math, 18 (2) (2025), 5859 8 of 8 Corollary 1. Let (T, [ ],≤, [ ]∗) be a commutative INPOTS. If F is a filter of T , then F = ⋃ a∈F S(a, 1). 4. Conclusions In this paper, we introduce the concept of filters in implicative negatively partially ordered ternary semigroups (Definition 4) and give a characterization of filters (Proposition 1). Then we consider the set S(a, b) := {c ∈ T : [aa[bbc]∗]∗ = 1} where a, b are elements of an implicative negatively partially ordered ternary semigroup (T, [ ],≤, [ ]∗). The main result obtained is that any filter can be represented by the union of such sets (Theorem 4), if (T, [ ],≤, [ ]∗) is commutative. Acknowledgements The Research on ”On filters of implicative negatively partially ordered ternary semi- groups” by Khon Kaen University has received funding support from the National Science, Research and Innovation Fund (NSRF). References [1] G. Birkhoff. Lattice theory. Amer. Math. Soc. Coll. Publ. Vol. XXV, Providence, 1967. [2] T. S. Blyth. Pseudo-residuals in semigroups. J. London Math. Soc., 1(1):441–454, 1965. [3] M. W. Chan and K. P. Shum. Homomorphisms of implicative semigroups. 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