In this article, we study some interesting properties of pseudo symmetric ideals and prime pseudo symmetric ideals in partially ordered ternary semigroup

Dattatray Shinde

Machchhindra Gophane

English

International Journal of Innovations in Engineering Research and Technology

A NOTE ON PSEUDO SYMMETRIC IDEALS OF PARTIALLY ORDERED TERNARY SEMIGROUPS1. Dattatray ShindeResearch Scholar, Department of Mathematics Shivaji University, Kolhapur- 416004, Maharashtra, India * shindednmaths@gmail.com2. Machchhindra  GophaneAssociate Professor, Department of Mathematics Shivaji University, Kolhapur- 416004, Maharashtra, India * mtg_maths@unishivaji.ac.inAbstract  In this article, we study some interesting properties of pseudo symmetric ideals and prime pseudo symmetric ideals in partially ordered ternary semigroup. Keywords: Partially ordered ternary semigroup, pseudo symmetric ideal, prime pseudo symmetric ideals.   2020 Mathematics Subject Classification: 06F99, 20M12, 20N99. Introduction  Lehmer D. H.  [3] studied the triplexes algebraic systems for commutative ternary groups. The ideal theory of 𝑛-ary semigroups and ternary semigroup was introduced by Sioson F. M.  [5] in 1965. Dixit V. N. and Dewan S.  [7] studied the properties of quasi-idealsand bi-ideals in ternary semigroups. Iampan A. [1, 2] has developed the theory of orderedternary semigroups along the line of the theory of ordered semigroups and ternarysemigroups. Daddi V. R. and Pawar Y. S.  [8] defined the notion of an ordered quasi-idealand an ordered bi-ideal in an ordered ternary semigroup.   In 2014, Siva Rami Reddy V.et al.  [9, 10] have established the ideal theory of a partially ordered ternary semigroup.They also defined and studied the notions of complete prime ideals, prime ideals,complete semiprime ideals and semiprime ideals of partially ordered ternary semigroups.Jyothi V. et al. in [6] introduced the notion of semipseudo symmetric ideals and pseudosymmetric ideals of partially ordered ternary semigroups. The notions of prime,semiprime pseudo symmetric ideals and irreducible pseudo symmetric ideals of partiallyordered ternary semigroups is defined by Shinde D. N. and Gophane M. T in [4].Preliminaries A non-empty set T with a ternary operation [  ] ∶  𝑇 ×   𝑇  ×   𝑇  → 𝑇 is called a ternary semigroup [3] if [ ] satisfies the associative law, [ 𝑝 𝑞 𝑟 𝑠 𝑡  ] = [ [ 𝑝 𝑞 𝑟  ]  𝑠 𝑡  ] =[ 𝑝  [ 𝑞  𝑟  𝑠  ]  𝑡  ] = [ 𝑝  𝑞  [ 𝑟  𝑠  𝑡  ]  ], for all 𝑝, 𝑞, 𝑟, 𝑠, 𝑡 ∈  𝑇. For non-empty subsets 𝑋, 𝑌 and  𝑍 of a ternary semigroup T, [𝑋𝑌𝑍] = {[𝑥𝑦𝑧]: 𝑥 ∈ 𝑋,𝑦 ∈ 𝑌 and 𝑧 ∈ 𝑍}. For easiness, we write, [𝑋𝑌𝑍]  as  𝑋𝑌𝑍, [𝑥𝑦𝑧] = 𝑥𝑦𝑧 and [𝑋𝑋𝑋] =𝑋3.A ternary semigroup T is said to be a partially ordered ternary semigroup [1] if there exist a partially ordered relation  ≤  on T such that, 𝑎 ≤  𝑏 ⇒  𝑥𝑦𝑎 ≤  𝑥𝑦𝑏, 𝑥𝑎𝑦 ≤  𝑥𝑏𝑦, 𝑎𝑥𝑦 ≤Proceedings of “National Conference on Emerging Trends in Science and Advances inEngineering”Organized by Phaltan Education Society’s College of Engineering Phaltan,www.coephaltan.edu.inInternational Journal of Innovations in Engineering Research and Technology [IJIERT]ISSN: 2394-3696, Website: www.ijiert.org, July, 2023217 𝑏𝑥𝑦  for all 𝑎, 𝑏, 𝑥, 𝑦 ∈  𝑇. In this article, we write T for a partially ordered ternary semigroup, unless otherwise specified. A partially ordered ternary semigroup 𝑇 is said to be commutative [2] if 𝑥𝑦𝑧 = 𝑧𝑥𝑦 =𝑦𝑧𝑥 = 𝑦𝑥𝑧 = 𝑧𝑦𝑥 = 𝑥𝑧𝑦, for all   𝑥, 𝑦, 𝑧 ∈  𝑇. Let 𝑋 be a non-empty subset of T. We denote, {𝑡 ∈  𝑇 ∶   𝑡 ≤  𝑥, for some  𝑥 ∈  𝑋 } by (𝑋]. A non-empty subset 𝐼 of T is said to be a left (respectively, right, lateral) ideal [8] of T if  𝑇𝑇𝐼 ⊆  𝐼 (respectively, 𝐼𝑇𝑇 ⊆  𝐼, 𝑇𝐼𝑇 ⊆  𝐼 ) and (𝐼]  =  𝐼. A non-empty subset 𝐼 of T is said to be ideal [8] of T if it is a left ideal, a right ideal and a lateral ideal of T.  An ideal 𝐼 of T is said to be a pseudo symmetric ideal [6] if  𝑥, 𝑦, 𝑧 ∈ 𝑇, 𝑥𝑦𝑧 ∈ 𝐼 implies 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼 ∀ 𝑠, 𝑡 ∈ 𝑇. A pseudo symmetric ideal I of T is said to be proper pseudo symmetric ideal of T if it different from 𝑇. A proper pseudo symmetric ideal 𝐼 of T is said to be a prime pseudo symmetric ideal [4] of T if  𝐼1 𝐼2 𝐼3 ⊆  𝐼 ⇒  𝐼1  ⊆  𝐼 or 𝐼2 ⊆  𝐼 or 𝐼3 ⊆  𝐼 where 𝐼1, 𝐼2, 𝐼3 are pseudo symmetric ideals of T.  Result: [8] Let 𝐴, 𝐵 and 𝐶 be non-empty subset of 𝑇 then the following statements hold, (1) 𝐴 ⊆  (𝐴].  (2) ((𝐴]] =  (𝐴]. (3) (𝐴](𝐵](𝐶] ⊆  (𝐴𝐵𝐶]. (4) If  𝐴 ⊆  𝐵 then (𝐴] ⊆   (𝐵].  Main Results:  Pseudo Symmetric Ideals  Theorem 1: If 𝐼1 and 𝐼2 are two pseudo symmetric ideal of 𝑇 then 𝐼1 ∩ 𝐼2 is also pseudo symmetric ideal of 𝑇, provided  𝐼1 ∩ 𝐼2 ≠ ∅. Proof: Since 𝐼1  and 𝐼2 are ideal of T.  So, 𝐼1  ∩ 𝐼2 is ideal of T. Let 𝑥𝑦𝑧 ∈ 𝐼1 ∩ 𝐼2 ∀ 𝑥, 𝑦, 𝑧 ∈ 𝑇 ⇒ 𝑥𝑦𝑧 ∈ 𝐼1 & 𝑥𝑦𝑧 ∈ 𝐼2. If 𝑥𝑦𝑧 ∈ 𝐼1 and 𝐼1 is pseudo symmetric ideal of 𝑇 then 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼1 ∀ 𝑠, 𝑡 ∈ 𝑇. If 𝑥𝑦𝑧 ∈ 𝐼2 and 𝐼2 is pseudo symmetric    ideal of 𝑇 then 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼2 ∀ 𝑠, 𝑡 ∈ 𝑇. So, 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼1 ∩ 𝐼2. Hence 𝐼1 ∩ 𝐼2 is a pseudo symmetric ideal of 𝑇.    Theorem 2: Arbitrary intersection of pseudo symmetric ideal of 𝑇 is a pseudo symmetric ideal of 𝑇, provided it is non-empty.  Proof: Let {𝐼𝑖}𝑖∈∆ be a family of pseudo symmetric ideals of 𝑇. Let 𝐼 =  ⋂ 𝐼𝑖𝑖∈∆ ≠ ∅, be the intersection of this family of T. Here 𝐼 is an ideal. Let 𝑥𝑦𝑧 ∈ 𝐼 ∀ 𝑥, 𝑦, 𝑧 ∈ 𝑇 ⇒ 𝑥𝑦𝑧 ∈𝐼𝑖, 𝑖 ∈ ∆ and 𝐼𝑖 is pseudo symmetric ideal of T then 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼𝑖, 𝑖 ∈ ∆  ∀ 𝑠, 𝑡 ∈ 𝑇 ⇒𝑥𝑠𝑦𝑡𝑧 ∈ ⋂ 𝐼𝑖𝑖∈∆ , ∀ 𝑠, 𝑡 ∈ 𝑇 ⇒ 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼  ∀ 𝑠, 𝑡 ∈ 𝑇. Hence 𝐼 is a pseudo symmetric ideal of T.    Theorem 3: If 𝐼1 and 𝐼2 are two pseudo symmetric ideal of partially ordered ternary semigroup 𝑇 then 𝐼1 ∪ 𝐼2 is also pseudo symmetric ideal of 𝑇. 218Proof:  Since, 𝐼1 and 𝐼2 are ideal of  T.  So, 𝐼1 ∪ 𝐼2 is  ideal of T. Let 𝑥𝑦𝑧 ∈ 𝐼1 ∪  𝐼2 ∀ 𝑥, 𝑦, 𝑧 ∈ 𝑇 ⇒ either 𝑥𝑦𝑧 ∈ 𝐼1 or 𝑥𝑦𝑧 ∈ 𝐼2. If 𝑥𝑦𝑧 ∈ 𝐼1 and 𝐼1 is pseudo symmetric ideal of 𝑇 then 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼1 ∀ 𝑠, 𝑡 ∈ 𝑇 ⇒ 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼1 ∪ 𝐼2. If 𝑥𝑦𝑧 ∈ 𝐼2 and 𝐼2 is pseudo symmetric ideal of 𝑇 then ∈ 𝐼2 ∀ 𝑠, 𝑡 ∈ 𝑇 ⇒    𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼1 ∪ 𝐼2 . Hence 𝐼1 ∪ 𝐼2 is a pseudo symmetric ideal of 𝑇.    Theorem 4: Arbitrary union of pseudo symmetric ideal of 𝑇 is a pseudo symmetric ideal of 𝑇. Proof: Let {𝐼𝑖}𝑖∈∆ be a family of pseudo symmetric ideals of T. Let 𝐼 = ⋃ 𝐼𝑖𝑖∈∆  be the union of this family of pseudo symmetric ideal of T. Hence 𝐼 is an ideal of 𝑇. Now, let 𝑥𝑦𝑧 ∈ 𝐼 ∀ 𝑥, 𝑦, 𝑧 ∈ 𝑇 ⇒ 𝑥𝑦𝑧 ∈ 𝐼𝑖 for some 𝑖 ∈ ∆ and 𝐼𝑖 is pseudo symmetric ideal of 𝑇 then 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼𝑖 for some 𝑖 ∈ ∆  ∀ 𝑠, 𝑡 ∈ 𝑇 ⇒ 𝑥𝑠𝑦𝑡𝑧 ∈ ⋃ 𝐼𝑖𝑖∈∆   ∀ 𝑠, 𝑡 ∈ 𝑇 𝑥𝑠𝑦𝑡𝑧 ∈𝐼  ∀ 𝑠, 𝑡 ∈ 𝑇. Hence I is a pseudo symmetric ideal of T.     Theorem 5: The collection of all pseudo symmetric ideals of partially ordered ternary semigroup T forms a poset with respect to the partial ordering relation ⊆.  Proof: Let ℐ = {𝐼𝑖 / 𝑖 ∈ ∆, ∆ is any indexing set} be a family of all pseudo symmetric ideals of partially ordered ternary semigroup T. (1) Reflexive- For any 𝐼𝑖 ∈ ℐ, 𝑖 ∈ ∆,  𝐼𝑖 ⊆ 𝐼𝑖 ⟹ ⊆ is reflexive. (2) Antisymmetric- For any 𝐼𝑖 , 𝐼𝑗 ∈ ℐ and 𝑖, 𝑗 ∈ ∆. If 𝐼𝑖 ⊆ 𝐼𝑗 , 𝐼𝑗 ⊆ 𝐼𝑖 then 𝐼𝑖 = 𝐼𝑗 ⟹ ⊆is antisymmetric. (3) Transitive- For any 𝐼𝑖, 𝐼𝑗 , 𝐼𝑘 ∈ ℐ and 𝑖, 𝑗, 𝑘 ∈ ∆. If 𝐼𝑖 ⊆ 𝐼𝑗 , 𝐼𝑗 ⊆ 𝐼𝑘then 𝐼𝑖 ⊆ 𝐼𝑘 ⟹ ⊆is transitive. Therefore ℐ forms a poset.  Theorem 6: Let 𝐼 be an ideal of commutative partially ordered ternary semigroup 𝑇, then 𝐼  is a pseudo symmetric ideal. Proof:  Let 𝑇 be the commutative partially ordered ternary semigroup and 𝐼 be any ideal of T. Let 𝑥, 𝑦, 𝑧 ∈ 𝑇, 𝑥𝑦𝑧 ∈ 𝐼 and 𝑠, 𝑡 ∈ 𝑇 then 𝑥𝑠𝑦𝑡𝑧 =   𝑥𝑦𝑠𝑡𝑧 = 𝑥𝑦𝑠𝑧𝑡 = 𝑥𝑦𝑧𝑠𝑡 =(𝑥𝑦𝑧)𝑠𝑡 ∈ 𝐼𝑇𝑇 ⊆ 𝐼 ⟹ 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼 . Hence 𝐼 is a pseudo symmetric ideal of T. Corollary 7: If 𝐼 is an ideal of commutative partially ordered ternary semigroup 𝑇 then ( 𝐼 ]  is a pseudo symmetric ideal of 𝑇. Note: In a commutative partially ordered ternary semigroup, a left, a right and a lateral ideal coincide. Theorem 8: The intersection of a left, a right and a lateral ideal of commutative partially ordered ternary semigroup 𝑇 is a pseudo symmetric ideal of 𝑇.  Proof: According to above note, a left, a right and a lateral ideal are coinciding and every ideal of commutative partially ordered ternary semigroup is pseudo symmetric ideal.   Theorem 8: If 𝐼 is an ideal of 𝑇 and 𝑃 is a pseudo symmetric ideal of 𝑇, then 𝐼 ∩ 𝑃 is a pseudo symmetric ideal of 𝐼, providing 𝐼 as a partially ordered ternary semigroup. 219Proof: Since, 𝐼 is an ideal of 𝑇 and 𝑃 is a pseudo symmetric ideal of T then  𝐼 ∩ 𝑃 is an ideal of 𝐼. Let 𝑥, 𝑦, 𝑧 ∈ 𝐼, 𝑥𝑦𝑧 ∈ 𝐼 ∩ 𝑃 ⟹ 𝑥𝑦𝑧 ∈ 𝐼 and 𝑥𝑦𝑧 ∈ 𝑃. If 𝑥𝑦𝑧 ∈ 𝐼 and I is an ideal of T then, for all 𝑠, 𝑡 ∈ 𝐼, consider 𝑥𝑠𝑦𝑡𝑧 = (𝑥𝑠𝑦)𝑡𝑧 ∈ 𝐼𝐼𝐼 ⊆ 𝐼 (since 𝐼 is an ideal of 𝑇 ) ⟹ 𝑥𝑠𝑦𝑡𝑧 ∈ 𝐼. If 𝑥𝑦𝑧 ∈ 𝑃 and P is a pseudo symmetric ideal of T then, for all 𝑠, 𝑡 ∈ 𝐼 ⊆ 𝑇 ⟹ 𝑥𝑠𝑦𝑡𝑧 ∈ 𝑃 ∀  𝑠, 𝑡 ∈ 𝑇 . Therefore 𝑥𝑠𝑦𝑡𝑧 ∈  𝐼 ∩ 𝑃. Hence  𝐼 ∩ 𝑃 is a pseudo symmetric ideal of 𝐼.  Theorem: Every pseudo symmetric ideal of T is a bi- ideal of T.  Proof: Let 𝐼 be a pseudo symmetric ideal of 𝑇. Consider  𝐼𝑇𝐼𝑇𝐼 = 𝐼(𝑇𝐼𝑇)𝐼 ⊆ 𝐼𝐼𝐼 ⊆𝑇𝑇𝐼 ⊆ 𝐼 (since 𝐼 is a pseudo symmetric ideal of 𝑇) and (𝐼] ⊆ 𝐼. Hence, 𝐼 is a bi- ideal of 𝑇.                                                                                                                                  Theorem: Every pseudo symmetric ideal of T is a quasi- ideal of T. Proof: Let 𝐼 be a pseudo symmetric ideal of 𝑇. Consider (𝑇𝑇𝐼] ∩ (𝑇𝐼𝑇 ∪ 𝑇𝑇𝐼𝑇𝑇] ∩(𝐼𝑇𝑇] ⊆ (𝐼] ∩ (𝐼] ∩ (𝐼] = (𝐼] ⊆ 𝐼 . Therefore (𝑇𝑇𝐼] ∩ (𝑇𝐼𝑇 ∪ 𝑇𝑇𝐼𝑇𝑇] ∩ (𝐼𝑇𝑇] ⊆ 𝐼 and (𝐼] ⊆ 𝐼 (since 𝐼 is a pseudo symmetric ideal of 𝑇).  Hence 𝐼 is quasi- ideal of T.                                                                 Prime Pseudo Symmetric Ideals  Theorem: A proper pseudo symmetric ideal 𝐼 of 𝑇 is prime if and only if 𝐼1, 𝐼2, 𝐼3, … … , 𝐼𝑛 are pseudo symmetric ideals of 𝑇, where 𝑛 is an odd natural number, 𝐼1 𝐼2 𝐼3 … … 𝐼𝑛 ⊆ 𝐼 implies 𝐼𝑖 ⊆ 𝐼 for some 𝑖 = 1, 2, 3, … … , 𝑛.  Proof: Suppose 𝐼 is a prime pseudo symmetric ideal of 𝑇. Let 𝐼1, 𝐼2, 𝐼3, … … , 𝐼𝑛 be pseudo symmetric ideals of 𝑇 such that 𝐼1𝐼2𝐼3 … … 𝐼𝑛 ⊆ 𝐼, where 𝑛 is an odd natural number.  If 𝑛 = 1 then 𝐼1 ⊆ 𝐼. If 𝑛 = 3 then 𝐼1𝐼2𝐼3 ⊆ 𝐼 ⇒ 𝐼1 ⊆ 𝐼 or 𝐼2 ⊆ 𝐼 or 𝐼3 ⊆ 𝐼 (since 𝐼 is a prime pseudo symmetric ideal of 𝑇). Hence 𝐼𝑖 ⊆ 𝐼 for some 𝑖 = 1, 2, 3. If 𝑛 = 5 then 𝐼1𝐼2𝐼3𝐼4𝐼5 ⊆ 𝐼 ⇒ 𝐼1𝐼2𝐼3 ⊆ 𝐼 or 𝐼4 ⊆ 𝐼 or 𝐼5 ⊆ 𝐼 ⇒ 𝐼1 ⊆ 𝐼 or 𝐼2 ⊆ 𝐼 or 𝐼3 ⊆ 𝐼 or 𝐼4 ⊆ 𝐼 or 𝐼5 ⊆ 𝐼   (since 𝐼 is a prime pseudo symmetric ideal of 𝑇). Hence 𝐼𝑖 ⊆ 𝐼 for some 𝑖 = 1, 2, 3, 4, 5. Therefore, by induction on 𝑛, 𝐼1𝐼2𝐼3 … … 𝐼𝑛 ⊆ 𝐼 ⇒ 𝐼𝑖 ⊆ 𝐼 for some 𝑖 = 1, 2, 3, … … , 𝑛. Conversely, let 𝐼1, 𝐼2, 𝐼3, … … , 𝐼𝑛 be pseudo symmetric ideals of 𝑇, where 𝑛 is an odd natural number such that 𝐼1𝐼2𝐼3 … … 𝐼𝑛 ⊆ 𝐼 ⇒ 𝐼𝑖 ⊆ 𝐼 for some 𝑖 = 1, 2, 3, … … , 𝑛. By the definition of prime pseudo symmetric ideal, 𝐼 is a prime pseudo symmetric ideal of 𝑇.  Definition: [4] A proper pseudo symmetric ideal 𝐼 of 𝑇 is said to be a maximal pseudo symmetric ideal of 𝑇 if 𝐼 is not properly contained in any proper pseudo symmetric ideal of 𝑇. Theorem: If  𝑇 is a partially ordered ternary semigroup such that 𝑇3 = 𝑇 then every maximal pseudo symmetric ideal of 𝑇 is a prime pseudo symmetric ideal of 𝑇.                                                                                                Proof: Let 𝐼 be a maximal pseudo symmetric ideal of 𝑇. Let 𝐼1, 𝐼2 and 𝐼3 be pseudo symmetric ideals of 𝑇 such that 𝐼1 𝐼2 𝐼3 ⊆  𝐼. Suppose that 𝐼1  ⊈  𝐼,  𝐼2 ⊈  𝐼, 𝐼3 ⊈  𝐼. If  220𝐼1  ⊈  𝐼 ⇒ 𝐼1 ⋃ 𝐼 is a pseudo symmetric ideal of 𝑇 and 𝐼 ⊂  𝐼1 ⋃ 𝐼 ⊆ 𝑇.  Since 𝐼 is a maximal pseudo symmetric ideal of 𝑇, 𝐼1 ⋃ 𝐼 = 𝑇. Similarly, we can prove that 𝐼2 ⋃ 𝐼 =𝑇 and 𝐼3 ⋃ 𝐼 = 𝑇. Now, 𝑇 = 𝑇3 = 𝑇𝑇𝑇 = (𝐼1 ⋃ 𝐼)(𝐼2 ⋃ 𝐼)(𝐼3 ⋃ 𝐼) ⊆ 𝐼 ⇒ 𝑇 ⊆ 𝐼. Thus 𝐼 = 𝑇. Which is contradiction. Therefore either  𝐼1 ⊆ 𝐼 or 𝐼2 ⊆ 𝐼 or 𝐼3 ⊆ 𝐼 . Hence 𝐼 is a prime pseudo symmetric ideal of 𝑇.  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A NOTE ON PSEUDO SYMMETRIC IDEALS OF PARTIALLY ORDERED TERNARY SEMIGROUPS

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A NOTE ON PSEUDO SYMMETRIC IDEALS OF PARTIALLY ORDERED TERNARY SEMIGROUPS

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