Microsoft Word - 120-127   120  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020       On SAH – Ideal of BH – Algebra Alaa Saleh Abed Alaas.abed@uokufa.edu.iq Article history: Received 23 June 2019, Accepted 8 September 2019, Published in April 2020. Abstract The aim of this investigation is to present the idea of SAH – ideal, closed SAH – ideal and closed SAH – ideal with respect to an element, 𝑆𝐴𝐻 – ideal and s- 𝑆𝐴𝐻 – ideal of BH – algebra. We detail and show theorems which regulate the relationship between these ideas and provide some examples in BH – algebra. Keywords: BH – algebra, SAH – ideal of BH – algebra, closed SAH – ideal with respect to an element of BH – algebra, 𝑆𝐴𝐻 – ideal . 1. Introduction After founding of fuzzy subset by Zadeh L. A [1]. Several researchers presented the generalizations of the idea of fuzzy subsets. Imai and Iseki K. established two classes BCK algebra and BCI – algebra [2, 3]. Jun Y. B., Rogh E. H. And Kin H. S. produced a new concept, named a BH – algebra [4]. In this paper, we will recall some basic definitions. A BH – algebra is a nonempty set Ψ with a binary operation * satisfies the conditions: ж ∗ ж 0 , for all ж ∈ Ψ , ж ∗ ц 0 and ц ∗ ж 0→ ж ц for all ж , ц ∈ Ψ and ж ∗ 0 ж , for all ж ∈ Ψ 4 . we will use Ψ for representing a BH – algebra Ψ ; ∗ ,0 . Let 𝔖 a nonempty subset of Ψ . then 𝔖 is named an ideal of Ψ if it holds: 0 ∈ 𝔖 ; ж ∗ ц ∈ 𝔖 and ц ∈ 𝔖 → ж ∈ 𝔖 4 . Let Ψ and Φ be BH – algebras. A mapping δ : Ψ → Φ is named ahomomorphism if: δ (ж ∗ ц ) δ (ж ∗ δ ц , ∀ ж,ц ∈ Ψ . A homomorphism δ is titled a monomerphism (resp, epimorphism) if it injective (resp., surjective). A bijective homomorphism is titled an isomorphism. Two BH – algebras Ψ and Φ are said to be isomorphic, written Ψ ≅ Φ , if there exists an isomorphism δ : Ψ → Φ . For any homomorphism : Ψ → Φ , the set ж ∈ Ψ : δ (ж) = 0' is titled the kernel of δ , symbolized by ker δ , and the set δ ж : ж ∈ Ψ is named the image of δ, represented by Im δ . Sign that δ (0) = 0' , ∀ homomorphism δ [5]. An ideal 𝔖 of Ψ is known as closed ideal of Ψ if: for each ж ∈ 𝔖 . Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.2.2433 Department of Mathematics, Faculty of Education for Girls, University of kufa, Najaf , Iraq   121  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 We requisite 0 ∗ ж ∈ 𝔖 [6]. Let 𝔖 be an ideal of Ψ . It is named a closed ideal with respect to an element s ∈ Ψ (symbolized by s closed ideal) if s ∗ 0 ∗ ж ∈ 𝔖, ∀ ж ∈ 𝔖 [7]. An ideal 𝔖 of Ψ is known as completely closed ideal if ж ∗ ц ∈ 𝔖 , ∀, ц ∈ 𝔖 7 . Let 𝔖 be an ideal of Ψ and s ∈ 𝔖 . It is named a completely closed with respect to an element s (know by s completely closed ideal) if: s ∗ ж ∗ ц ∈ 𝔖 , ∀ ж, ц ∈ 𝔖 [7]. In the next parts of our research, we will symbolize to BH- algebra € ; ∗ ,0 by € . 2. Closed SAH – Ideal with Respect to an Element of BH – Algebra Definition (1) An ideal 𝔜 of € is named a SAH – ideal of € if it fillfulls the requirement: ∀ς , ζ ∈ 𝔜 , if ς∗ ∗ ζ ∈ 𝔜 , ζ∗ ∈ 𝔜 → ζ∗ ∗ ς ∈ 𝔜 , where ς∗ e ∗ ς , and e is unit number, i.e: ς ∗ e = 0 Example (2) Assume € 0, 𝔴, 𝔳 with the binary operation ∗ symbolized by the subsequent table: Table 1. * 0 𝔴 𝔳 0 0 0 0 𝔴 𝔴 0 0 𝔳 𝔳 𝔳 0 Then the ideal 𝔜 0, 𝔳 is a SAH – ideal of €. Definition (3) Assume 𝔜 is SAH – ideal of € , then 𝔜 is known as closed SAH – ideal if it fulfills the requirement: ∀ ς , ζ ∈ 𝔜 if 0 ∗ ς∗ ∗ ζ ∈ 𝔜 ∧ 0 ∗ ζ∗ ∈ 𝔜 → 0 ∗ ζ∗ ∗ ς ∈ 𝔜 Example (4) Assume € 0,1,2,3 with the binary operation ∗ definition by the ensuing table: Table 2 * 0 1 2 3 0 0 1 0 0 1 1 0 1 0 2 2 2 0 0 3 3 3 3 0 Then, the ideal 𝔜 0,3 is a closed SAH – ideal of €.   122  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Remark (5) We know that every SAH – ideal in € is closed SAH – ideal. But the converse not correct. Example (6) Consider € 0,1,2,3 with a binary operation ∗ connoted by the ensuing table: Table 3. * 0 1 2 3 0 0 0 0 0 1 1 0 0 0 2 2 2 0 0 3 3 2 2 0 𝔜 0,3 is a closed SAH – ideal of € but 𝔜 doesn't SAH – ideal, because: when ς 2 , ζ 1 → ς∗ 2 , ζ∗ 2 0 ∗ 2 0 ∈ 𝔜 , 0 ∗ 2 0 ∈ 𝔜 → 0 ∗ 0 0 ∈ 𝔜 , while 2 ∗ 1 2 ∉ 𝔜 , 2 ∉ 𝔜 → 2 ∗ 2 0 ∈ 𝔜 Theorem (7) Assume 𝔜 , λ ∈ Λ is a collocation of closed SAH – ideal of €. Then   𝔜 is a closed SAH – ideal of € . Proof ∀ ς , ζ ∈   𝔜 ∴ ς , ζ ∈ 𝔜 , ∀ λ ∈ Λ ⟹ 0 ∗ ς∗ ∗ ζ ∈ 𝔜 and 0 ∗ ζ∗ ∈ 𝔜 then 0 ∗ ζ∗ ∗ ς ∈ 𝔜 , ∀λ ∈ Λ Since each 𝔜 is closed SAH – ideal ∀λ ∈ Λ ⇒ 0 ∗ ς∗ ∗ ζ ∈   𝔜 and 0 ∗ ζ∗ ∈   𝔜 then 0 ∗ ζ∗ ∗ ς ∈   𝔜 ∴   𝔜 is closed SAH – ideal of BH – algebra € . ∎   123  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Theorem (8) Assume 𝔜 , λ ∈ Λ is a collocation of closed SAH – ideals of €. Then   𝔜 is a closed SAH – ideal of € . Proof To prove that   𝔜 is closed SAH – ideal ∀ ς , ζ ∈   𝔜 ⇒ ∃ 𝔜 ∈ 𝔜 ∈ is a c SAH – ideal Such that ∀ ς , ζ ∈ 𝔜 ⟹ 0 ∗ ς∗ ∗ ζ ∈ 𝔜 and 0 ∗ ζ∗ ∈ 𝔜 so 0 ∗ ζ∗ ∗ ς ∈ 𝔜 ⇒ 0 ∗ ζ∗ ∗ ς ∈   𝔜 ⇒   𝔜 is closed SAH – ideal of € . ∎ Theorem (9) Assume € ∈ is a collocation of € and 𝔜 be a closed SAH – ideal of €, ∀λ ∈ Λ . Then   𝔜 is a closed SAH – ideal of the direct product of €. Proof ∀ ς , ζ ∈ 𝔜 0 ς∗ ζ𝛌 ∈   𝔜 ∧ 0 ζ∗ ∈   𝔜 ⇒ 0 ∗ ς∗ ∗ ζ ∈   𝔜 ∧ 0 ∗ ζ∗ ∈   𝔜 0 ∗ ς∗ ∗ ζ ∈ 𝔜 ∧ 0 ∗ ζ∗ ∈ 𝔜 and Since 𝔜 is closed SAH – ideal ∀λ ∈ Λ ,then ∴ 0 ∗ ζ∗ ∗ ς ∈ 𝔜 , ∀λ ∈ Λ ⟹ 0 ∗ ζ∗ ∗ ς ∈   𝔜 ⟹   𝔜 is closed SAH – ideal of € . ∎   124  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Definition (10) Assume 𝔜 is a closed SAH – ideal of € . Then 𝔜 is named closed SAH – ideal with respect to an element s ∈ € (represented by s closed SAH – ideal) if: s ∗ 0 ∗ ς∗ ∗ ζ ∈ 𝔜 ∧ s ∗ 0 ∗ ζ∗ ∈ 𝔜 . Then s ∗ 0 ∗ ζ∗ ∗ ς ∈ 𝔜 Example (11) Consider € 0,1,2,3 with binary operation ∗ defined by the ensuing table: Table 4. * 0 1 2 3 0 0 0 0 0 1 1 0 0 1 2 2 3 0 3 3 3 0 0 0 𝔜 0,2 , s 3 and 𝔜 is 3 – closed SAH – ideal of €. 3. Completely Closed SAH – Ideal with Respect to an Element of BH – Algebra Definition (12) A SAH – ideal 𝔜 of € is known as completely closed SAH – ideal if ς ∗ ζ ∈ 𝔜 , ∀ ς , ζ ∈ 𝔜 (represented by 𝑆𝐴𝐻 –ideal). Example (13) In example (11), we have 𝔜 is 𝑆𝐴𝐻 – ideal of € since: 0 ∗ 0 0 ∈ 𝔜 , 0 ∗ 2 0 ∈ 𝔜 2 ∗ 0 2 ∈ 𝔜 , 2 ∗ 2 0 ∈ 𝔜 Definition (14) A SAH – ideal 𝔜 of € and s ∈ € , then 𝔜 is named a completely closed SAH – ideal with respect to an element s ∈ € (represented by s 𝑆𝐴𝐻 – ideal ) If s ∗ 0 ∗ ς ∗ ζ ∈ 𝔜 , ∀ ς , ζ ∈ 𝔜 Example (15) In example (11), we have: 𝔜 0,2 and s 2 , then 𝔜 is 2 – 𝑆𝐴𝐻 – ideal since: 2 ∗ 0 ∗ 0 ∗ 0 2 ∈ 𝔜 , 2 ∗ 0 ∗ 0 ∗ 2 2 ∈ 𝔜 2 ∗ 0 ∗ 2 ∗ 0 2 ∈ 𝔜 , 2 ∗ 2 ∗ 2 ∗ 2 2 ∈ 𝔜 Remark (16) In € every s 𝑆𝐴𝐻 – ideal is a s closed SAH – ideal.   125  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Proposition (17) Assume 𝔜 is a 𝑆𝐴𝐻 – ideal of € . Then 𝔜 is a s 𝑆𝐴𝐻 – ideal, ∀s ∈ 𝔜 . Proof Assume ∀ ς , ζ ∈ 𝔜 Mean while 𝔜 is 𝑆𝐴𝐻 – ideal and s ∈ 𝔜 Then 𝑠 ∗ 0 ∗ ς ∗ ζ ∈ 𝔜 . ∎ Theorem (18) Assume €;∗ ,0 and ₡;⊛, 0 are BH – algebras and 𝔥 : € → ₡ is a BH – epimorphism and 𝔜 is a SAH – ideal in €, then 𝔥 𝔜 is a SAH – ideal in ₡. Proof Assume ς∗ ⊛ ζ ∈ 𝔥 𝔜 ∧ ζ∗ ∈ 𝔥 𝔜 to prove ζ∗ ⊛ ς ∈ 𝔥 𝔜 , ∀ς , ζ ∈ 𝔜 ⟹ ∃ a, b ∈ 𝔜 such that 𝔥 a ς , 𝔥 b ζ , 𝔥 a ∗ ⊛ 𝔥 b ∈ 𝔥 𝔜 ∧ 𝔥 b ∗ ∈ 𝔥 𝔜 𝔥 a ∗ ⊛ 𝔥 b ∈ 𝔥 𝔜 ∧ 𝔥 b ∗ ∈ 𝔥 𝔜 𝔥 a∗ ∗ b ∈ 𝔥 𝔜 ∧ 𝔥 b∗ ∈ 𝔥 𝔜 ⟹ a∗ ∗ b ∈ 𝔜 ∧ b∗ ∈ 𝔜 → b∗ ∗ a ∈ 𝔜 → 𝔥 b∗ ∗ a ∈ 𝔥 𝔜 ∵ 𝔥 is epimorphism ⟹ 𝔥 b∗ ⊛ 𝔥 a ∈ 𝔥 𝔜 ⟹ 𝔥 b ∗ ⊛ 𝔥 a ∈ 𝔥 𝔜 ζ∗ ⊛ ς ∈ 𝔥 𝔜 ∴ 𝔥 𝔜 is SAH – ideal in ₡ . ∎ Theorem (19) Assume €;∗ ,0 and ₡;⊛, 0 are BH – algebras and 𝔥 : € → ₡ an epimorphism and 𝔜 is a SAH – ideal in € . Then 𝔥 𝔜 is a closed SAH – ideal in ₡. Proof Assume 𝔜 is a SAH – ideal in € 𝔥 𝔜 is SAH – ideal (theorem (18)) And by using remark (5)   126  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 𝔥 𝔜 is a closed SAH – ideal in ₡ . ∎ Remark (20) Now each SAH – ideal of € is a s 𝑆𝐴𝐻 – ideal of €, ∀s ∈ 𝔜 . Theorem (21) Assume €;∗ ,0 and ₡;⊛, 0 are BH – algebras and : € → ₡ is a epimorphism , if 𝔜 is a s 𝑆𝐴𝐻 – ideal in € , then 𝔥 𝔜 is a 𝔥 s 𝑆𝐴𝐻 – ideal in ₡ . Proof Assume 𝔜 is a s 𝑆𝐴𝐻 – ideal in € , then s ∗ a ∗ c ∈ 𝔜 , ∀ a , c ∈ 𝔜 Since 𝔜 is SAH – ideal , then 𝔥 𝔜 is a SAH – ideal (theorem 18) Assume ς , ζ ∈ 𝔥 𝔜 ⟹ ∃ m, n ∈ 𝔜 such that 𝔥 m ς , 𝖍 n ζ 𝔥 s ⊛ ς ⊛ ζ 𝖍 s ⊛ 𝔥 m ⊛ 𝔥 n 𝔥 s ⊛ 𝔥 m ∗ n 𝔥 s ∗ m ∗ n ∈ 𝔥 𝔜 [since s ∗ m ∗ n ∈ 𝔜] ∴ 𝔥 𝔜 is a 𝔥 s 𝑆𝐴𝐻 – ideal. ∎ Proposition (22) Assume 𝔜 is a SAH – ideal of € such that 𝔜 ⊆ € . Then 𝔜 is s closed SAH – ideal ∀s ∈ 𝔜 . Where € ς ∈ €: 0 ∗ ς 0 . Proof Assume s ∈ 𝔜 and ⊆ € . Then s ∗ 0 ∗ ς s ∗ 0 [ since 𝔜 ⊆ € ] = s∈ 𝔜 ∴ 𝔜 is s closed SAH – ideal . ∎ 4. Conclusion In this paper , we constructed the idea of SAH – ideal , closed SAH – ideal ,s- closed SAH – ideal , 𝑆𝐴𝐻 – ideal and s- 𝑆𝐴𝐻 – ideal of BH – algebra which are presented with some of their properties , examples and theorems . In our future work, we introduce the concept of fuzzy SAH – ideal of BH – algebra. It is our optimism that this effort grows into other fundamentals for further study of ideas of BH-algebra. References 1. Zadah, L. A. 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