id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-6909	Ramesh, D. ; Baji, Shake ; Iampan, Aiyared; Satyanarayana, B. 	Bipolar Fuzzy Commutative Hyper BCK-ideals in Hyper BCK-algebras	2025	16	.pdf	application/pdf	7371	457	81	= {ℏ1}, (P11) 0 ◦ H1 = {0}, (P12) ℏ1 ≪ {0} ⇒ ℏ1 = {0}, (P13) H1 ◦ H2 ≪ H1, (P14) ℏ1 ∈ ℏ1 ◦ 0, (P15) ℏ1 ◦ 0 ≪ {ℏ2} ⇒ ℏ1 ≪ ℏ2, (P16) ℏ2 ≪ ℏ3 ⇒ ℏ1 ◦ ℏ3 ≪ ℏ1 ◦ ℏ2, (P17) ℏ1 ◦ ℏ2 = {0} ⇒ (ℏ1 ◦ ℏ3) ◦ (ℏ2 ◦ ℏ3) ◦ ℏ3 = (ℏ1 ◦ ℏ3) ◦ ℏ2, (HBCKA-3) ℏ1 ◦ H ≪ {ℏ1}, (HBCKA-4) ℏ1 ≪ ℏ2 and ℏ2 ≪ ℏ1 ⇒ ℏ1 = ℏ2, for all ℏ1, ℏ2, ℏ3 ∈ H. We denote a relationship ≪ on H by letting ℏ1 ≪ ℏ2 ⇔ 0 ∈ ℏ1 ◦ ℏ2 and every H1,H2 ⊆ H,H1 ≪ H2 is described by ∀r ∈ H1, ∃j ∈ H2 such that r ≪ j.	cache/ejpam-6909.pdf	txt/ejpam-6909.txt