id	sid	eid	entity	type
ejpam-5490	1	1	european journal	ORG
ejpam-5490	1	2	2025	DATE
ejpam-5490	1	3	18	CARDINAL
ejpam-5490	1	4	1	CARDINAL
ejpam-5490	1	5	5490	CARDINAL
ejpam-5490	1	6	1307-5543	CARDINAL
ejpam-5490	1	7	new york	GPE
ejpam-5490	1	8	jogarao gunda1	PERSON
ejpam-5490	1	9	sirisetti2	PERSON
ejpam-5490	1	10	ravikumar bandaru3	PERSON
ejpam-5490	1	11	1	CARDINAL
ejpam-5490	1	12	bs & h	ORG
ejpam-5490	1	13	aditya institute of technology	ORG
ejpam-5490	1	14	srikakulam	NORP
ejpam-5490	1	15	andhra pradesh-530021	ORG
ejpam-5490	1	16	india 2	ORG
ejpam-5490	1	17	visakhapatnam	GPE
ejpam-5490	1	18	andhra pradesh-530045	ORG
ejpam-5490	1	19	india 3	ORG
ejpam-5490	1	20	vit-ap university	ORG
ejpam-5490	1	21	pradesh-522237	DATE
ejpam-5490	1	22	india	GPE
ejpam-5490	1	23	4	CARDINAL
ejpam-5490	1	24	walter sisulu university	ORG
ejpam-5490	1	25	5117	DATE
ejpam-5490	1	26	south africa	GPE
ejpam-5490	8	1	2020	DATE
ejpam-5490	8	2	06d05	CARDINAL
ejpam-5490	8	3	06d15	CARDINAL
ejpam-5490	8	4	1	CARDINAL
ejpam-5490	10	1	ω	ORDINAL
ejpam-5490	12	1	j. gunda	PERSON
ejpam-5490	12	2	r.	NORP
ejpam-5490	12	3	r. bandaru	PERSON
ejpam-5490	12	4	r. shukla	PERSON
ejpam-5490	12	5	1	CARDINAL
ejpam-5490	13	1	2025	CARDINAL
ejpam-5490	14	1	j. gunda et al	PERSON
ejpam-5490	15	1	j. pure	PERSON
ejpam-5490	15	2	18	CARDINAL
ejpam-5490	15	3	1	CARDINAL
ejpam-5490	15	4	2025	CARDINAL
ejpam-5490	15	5	5490 2	CARDINAL
ejpam-5490	15	6	11	CARDINAL
ejpam-5490	16	1	1	CARDINAL
ejpam-5490	16	2	8	CARDINAL
ejpam-5490	16	3	3	CARDINAL
ejpam-5490	22	1	zero	CARDINAL
ejpam-5490	25	1	2.	CARDINAL
ejpam-5490	28	1	4	CARDINAL
ejpam-5490	28	2	kumar	PERSON
ejpam-5490	28	3	rao	PERSON
ejpam-5490	30	1	1	CARDINAL
ejpam-5490	31	1	k	PERSON
ejpam-5490	31	2	kdd = k. lemma	PERSON
ejpam-5490	32	1	1	CARDINAL
ejpam-5490	33	1	kdd	PERSON
ejpam-5490	34	1	kdd	PERSON
ejpam-5490	34	2	k. lemma	PERSON
ejpam-5490	35	1	2	CARDINAL
ejpam-5490	35	2	a.	PERSON
ejpam-5490	37	1	∈	ORG
ejpam-5490	39	1	1	CARDINAL
ejpam-5490	41	1	k ∨∗ g	ORG
ejpam-5490	41	2	k	PERSON
ejpam-5490	42	1	j. gunda et al	PERSON
ejpam-5490	43	1	j. pure	PERSON
ejpam-5490	43	2	18	CARDINAL
ejpam-5490	43	3	1	CARDINAL
ejpam-5490	43	4	2025	CARDINAL
ejpam-5490	43	5	5490 3	DATE
ejpam-5490	43	6	11	CARDINAL
ejpam-5490	43	7	2	CARDINAL
ejpam-5490	44	1	w ∈	CARDINAL
ejpam-5490	44	2	w ∈ k	PERSON
ejpam-5490	45	1	a. theorem 2	PERSON
ejpam-5490	46	1	k ∩	ORG
ejpam-5490	47	1	k ⊆ u	ORG
ejpam-5490	49	1	∈	ORG
ejpam-5490	49	2	k ∈	ORG
ejpam-5490	49	3	g1 ⊆	ORG
ejpam-5490	49	4	⊆	CARDINAL
ejpam-5490	49	5	g3	ORG
ejpam-5490	49	6	⊆	PRODUCT
ejpam-5490	49	7	1	CARDINAL
ejpam-5490	49	8	2	DATE
ejpam-5490	49	9	3	CARDINAL
ejpam-5490	50	1	∈	ORG
ejpam-5490	52	1	= ⊔(i ∩	PERSON
ejpam-5490	53	1	∈ q.	PERSON
ejpam-5490	54	1	zorn	ORG
ejpam-5490	54	2	lemma states	ORG
ejpam-5490	54	3	∩	PERSON
ejpam-5490	54	4	k ⊆ u .	ORG
ejpam-5490	55	1	w ∈	CARDINAL
ejpam-5490	55	2	w /∈ u .	NORP
ejpam-5490	56	1	⊆	CARDINAL
ejpam-5490	60	1	⊆	CARDINAL
ejpam-5490	62	1	ud∧∗	ORG
ejpam-5490	66	1	dd∩i}∩{u∨∗[w	GPE
ejpam-5490	67	1	[w)]}dd ∩	PERSON
ejpam-5490	67	2	w)}dd	PERSON
ejpam-5490	67	3	i.	PERSON
ejpam-5490	68	1	w ∈	PERSON
ejpam-5490	69	1	w ∈ i.	PERSON
ejpam-5490	70	1	∈	ORG
ejpam-5490	70	2	w ∈	ORG
ejpam-5490	72	1	1	CARDINAL
ejpam-5490	75	1	a.	PERSON
ejpam-5490	75	2	k ⊆ u	ORG
ejpam-5490	76	1	3	CARDINAL
ejpam-5490	76	2	kdd	PERSON
ejpam-5490	76	3	k. proof	PERSON
ejpam-5490	77	1	/∈ kdd	ORG
ejpam-5490	80	1	u ̸= ϕ	ORG
ejpam-5490	80	2	kdd ∈	PERSON
ejpam-5490	81	1	⊆	CARDINAL
ejpam-5490	82	1	dd]dd	ORG
ejpam-5490	82	2	∨∗ki	GPE
ejpam-5490	83	1	⊔ki	PERSON
ejpam-5490	84	1	k ⊆	PERSON
ejpam-5490	84	2	∩	PERSON
ejpam-5490	85	1	⊔ki ∈	PRODUCT
ejpam-5490	85	2	⊔ki	DATE
ejpam-5490	86	1	zorn	ORG
ejpam-5490	86	2	lemma states	ORG
ejpam-5490	87	1	w ∈	CARDINAL
ejpam-5490	87	2	w /∈ u	PERSON
ejpam-5490	87	3	∩	PERSON
ejpam-5490	88	1	∈	PERSON
ejpam-5490	88	2	⊆	CARDINAL
ejpam-5490	88	3	⊆	CARDINAL
ejpam-5490	89	1	∈	ORG
ejpam-5490	92	1	2	CARDINAL
ejpam-5490	93	1	3	CARDINAL
ejpam-5490	94	1	d. lemma	PERSON
ejpam-5490	94	2	3	CARDINAL
ejpam-5490	95	1	at least one	CARDINAL
ejpam-5490	96	1	j. gunda et al	PERSON
ejpam-5490	97	1	j. pure	PERSON
ejpam-5490	97	2	18	CARDINAL
ejpam-5490	97	3	1	CARDINAL
ejpam-5490	97	4	2025	CARDINAL
ejpam-5490	97	5	5490 4	DATE
ejpam-5490	97	6	11	CARDINAL
ejpam-5490	97	7	4	CARDINAL
ejpam-5490	100	1	5	CARDINAL
ejpam-5490	103	1	k ⊆ kdd	ORG
ejpam-5490	104	1	kdd	PERSON
ejpam-5490	105	1	kdd	PERSON
ejpam-5490	105	2	= d. therefore k	PERSON
ejpam-5490	106	1	k = kdd	PERSON
ejpam-5490	107	1	3.	CARDINAL
ejpam-5490	110	1	w ∈ a. lemma 4	PERSON
ejpam-5490	112	1	∈	ORG
ejpam-5490	112	2	∈	ORG
ejpam-5490	112	3	5	CARDINAL
ejpam-5490	112	4	∈	ORG
ejpam-5490	112	5	1	CARDINAL
ejpam-5490	113	1	⊆	CARDINAL
ejpam-5490	113	2	2	CARDINAL
ejpam-5490	113	3	⊆	CARDINAL
ejpam-5490	113	4	⊆	CARDINAL
ejpam-5490	113	5	3	CARDINAL
ejpam-5490	113	6	w)d ∩	PERSON
ejpam-5490	113	7	w	ORG
ejpam-5490	113	8	4	CARDINAL
ejpam-5490	113	9	1	CARDINAL
ejpam-5490	114	1	w ∈	CARDINAL
ejpam-5490	115	1	w.	PERSON
ejpam-5490	115	2	1	CARDINAL
ejpam-5490	116	1	0	CARDINAL
ejpam-5490	116	2	1	CARDINAL
ejpam-5490	116	3	j. gunda et al	PERSON
ejpam-5490	117	1	j. pure	PERSON
ejpam-5490	117	2	18	CARDINAL
ejpam-5490	117	3	1	CARDINAL
ejpam-5490	117	4	2025	CARDINAL
ejpam-5490	117	5	5490 5	DATE
ejpam-5490	117	6	11	CARDINAL
ejpam-5490	117	7	0	CARDINAL
ejpam-5490	117	8	1 ̸= x.	QUANTITY
ejpam-5490	117	9	6	DATE
ejpam-5490	119	1	2	CARDINAL
ejpam-5490	122	1	7	CARDINAL
ejpam-5490	126	1	∈	PRODUCT
ejpam-5490	127	1	∩	PERSON
ejpam-5490	127	2	0)d	LAW
ejpam-5490	129	1	8	CARDINAL
ejpam-5490	132	1	7	CARDINAL
ejpam-5490	132	2	∈	ORG
ejpam-5490	133	1	x)d	ORG
ejpam-5490	133	2	∈	ORG
ejpam-5490	133	3	0)d	LAW
ejpam-5490	133	4	∩	PERSON
ejpam-5490	137	1	4	CARDINAL
ejpam-5490	141	1	9	CARDINAL
ejpam-5490	142	1	a. lemma	PERSON
ejpam-5490	142	2	6	CARDINAL
ejpam-5490	143	1	w)d =⇒	PERSON
ejpam-5490	144	1	w)d =⇒	PERSON
ejpam-5490	144	2	10	CARDINAL
ejpam-5490	146	1	j. gunda et al	PERSON
ejpam-5490	147	1	j. pure	PERSON
ejpam-5490	147	2	18	CARDINAL
ejpam-5490	147	3	1	CARDINAL
ejpam-5490	147	4	2025	CARDINAL
ejpam-5490	147	5	5490 6	CARDINAL
ejpam-5490	147	6	⊆	CARDINAL
ejpam-5490	147	7	w ∈ k	PERSON
ejpam-5490	148	1	⋃	PERSON
ejpam-5490	148	2	5	CARDINAL
ejpam-5490	150	1	11	CARDINAL
ejpam-5490	151	1	0)d	LAW
ejpam-5490	151	2	∈	ORG
ejpam-5490	151	3	d.	NORP
ejpam-5490	152	1	∈	ORG
ejpam-5490	153	1	k =	PERSON
ejpam-5490	153	2	y ∈ a.	PERSON
ejpam-5490	153	3	∈	ORG
ejpam-5490	155	1	⊆	CARDINAL
ejpam-5490	156	1	x∨∗ y	PERSON
ejpam-5490	157	1	therefore	PERSON
ejpam-5490	157	2	⊆	CARDINAL
ejpam-5490	158	1	k ⊔g	ORG
ejpam-5490	160	1	0)d	LAW
ejpam-5490	160	2	0)d	LAW
ejpam-5490	161	1	w ∈	PERSON
ejpam-5490	162	1	y ∈	PERSON
ejpam-5490	165	1	a. lemma 7	PERSON
ejpam-5490	165	2	0	CARDINAL
ejpam-5490	167	1	∈	ORG
ejpam-5490	167	2	12	CARDINAL
ejpam-5490	168	1	w)/ψ	PERSON
ejpam-5490	169	1	13	CARDINAL
ejpam-5490	170	1	1	CARDINAL
ejpam-5490	170	2	2	CARDINAL
ejpam-5490	170	3	3	CARDINAL
ejpam-5490	170	4	4	CARDINAL
ejpam-5490	172	1	1	CARDINAL
ejpam-5490	172	2	2	CARDINAL
ejpam-5490	174	1	0)d	LAW
ejpam-5490	174	2	w)d.	GPE
ejpam-5490	176	1	2	CARDINAL
ejpam-5490	176	2	3	CARDINAL
ejpam-5490	176	3	∈	PRODUCT
ejpam-5490	177	1	∈	PRODUCT
ejpam-5490	177	2	j. gunda et al	PERSON
ejpam-5490	178	1	j. pure	PERSON
ejpam-5490	178	2	18	CARDINAL
ejpam-5490	178	3	1	CARDINAL
ejpam-5490	178	4	2025	CARDINAL
ejpam-5490	178	5	5490 7	DATE
ejpam-5490	178	6	11	CARDINAL
ejpam-5490	178	7	0)d	LAW
ejpam-5490	178	8	∩	PERSON
ejpam-5490	178	9	w)d.	GPE
ejpam-5490	180	1	0	CARDINAL
ejpam-5490	181	1	0	CARDINAL
ejpam-5490	181	2	w)/ψ	PERSON
ejpam-5490	181	3	v∨∗w)/ψ	PERSON
ejpam-5490	183	1	3	CARDINAL
ejpam-5490	183	2	4	DATE
ejpam-5490	183	3	0	CARDINAL
ejpam-5490	183	4	0	CARDINAL
ejpam-5490	184	1	0	CARDINAL
ejpam-5490	185	1	0	CARDINAL
ejpam-5490	185	2	w.	PERSON
ejpam-5490	186	1	4	CARDINAL
ejpam-5490	186	2	1	CARDINAL
ejpam-5490	187	1	0	CARDINAL
ejpam-5490	188	1	0	CARDINAL
ejpam-5490	190	1	4	CARDINAL
ejpam-5490	193	1	6	CARDINAL
ejpam-5490	195	1	w.	PERSON
ejpam-5490	196	1	2	CARDINAL
ejpam-5490	196	2	example 1	DATE
ejpam-5490	196	3	1	CARDINAL
ejpam-5490	196	4	0	CARDINAL
ejpam-5490	197	1	a. example 3	PERSON
ejpam-5490	199	1	2	CARDINAL
ejpam-5490	201	1	example 1	DATE
ejpam-5490	202	1	8	CARDINAL
ejpam-5490	203	1	w ∈	CARDINAL
ejpam-5490	204	1	j. gunda et al	PERSON
ejpam-5490	205	1	j. pure	PERSON
ejpam-5490	205	2	18	CARDINAL
ejpam-5490	205	3	1	CARDINAL
ejpam-5490	205	4	2025	CARDINAL
ejpam-5490	205	5	5490 8	DATE
ejpam-5490	206	1	∈	ORG
ejpam-5490	206	2	∈	ORG
ejpam-5490	207	1	0	CARDINAL
ejpam-5490	208	1	9	CARDINAL
ejpam-5490	209	1	w ∈	PERSON
ejpam-5490	209	2	w ∈ a. (ii	PERSON
ejpam-5490	210	1	w ∈	PERSON
ejpam-5490	210	2	w ∈ a.	PERSON
ejpam-5490	210	3	w ∈ a.	ORG
ejpam-5490	210	4	w gg ∈	PERSON
ejpam-5490	210	5	w ∈ a. proof	PERSON
ejpam-5490	211	1	w ∈ a.	ORG
ejpam-5490	212	1	w ∈	PERSON
ejpam-5490	212	2	d.	NORP
ejpam-5490	212	3	w.	PERSON
ejpam-5490	212	4	vgg	PERSON
ejpam-5490	212	5	∈	ORG
ejpam-5490	212	6	w ∈	PERSON
ejpam-5490	212	7	d.	NORP
ejpam-5490	212	8	w	GPE
ejpam-5490	212	9	v gg ∈	PERSON
ejpam-5490	213	1	w gg ∈	PERSON
ejpam-5490	214	1	∈	ORG
ejpam-5490	214	2	lemma 8(iv	ORG
ejpam-5490	214	3	w gg ∈	PERSON
ejpam-5490	215	1	∈	ORG
ejpam-5490	215	2	lemma 8(iv	ORG
ejpam-5490	216	1	∈	ORG
ejpam-5490	216	2	d. lemma	PERSON
ejpam-5490	216	3	10	CARDINAL
ejpam-5490	217	1	w ∈	CARDINAL
ejpam-5490	217	2	w	LANGUAGE
ejpam-5490	218	1	w ∈	CARDINAL
ejpam-5490	218	2	w ≥	GPE
ejpam-5490	218	3	w.	PERSON
ejpam-5490	224	1	w	ORG
ejpam-5490	225	1	v g	PERSON
ejpam-5490	225	2	w	ORG
ejpam-5490	225	3	∈	ORG
ejpam-5490	226	1	w g.	PERSON
ejpam-5490	226	2	w g. (ii	PERSON
ejpam-5490	226	3	w.	PERSON
ejpam-5490	227	1	w g. j. gunda et al	PERSON
ejpam-5490	228	1	j. pure	PERSON
ejpam-5490	228	2	18	CARDINAL
ejpam-5490	228	3	1	CARDINAL
ejpam-5490	228	4	2025	CARDINAL
ejpam-5490	228	5	5490 9	DATE
ejpam-5490	230	1	∈	ORG
ejpam-5490	231	1	9	CARDINAL
ejpam-5490	234	1	⇒ w	ORG
ejpam-5490	234	2	v gg	PERSON
ejpam-5490	234	3	w gg	PERSON
ejpam-5490	240	1	w	LANGUAGE
ejpam-5490	240	2	14	DATE
ejpam-5490	243	1	w ∈	PERSON
ejpam-5490	244	1	∈	ORG
ejpam-5490	244	2	w ∈	PERSON
ejpam-5490	246	1	∧∗[{v	ORG
ejpam-5490	247	1	w ∈	PERSON
ejpam-5490	247	2	d.	NORP
ejpam-5490	247	3	∈	ORG
ejpam-5490	248	1	∈	ORG
ejpam-5490	248	2	∈	ORG
ejpam-5490	249	1	∈	ORG
ejpam-5490	249	2	d.	NORP
ejpam-5490	250	1	15	CARDINAL
ejpam-5490	251	1	0	CARDINAL
ejpam-5490	253	1	0	CARDINAL
ejpam-5490	254	1	∈	ORG
ejpam-5490	254	2	∈	ORG
ejpam-5490	254	3	w ∈	PERSON
ejpam-5490	255	1	w.	PERSON
ejpam-5490	256	1	0	CARDINAL
ejpam-5490	257	1	0	CARDINAL
ejpam-5490	258	1	0	CARDINAL
ejpam-5490	258	2	a. therefore v ∈	PERSON
ejpam-5490	259	1	0	CARDINAL
ejpam-5490	260	1	0	CARDINAL
ejpam-5490	260	2	0	CARDINAL
ejpam-5490	261	1	a.	PERSON
ejpam-5490	261	2	0	CARDINAL
ejpam-5490	262	1	0	CARDINAL
ejpam-5490	263	1	w.	PERSON
ejpam-5490	263	2	w.	PERSON
ejpam-5490	263	3	∈	ORG
ejpam-5490	263	4	w ∈	PERSON
ejpam-5490	263	5	d.	NORP
ejpam-5490	263	6	16	DATE
ejpam-5490	266	1	∈	ORG
ejpam-5490	266	2	∈	ORG
ejpam-5490	266	3	∈	ORG
ejpam-5490	267	1	w ∈ a.	ORG
ejpam-5490	267	2	j. gunda et al	PERSON
ejpam-5490	268	1	j. pure	PERSON
ejpam-5490	268	2	18	CARDINAL
ejpam-5490	268	3	1	CARDINAL
ejpam-5490	268	4	2025	CARDINAL
ejpam-5490	268	5	5490 10	DATE
ejpam-5490	278	1	w.	PERSON
ejpam-5490	278	2	⊆	CARDINAL
ejpam-5490	286	1	∈	ORG
ejpam-5490	287	1	v]∨∗	PERSON
ejpam-5490	288	1	a.	GPE
ejpam-5490	290	1	⊆	CARDINAL
ejpam-5490	293	1	w.	PERSON
ejpam-5490	293	2	w.	PERSON
ejpam-5490	293	3	⊆	CARDINAL
ejpam-5490	294	1	⊆	CARDINAL
ejpam-5490	297	1	7	CARDINAL
ejpam-5490	298	1	∈	ORG
ejpam-5490	298	2	0	CARDINAL
ejpam-5490	298	3	17	CARDINAL
ejpam-5490	301	1	a.	PERSON
ejpam-5490	302	1	w ∈	PERSON
ejpam-5490	302	2	0	CARDINAL
ejpam-5490	309	1	∈	ORG
ejpam-5490	312	1	5	CARDINAL
ejpam-5490	315	1	j. gunda et al	PERSON
ejpam-5490	316	1	j. pure	PERSON
ejpam-5490	316	2	18	CARDINAL
ejpam-5490	316	3	1	CARDINAL
ejpam-5490	316	4	2025	CARDINAL
ejpam-5490	316	5	5490 11	DATE
ejpam-5490	316	6	11	CARDINAL
ejpam-5490	317	1	walter sisulu university	ORG
ejpam-5490	317	2	south africa	GPE
ejpam-5490	320	1	1	CARDINAL
ejpam-5490	322	1	amer	PERSON
ejpam-5490	324	1	collequium pub, 1967	ORG
ejpam-5490	325	1	2	CARDINAL
ejpam-5490	326	1	australian	NORP
ejpam-5490	326	2	14:200	CARDINAL
ejpam-5490	326	3	215	CARDINAL
ejpam-5490	326	4	1972	DATE
ejpam-5490	327	1	3	CARDINAL
ejpam-5490	329	1	15:501–511, 1974	DATE
ejpam-5490	330	1	4	CARDINAL
ejpam-5490	330	2	s rao	PERSON
ejpam-5490	332	1	palestine journal of mathematics	ORG
ejpam-5490	332	2	11:143–151	CARDINAL
ejpam-5490	332	3	2022	CARDINAL
ejpam-5490	333	1	5	CARDINAL
ejpam-5490	333	2	s rao	PERSON
ejpam-5490	333	3	k s babu	PERSON
ejpam-5490	334	1	generalized prime	ORG
ejpam-5490	335	1	archivum mathematicum	PERSON
ejpam-5490	335	2	57:157–174	CARDINAL
ejpam-5490	335	3	2021	CARDINAL
ejpam-5490	336	1	6	CARDINAL
ejpam-5490	336	2	s rao	PERSON
ejpam-5490	338	1	southeast asian	NORP
ejpam-5490	338	2	40:251–264	DATE
ejpam-5490	338	3	2016	DATE
ejpam-5490	339	1	s rao	PERSON
ejpam-5490	339	2	ch v rao	PERSON
ejpam-5490	341	1	2022	CARDINAL
ejpam-5490	342	1	8	CARDINAL
ejpam-5490	344	1	american mathematical society	ORG
ejpam-5490	344	2	1971	DATE