id	sid	eid	entity	type
ejpam-5618	1	1	european journal	ORG
ejpam-5618	1	2	2025	DATE
ejpam-5618	1	3	18	CARDINAL
ejpam-5618	1	4	1	CARDINAL
ejpam-5618	1	5	5618	CARDINAL
ejpam-5618	1	6	1307-5543	DATE
ejpam-5618	1	7	new york	GPE
ejpam-5618	1	8	saud m. alammar1,∗	PERSON
ejpam-5618	1	9	m. a. al shumrani2	PERSON
ejpam-5618	1	10	özel2 1	DATE
ejpam-5618	1	11	2440	DATE
ejpam-5618	1	12	saudi arabia	GPE
ejpam-5618	1	13	2	CARDINAL
ejpam-5618	1	14	abdulaziz university	PERSON
ejpam-5618	1	15	80203 jeddah 21589	DATE
ejpam-5618	1	16	saudi arabia	GPE
ejpam-5618	10	1	2020	DATE
ejpam-5618	10	2	54h11	CARDINAL
ejpam-5618	10	3	22a05	CARDINAL
ejpam-5618	10	4	1	CARDINAL
ejpam-5618	12	1	11	CARDINAL
ejpam-5618	12	2	13	CARDINAL
ejpam-5618	13	1	1990	DATE
ejpam-5618	13	2	jankovic	PERSON
ejpam-5618	13	3	hamlett	PERSON
ejpam-5618	13	4	9	CARDINAL
ejpam-5618	13	5	10	CARDINAL
ejpam-5618	14	1	abd el-monsef	PERSON
ejpam-5618	16	1	1943	DATE
ejpam-5618	16	2	hewitt	PERSON
ejpam-5618	19	1	3	CARDINAL
ejpam-5618	20	1	dontchev	PERSON
ejpam-5618	20	2	5	CARDINAL
ejpam-5618	21	1	2020	DATE
ejpam-5618	21	2	jafari	GPE
ejpam-5618	21	3	8	CARDINAL
ejpam-5618	24	1	2	CARDINAL
ejpam-5618	26	1	s.alammar@uoh.edu.sa	ORG
ejpam-5618	26	2	s. alammar	PERSON
ejpam-5618	26	3	maalshmrani1@kau.edu.sa	PERSON
ejpam-5618	26	4	m. al shumrani	PERSON
ejpam-5618	27	1	1	CARDINAL
ejpam-5618	27	2	2025	CARDINAL
ejpam-5618	28	1	s. alammar	PERSON
ejpam-5618	28	2	m. al shumrani	PERSON
ejpam-5618	29	1	j. pure	PERSON
ejpam-5618	29	2	18	CARDINAL
ejpam-5618	29	3	1	CARDINAL
ejpam-5618	29	4	2025	CARDINAL
ejpam-5618	29	5	5618 2	DATE
ejpam-5618	29	6	13 2	CARDINAL
ejpam-5618	31	1	6	CARDINAL
ejpam-5618	31	2	12	CARDINAL
ejpam-5618	31	3	4	CARDINAL
ejpam-5618	32	1	1	CARDINAL
ejpam-5618	33	1	13	CARDINAL
ejpam-5618	33	2	two	CARDINAL
ejpam-5618	34	1	∈	ORG
ejpam-5618	35	1	∈	ORG
ejpam-5618	36	1	2	CARDINAL
ejpam-5618	37	1	11	CARDINAL
ejpam-5618	41	1	3	CARDINAL
ejpam-5618	42	1	10	CARDINAL
ejpam-5618	43	1	⊆	CARDINAL
ejpam-5618	46	1	1	CARDINAL
ejpam-5618	47	1	1	CARDINAL
ejpam-5618	48	1	two	CARDINAL
ejpam-5618	50	1	4	CARDINAL
ejpam-5618	51	1	5	CARDINAL
ejpam-5618	54	1	⊂	GPE
ejpam-5618	55	1	6	CARDINAL
ejpam-5618	56	1	1	CARDINAL
ejpam-5618	56	2	x.	GPE
ejpam-5618	56	3	7	CARDINAL
ejpam-5618	57	1	1	CARDINAL
ejpam-5618	57	2	f(x	ORG
ejpam-5618	57	3	f(u	GPE
ejpam-5618	58	1	1	CARDINAL
ejpam-5618	59	1	1	CARDINAL
ejpam-5618	61	1	s. alammar	PERSON
ejpam-5618	61	2	m. al shumrani	PERSON
ejpam-5618	62	1	j. pure	PERSON
ejpam-5618	62	2	18	CARDINAL
ejpam-5618	62	3	1	CARDINAL
ejpam-5618	62	4	2025	CARDINAL
ejpam-5618	62	5	5618 3	DATE
ejpam-5618	62	6	x.	GPE
ejpam-5618	62	7	8	DATE
ejpam-5618	63	1	1	CARDINAL
ejpam-5618	63	2	f(u	PERSON
ejpam-5618	64	1	9	CARDINAL
ejpam-5618	65	1	5	CARDINAL
ejpam-5618	65	2	σ	PERSON
ejpam-5618	65	3	x.	GPE
ejpam-5618	65	4	10	CARDINAL
ejpam-5618	66	1	8	CARDINAL
ejpam-5618	66	2	σ	PERSON
ejpam-5618	67	1	11	CARDINAL
ejpam-5618	68	1	8	CARDINAL
ejpam-5618	68	2	f(x	ORG
ejpam-5618	70	1	12	CARDINAL
ejpam-5618	71	1	two	CARDINAL
ejpam-5618	71	2	two	CARDINAL
ejpam-5618	72	1	1	CARDINAL
ejpam-5618	75	1	1	CARDINAL
ejpam-5618	81	1	wy	GPE
ejpam-5618	81	2	vy ∩ a.	ORG
ejpam-5618	81	3	wy	GPE
ejpam-5618	81	4	wy ⊂ a.	ORG
ejpam-5618	82	1	w	ORG
ejpam-5618	83	1	w = a. hence	PERSON
ejpam-5618	84	1	13	CARDINAL
ejpam-5618	85	1	3	CARDINAL
ejpam-5618	85	2	one	CARDINAL
ejpam-5618	88	1	2	CARDINAL
ejpam-5618	89	1	1	CARDINAL
ejpam-5618	91	1	2	CARDINAL
ejpam-5618	92	1	1	CARDINAL
ejpam-5618	92	2	σ	PERSON
ejpam-5618	92	3	two	CARDINAL
ejpam-5618	94	1	3	CARDINAL
ejpam-5618	95	1	5	CARDINAL
ejpam-5618	95	2	σ	PERSON
ejpam-5618	95	3	σ	PERSON
ejpam-5618	95	4	two	CARDINAL
ejpam-5618	97	1	s. alammar	PERSON
ejpam-5618	97	2	m. al shumrani	PERSON
ejpam-5618	98	1	j. pure	PERSON
ejpam-5618	98	2	18	CARDINAL
ejpam-5618	98	3	1	CARDINAL
ejpam-5618	98	4	2025	CARDINAL
ejpam-5618	98	5	5618 4	DATE
ejpam-5618	98	6	13	CARDINAL
ejpam-5618	98	7	4	CARDINAL
ejpam-5618	99	1	5	CARDINAL
ejpam-5618	99	2	σ	PERSON
ejpam-5618	101	1	5	CARDINAL
ejpam-5618	102	1	1	CARDINAL
ejpam-5618	102	2	∏n	DATE
ejpam-5618	102	3	⊂	GPE
ejpam-5618	103	1	1 ≤ α ≤	QUANTITY
ejpam-5618	104	1	x.	GPE
ejpam-5618	104	2	6	CARDINAL
ejpam-5618	105	1	1	CARDINAL
ejpam-5618	107	1	3	CARDINAL
ejpam-5618	109	1	14	DATE
ejpam-5618	113	1	1	CARDINAL
ejpam-5618	117	1	2	CARDINAL
ejpam-5618	117	2	r.	NORP
ejpam-5618	120	1	3	CARDINAL
ejpam-5618	125	1	two	CARDINAL
ejpam-5618	127	1	4	CARDINAL
ejpam-5618	131	1	s. alammar	PERSON
ejpam-5618	131	2	m. al shumrani	PERSON
ejpam-5618	132	1	j. pure	PERSON
ejpam-5618	132	2	18	CARDINAL
ejpam-5618	132	3	1	CARDINAL
ejpam-5618	132	4	2025	CARDINAL
ejpam-5618	132	5	5618 5	DATE
ejpam-5618	132	6	13	CARDINAL
ejpam-5618	132	7	5	CARDINAL
ejpam-5618	132	8	0	CARDINAL
ejpam-5618	132	9	1	DATE
ejpam-5618	132	10	2	CARDINAL
ejpam-5618	132	11	integers mod 3	PERSON
ejpam-5618	132	12	1	CARDINAL
ejpam-5618	132	13	2	CARDINAL
ejpam-5618	137	1	0	CARDINAL
ejpam-5618	137	2	1	CARDINAL
ejpam-5618	138	1	1	CARDINAL
ejpam-5618	145	1	first	ORDINAL
ejpam-5618	148	1	⊂	GPE
ejpam-5618	149	1	1	CARDINAL
ejpam-5618	154	1	8	CARDINAL
ejpam-5618	160	1	first	ORDINAL
ejpam-5618	162	1	⊂	GPE
ejpam-5618	166	1	9	CARDINAL
ejpam-5618	168	1	∈	ORG
ejpam-5618	173	1	gx	ORG
ejpam-5618	174	1	⊂	GPE
ejpam-5618	178	1	−1	DATE
ejpam-5618	181	1	9	CARDINAL
ejpam-5618	183	1	s. alammar	PERSON
ejpam-5618	183	2	m. al shumrani	PERSON
ejpam-5618	184	1	j. pure	PERSON
ejpam-5618	184	2	18	CARDINAL
ejpam-5618	184	3	1	CARDINAL
ejpam-5618	184	4	2025	CARDINAL
ejpam-5618	184	5	5618 6	CARDINAL
ejpam-5618	184	6	13	CARDINAL
ejpam-5618	184	7	1	CARDINAL
ejpam-5618	188	1	g ∈	PERSON
ejpam-5618	190	1	ga	ORG
ejpam-5618	191	1	ga	ORG
ejpam-5618	194	1	9	CARDINAL
ejpam-5618	195	1	ab = ∪b∈bab	ORG
ejpam-5618	195	2	ab	ORG
ejpam-5618	197	1	10	CARDINAL
ejpam-5618	199	1	inv(x	ORG
ejpam-5618	200	1	x−1	PERSON
ejpam-5618	206	1	2	CARDINAL
ejpam-5618	209	1	3	CARDINAL
ejpam-5618	211	1	u ⊂ a. proof.	ORG
ejpam-5618	212	1	2	CARDINAL
ejpam-5618	215	1	u ⊂ a. theorem 11	ORG
ejpam-5618	216	1	ga	ORG
ejpam-5618	220	1	1	CARDINAL
ejpam-5618	221	1	2	CARDINAL
ejpam-5618	221	2	12	CARDINAL
ejpam-5618	225	1	s. alammar	PERSON
ejpam-5618	225	2	m. al shumrani	PERSON
ejpam-5618	226	1	j. pure	PERSON
ejpam-5618	226	2	18	CARDINAL
ejpam-5618	226	3	1	CARDINAL
ejpam-5618	226	4	2025	CARDINAL
ejpam-5618	226	5	5618 7	DATE
ejpam-5618	226	6	13	CARDINAL
ejpam-5618	227	1	h(x	GPE
ejpam-5618	228	1	first	ORDINAL
ejpam-5618	230	1	g×g	ORG
ejpam-5618	231	1	h−1(u	ORG
ejpam-5618	231	2	u	NORP
ejpam-5618	231	3	inv−1(v	GPE
ejpam-5618	232	1	inv−1(v	GPE
ejpam-5618	233	1	f(x	ORG
ejpam-5618	236	1	3	CARDINAL
ejpam-5618	237	1	13	CARDINAL
ejpam-5618	240	1	2 ⊂	PERCENT
ejpam-5618	241	1	−1 ⊂	PERCENT
ejpam-5618	243	1	⊂	GPE
ejpam-5618	244	1	w	GPE
ejpam-5618	244	2	∩	PERSON
ejpam-5618	246	1	e. (ii	PERSON
ejpam-5618	246	2	∩	PERSON
ejpam-5618	246	3	w ⊂ u	ORG
ejpam-5618	248	1	w	ORG
ejpam-5618	249	1	14	DATE
ejpam-5618	251	1	2 ⊂	PERCENT
ejpam-5618	252	1	−1 ⊂	PERCENT
ejpam-5618	254	1	⊂	GPE
ejpam-5618	255	1	g×g	ORG
ejpam-5618	257	1	⊂	GPE
ejpam-5618	259	1	13	CARDINAL
ejpam-5618	261	1	12	CARDINAL
ejpam-5618	261	2	f(x	ORG
ejpam-5618	261	3	= xy−1	PERSON
ejpam-5618	262	1	g×g	ORG
ejpam-5618	263	1	⊂	GPE
ejpam-5618	265	1	1	CARDINAL
ejpam-5618	266	1	⊂	GPE
ejpam-5618	267	1	⊂ f(a×b	ORG
ejpam-5618	268	1	⊂	GPE
ejpam-5618	269	1	s. alammar	PERSON
ejpam-5618	269	2	m. al shumrani	PERSON
ejpam-5618	270	1	j. pure	PERSON
ejpam-5618	270	2	18	CARDINAL
ejpam-5618	270	3	1	CARDINAL
ejpam-5618	270	4	2025	CARDINAL
ejpam-5618	270	5	5618 8 of 13	DATE
ejpam-5618	270	6	⊂	GPE
ejpam-5618	273	1	rg−1 ◦	ORG
ejpam-5618	276	1	2	CARDINAL
ejpam-5618	277	1	rg−1 ◦	LOC
ejpam-5618	278	1	⊂	GPE
ejpam-5618	279	1	⊂	GPE
ejpam-5618	279	2	15	CARDINAL
ejpam-5618	283	1	x−1y	PERSON
ejpam-5618	286	1	x−1y	ORG
ejpam-5618	289	1	16	CARDINAL
ejpam-5618	292	1	g −→ g.	PERSON
ejpam-5618	293	1	⊂	GPE
ejpam-5618	294	1	u ⊂ w	ORG
ejpam-5618	295	1	⊂	GPE
ejpam-5618	296	1	⊂	GPE
ejpam-5618	297	1	17	CARDINAL
ejpam-5618	300	1	e. proof.	ORG
ejpam-5618	301	1	g −→ g.	PERSON
ejpam-5618	303	1	u ⊂ w	ORG
ejpam-5618	304	1	⊂	GPE
ejpam-5618	306	1	e. theorem 18	ORG
ejpam-5618	308	1	f(u	GPE
ejpam-5618	311	1	∈	PRODUCT
ejpam-5618	312	1	yv ⊂	ORG
ejpam-5618	312	2	f(u	GPE
ejpam-5618	315	1	yv ⊂	ORG
ejpam-5618	315	2	xu	PERSON
ejpam-5618	316	1	x. s. alammar	PERSON
ejpam-5618	316	2	m. al shumrani	PERSON
ejpam-5618	317	1	j. pure	PERSON
ejpam-5618	317	2	18	CARDINAL
ejpam-5618	317	3	1	CARDINAL
ejpam-5618	317	4	2025	CARDINAL
ejpam-5618	317	5	5618 9	DATE
ejpam-5618	317	6	13	CARDINAL
ejpam-5618	317	7	19	DATE
ejpam-5618	322	1	∈	PRODUCT
ejpam-5618	323	1	ofh	GPE
ejpam-5618	323	2	yv ⊂	ORG
ejpam-5618	324	1	f(u	GPE
ejpam-5618	325	1	xu	PERSON
ejpam-5618	327	1	yv ⊂	ORG
ejpam-5618	327	2	x.	GPE
ejpam-5618	328	1	20	CARDINAL
ejpam-5618	330	1	u ⊂ h.	ORG
ejpam-5618	330	2	∈	ORG
ejpam-5618	330	3	= hu	PERSON
ejpam-5618	331	1	⋃	DATE
ejpam-5618	335	1	6	CARDINAL
ejpam-5618	336	1	example 1	DATE
ejpam-5618	337	1	r. moreover	PERSON
ejpam-5618	339	1	21	CARDINAL
ejpam-5618	343	1	first	ORDINAL
ejpam-5618	344	1	w	GPE
ejpam-5618	348	1	h × h	ORG
ejpam-5618	348	2	6	CARDINAL
ejpam-5618	349	1	h × h → g	ORG
ejpam-5618	351	1	⊂	GPE
ejpam-5618	352	1	5	CARDINAL
ejpam-5618	356	1	s. alammar	PERSON
ejpam-5618	356	2	m. al shumrani	PERSON
ejpam-5618	357	1	j. pure	PERSON
ejpam-5618	357	2	18	CARDINAL
ejpam-5618	357	3	1	CARDINAL
ejpam-5618	357	4	2025	CARDINAL
ejpam-5618	357	5	5618 10	DATE
ejpam-5618	357	6	13	CARDINAL
ejpam-5618	357	7	22	DATE
ejpam-5618	359	1	1	CARDINAL
ejpam-5618	360	1	g ∈	PERSON
ejpam-5618	361	1	g. corollary 5	PERSON
ejpam-5618	362	1	g. 4	GPE
ejpam-5618	366	1	15	CARDINAL
ejpam-5618	368	1	x.	GPE
ejpam-5618	368	2	16	DATE
ejpam-5618	369	1	x.	ORG
ejpam-5618	370	1	7	CARDINAL
ejpam-5618	373	1	8	CARDINAL
ejpam-5618	377	1	9	CARDINAL
ejpam-5618	377	2	0	CARDINAL
ejpam-5618	377	3	1	CARDINAL
ejpam-5618	377	4	integers mod 2	PERSON
ejpam-5618	378	1	1	CARDINAL
ejpam-5618	382	1	10	CARDINAL
ejpam-5618	383	1	example 1	DATE
ejpam-5618	386	1	17	CARDINAL
ejpam-5618	387	1	x∈c⊂s	PERSON
ejpam-5618	388	1	x. s. alammar	PERSON
ejpam-5618	388	2	m. al shumrani	PERSON
ejpam-5618	389	1	j. pure	PERSON
ejpam-5618	389	2	18	CARDINAL
ejpam-5618	389	3	1	CARDINAL
ejpam-5618	389	4	2025	CARDINAL
ejpam-5618	389	5	5618 11	DATE
ejpam-5618	389	6	13	CARDINAL
ejpam-5618	389	7	18	DATE
ejpam-5618	397	1	two	CARDINAL
ejpam-5618	397	2	∩	PERSON
ejpam-5618	400	1	4	CARDINAL
ejpam-5618	402	1	24	CARDINAL
ejpam-5618	405	1	∈	ORG
ejpam-5618	406	1	23	CARDINAL
ejpam-5618	407	1	a−1ha ⊂ h or	ORG
ejpam-5618	407	2	⊂	GPE
ejpam-5618	407	3	aha−1 = h. hence	PERSON
ejpam-5618	407	4	25	DATE
ejpam-5618	413	1	26	CARDINAL
ejpam-5618	417	1	y = ut	PERSON
ejpam-5618	417	2	x−1	PERSON
ejpam-5618	417	3	−1	DATE
ejpam-5618	419	1	x−1	PERSON
ejpam-5618	419	2	g. corollary 6	PERSON
ejpam-5618	423	1	26	CARDINAL
ejpam-5618	423	2	5	CARDINAL
ejpam-5618	424	1	s. alammar	PERSON
ejpam-5618	424	2	m. al shumrani	PERSON
ejpam-5618	425	1	j. pure	PERSON
ejpam-5618	425	2	18	CARDINAL
ejpam-5618	425	3	1	CARDINAL
ejpam-5618	425	4	2025	CARDINAL
ejpam-5618	425	5	5618 12	DATE
ejpam-5618	425	6	13	CARDINAL
ejpam-5618	425	7	2	CARDINAL
ejpam-5618	427	1	u n. proof	PERSON
ejpam-5618	428	1	3	CARDINAL
ejpam-5618	428	2	⊂	GPE
ejpam-5618	429	1	6	CARDINAL
ejpam-5618	430	1	u n. theorem 27	ORG
ejpam-5618	436	1	g−h	PERSON
ejpam-5618	437	1	28	CARDINAL
ejpam-5618	440	1	g. proof	PERSON
ejpam-5618	443	1	⊂	GPE
ejpam-5618	443	2	∈	ORG
ejpam-5618	445	1	∈	ORG
ejpam-5618	445	2	∈	ORG
ejpam-5618	445	3	= h. therefore	PERSON
ejpam-5618	447	1	27	CARDINAL
ejpam-5618	448	1	5	CARDINAL
ejpam-5618	457	1	s. alammar	PERSON
ejpam-5618	457	2	m. al shumrani	PERSON
ejpam-5618	458	1	j. pure	PERSON
ejpam-5618	458	2	18	CARDINAL
ejpam-5618	458	3	1	CARDINAL
ejpam-5618	458	4	2025	CARDINAL
ejpam-5618	458	5	5618 13	DATE
ejpam-5618	458	6	13	CARDINAL
ejpam-5618	459	1	1	CARDINAL
ejpam-5618	459	2	abd el-monsef	PERSON
ejpam-5618	461	1	kyungpook mathematical journal	ORG
ejpam-5618	461	2	1992	DATE
ejpam-5618	462	1	2	CARDINAL
ejpam-5618	462	2	ma alshumrani	PERSON
ejpam-5618	464	1	phd thesis	PERSON
ejpam-5618	464	2	msc thesis	PERSON
ejpam-5618	464	3	king abdulaziz university	ORG
ejpam-5618	464	4	2024	DATE
ejpam-5618	465	1	3	CARDINAL
ejpam-5618	465	2	arhangel’skĭı and p.j	ORG
ejpam-5618	465	3	collins	PERSON
ejpam-5618	467	1	64(3):219–241, 1995	DATE
ejpam-5618	468	1	4	CARDINAL
ejpam-5618	468	2	mikhail tkachenko	PERSON
ejpam-5618	469	1	1	CARDINAL
ejpam-5618	469	2	2008	DATE
ejpam-5618	470	1	5	CARDINAL
ejpam-5618	470	2	julian dontchev	PERSON
ejpam-5618	472	1	the new york academy of sciences	ORG
ejpam-5618	472	2	767(1):28–38, 1995	DATE
ejpam-5618	473	1	6	CARDINAL
ejpam-5618	474	1	sci.	ORG
ejpam-5618	474	2	warszawa	PERSON
ejpam-5618	474	3	1977	DATE
ejpam-5618	475	1	7	CARDINAL
ejpam-5618	475	2	edwin hewitt	PERSON
ejpam-5618	478	1	j.	PERSON
ejpam-5618	478	2	10:309–333, 1943	DATE
ejpam-5618	479	1	8	CARDINAL
ejpam-5618	479	2	s. hadi jafari	PERSON
ejpam-5618	479	3	neelamegarajan rajesh	PERSON
ejpam-5618	481	1	vixra	PERSON
ejpam-5618	481	2	2020	DATE
ejpam-5618	482	1	9	CARDINAL
ejpam-5618	482	2	hamlett	PERSON
ejpam-5618	484	1	american	NORP
ejpam-5618	484	2	monthly	DATE
ejpam-5618	484	3	97(4):295–310, 1990	DATE
ejpam-5618	485	1	10	CARDINAL
ejpam-5618	485	2	hamlett	PERSON
ejpam-5618	488	1	un	ORG
ejpam-5618	488	2	7(6-b):453–465, 1992	DATE
ejpam-5618	489	1	11	CARDINAL
ejpam-5618	489	2	k. kuratowski	PERSON
ejpam-5618	490	1	1	CARDINAL
ejpam-5618	491	1	new york	GPE
ejpam-5618	491	2	1966	DATE
ejpam-5618	492	1	12	CARDINAL
ejpam-5618	493	1	2000	DATE
ejpam-5618	494	1	13	CARDINAL
ejpam-5618	496	1	the indian academy of sciences-section	ORG
ejpam-5618	496	2	20	CARDINAL
ejpam-5618	496	3	india	GPE
ejpam-5618	496	4	1944	DATE