id sid eid entity type american_scientific_journal-707 1 1 american scientific research journal ORG american_scientific_journal-707 1 2 issn PERSON american_scientific_journal-707 1 3 2313 CARDINAL american_scientific_journal-707 1 4 2313 CARDINAL american_scientific_journal-707 1 5 addis* GPE american_scientific_journal-707 1 6 dilla university ORG american_scientific_journal-707 1 7 dilla PERSON american_scientific_journal-707 1 8 ethiopia GPE american_scientific_journal-707 1 9 homomorphisms ORG american_scientific_journal-707 3 1 1. CARDINAL american_scientific_journal-707 3 2 𝑔(π‘Ž + ker 𝑓 PERSON american_scientific_journal-707 7 1 3 CARDINAL american_scientific_journal-707 7 2 𝑓:𝐴 ORG american_scientific_journal-707 7 3 ∈ 𝐴 Γ— 𝐴 ∢ 𝑓(π‘Ž ORG american_scientific_journal-707 9 1 61 CARDINAL american_scientific_journal-707 9 2 http://asrjetsjournal.org/ mailto:buttu412@yahoo.com american scientific research journal ORG american_scientific_journal-707 9 3 2014 DATE american_scientific_journal-707 9 4 10 CARDINAL american_scientific_journal-707 9 5 61-73 CARDINAL american_scientific_journal-707 11 1 8 CARDINAL american_scientific_journal-707 12 1 5 CARDINAL american_scientific_journal-707 12 2 9 DATE american_scientific_journal-707 12 3 10 CARDINAL american_scientific_journal-707 13 1 subgroups(or ORG american_scientific_journal-707 15 1 one CARDINAL american_scientific_journal-707 16 1 one CARDINAL american_scientific_journal-707 16 2 only one CARDINAL american_scientific_journal-707 17 1 2 CARDINAL american_scientific_journal-707 18 1 𝐾𝑓 ORG american_scientific_journal-707 18 2 𝐴 Γ— 𝐴 ∢ 𝑓(π‘₯ ORG american_scientific_journal-707 18 3 𝐾𝑓 ORG american_scientific_journal-707 18 4 6 CARDINAL american_scientific_journal-707 18 5 11 CARDINAL american_scientific_journal-707 19 1 β„Ž(π‘Ž ORG american_scientific_journal-707 20 1 𝐾𝑓 ORG american_scientific_journal-707 21 1 first ORDINAL american_scientific_journal-707 21 2 algebras ORG american_scientific_journal-707 21 3 two CARDINAL american_scientific_journal-707 21 4 62 CARDINAL american_scientific_journal-707 21 5 american NORP american_scientific_journal-707 21 6 2014 DATE american_scientific_journal-707 21 7 10 CARDINAL american_scientific_journal-707 21 8 61-73 CARDINAL american_scientific_journal-707 22 1 homomorphisms ORG american_scientific_journal-707 23 1 3 CARDINAL american_scientific_journal-707 23 2 3.1 CARDINAL american_scientific_journal-707 24 1 algebras ORG american_scientific_journal-707 24 2 3.1.1 CARDINAL american_scientific_journal-707 25 1 fora nonempty ORG american_scientific_journal-707 25 2 𝐴0 ORG american_scientific_journal-707 25 3 𝐴𝑛is ORG american_scientific_journal-707 25 4 arity(or GPE american_scientific_journal-707 27 1 𝑓(π‘Ž1 ORG american_scientific_journal-707 30 1 𝐴is PERSON american_scientific_journal-707 30 2 zero CARDINAL american_scientific_journal-707 30 3 𝑓(βˆ… PERSON american_scientific_journal-707 30 4 𝐴0 DATE american_scientific_journal-707 30 5 𝑓(βˆ… PERSON american_scientific_journal-707 31 1 1 CARDINAL american_scientific_journal-707 31 2 2 DATE american_scientific_journal-707 31 3 3 CARDINAL american_scientific_journal-707 32 1 1 CARDINAL american_scientific_journal-707 33 1 algebras ORG american_scientific_journal-707 36 1 algebras ORG american_scientific_journal-707 36 2 𝐴,𝐹 CARDINAL american_scientific_journal-707 36 3 𝑓1 ORG american_scientific_journal-707 38 1 〈𝐴 PERSON american_scientific_journal-707 38 2 𝑓1 ORG american_scientific_journal-707 40 1 𝐴,𝐹 CARDINAL american_scientific_journal-707 40 2 3.2 CARDINAL american_scientific_journal-707 41 1 homomorphisms ORG american_scientific_journal-707 41 2 algebras ORG american_scientific_journal-707 41 3 3.2.1 CARDINAL american_scientific_journal-707 41 4 β„±. GPE american_scientific_journal-707 42 1 1 CARDINAL american_scientific_journal-707 42 2 4 &7] i. ORG american_scientific_journal-707 42 3 ∈ ORG american_scientific_journal-707 44 1 ∈ ORG american_scientific_journal-707 44 2 symbol,𝑛 ORG american_scientific_journal-707 45 1 ∈ 𝐴 ORG american_scientific_journal-707 45 2 𝛼(𝑓𝐴(π‘Ž1 ORG american_scientific_journal-707 47 1 3.2.1 CARDINAL american_scientific_journal-707 48 1 7 CARDINAL american_scientific_journal-707 49 1 63 CARDINAL american_scientific_journal-707 49 2 american scientific research journal ORG american_scientific_journal-707 49 3 2014 DATE american_scientific_journal-707 49 4 10 CARDINAL american_scientific_journal-707 49 5 61-73 CARDINAL american_scientific_journal-707 50 1 let𝑓 ∈ PERSON american_scientific_journal-707 51 1 ∈ 𝐴. ORG american_scientific_journal-707 57 1 ∘ 𝛼(π‘Ž1 DATE american_scientific_journal-707 58 1 𝐢is PERSON american_scientific_journal-707 59 1 3.2.2 CARDINAL american_scientific_journal-707 60 1 i. 𝛼 PERSON american_scientific_journal-707 62 1 𝛼 PERSON american_scientific_journal-707 64 1 homomorphism.[1 NORP american_scientific_journal-707 64 2 6 CARDINAL american_scientific_journal-707 64 3 3.3 CARDINAL american_scientific_journal-707 65 1 3.3.1 CARDINAL american_scientific_journal-707 66 1 𝑏2 GPE american_scientific_journal-707 67 1 ∈ 𝐴 ORG american_scientific_journal-707 70 1 𝑏2 GPE american_scientific_journal-707 73 1 1 CARDINAL american_scientific_journal-707 73 2 2 DATE american_scientific_journal-707 74 1 two CARDINAL american_scientific_journal-707 75 1 3.3.1 CARDINAL american_scientific_journal-707 78 1 ∈ 𝐴 Γ— 𝐴 ∢ 𝛼(π‘Ž ORG american_scientific_journal-707 78 2 𝛼(𝑏 PERSON american_scientific_journal-707 79 1 1 CARDINAL american_scientific_journal-707 79 2 2 DATE american_scientific_journal-707 79 3 6 CARDINAL american_scientific_journal-707 79 4 64 CARDINAL american_scientific_journal-707 79 5 american scientific research journal ORG american_scientific_journal-707 79 6 2014 DATE american_scientific_journal-707 79 7 10 CARDINAL american_scientific_journal-707 79 8 61-73 CARDINAL american_scientific_journal-707 83 1 𝑏2 GPE american_scientific_journal-707 84 1 ∈ 𝐴 ORG american_scientific_journal-707 89 1 𝑏2 GPE american_scientific_journal-707 92 1 𝑏2 GPE american_scientific_journal-707 93 1 ker𝛼 ORG american_scientific_journal-707 93 2 3.3.2 CARDINAL american_scientific_journal-707 94 1 ker𝛼. TIME american_scientific_journal-707 95 1 2 CARDINAL american_scientific_journal-707 95 2 ∈ 𝐴 Γ— 𝐴 ∢ 𝛼(π‘Ž ORG american_scientific_journal-707 96 1 𝛼(𝑏 PERSON american_scientific_journal-707 96 2 𝑏 ∈ 𝐴 ORG american_scientific_journal-707 97 1 β„±. GPE american_scientific_journal-707 97 2 𝛼(π‘Ž ORG american_scientific_journal-707 97 3 ∈ 𝐴 Γ— 𝐴 ∢ 𝛼(π‘Ž ORG american_scientific_journal-707 98 1 𝛼(𝑏 PERSON american_scientific_journal-707 98 2 ∈ 𝐴 Γ— 𝐴 ∢ ORG american_scientific_journal-707 98 3 ∈ 𝐴 Γ— 𝐴 ORG american_scientific_journal-707 98 4 ker𝛼 = πœƒ. ∎ ORG american_scientific_journal-707 98 5 two CARDINAL american_scientific_journal-707 99 1 4 CARDINAL american_scientific_journal-707 100 1 algebras 4.1 LAW american_scientific_journal-707 101 1 4.1.1 CARDINAL american_scientific_journal-707 101 2 β„±. let𝑔 PERSON american_scientific_journal-707 101 3 𝐴 β†’ 𝐡 ORG american_scientific_journal-707 101 4 65 CARDINAL american_scientific_journal-707 101 5 american scientific research journal ORG american_scientific_journal-707 101 6 2014 DATE american_scientific_journal-707 101 7 10 CARDINAL american_scientific_journal-707 101 8 61-73 CARDINAL american_scientific_journal-707 102 1 two CARDINAL american_scientific_journal-707 102 2 𝐢 β†’ 𝐴 ORG american_scientific_journal-707 102 3 2 CARDINAL american_scientific_journal-707 103 1 two CARDINAL american_scientific_journal-707 103 2 one CARDINAL american_scientific_journal-707 103 3 two CARDINAL american_scientific_journal-707 104 1 4.2 CARDINAL american_scientific_journal-707 105 1 algebras ORG american_scientific_journal-707 105 2 algebras ORG american_scientific_journal-707 105 3 𝐴 ORG american_scientific_journal-707 106 1 1 CARDINAL american_scientific_journal-707 106 2 𝑔:𝑋 ORG american_scientific_journal-707 106 3 𝑖. 𝑒. PERSON american_scientific_journal-707 108 1 homomorphisms ORG american_scientific_journal-707 108 2 2 CARDINAL american_scientific_journal-707 109 1 two CARDINAL american_scientific_journal-707 109 2 πœ‹ PERSON american_scientific_journal-707 109 3 𝑔:𝑋 ORG american_scientific_journal-707 109 4 66 CARDINAL american_scientific_journal-707 109 5 american scientific research journal ORG american_scientific_journal-707 109 6 2014 DATE american_scientific_journal-707 109 7 10 CARDINAL american_scientific_journal-707 109 8 61-73 CARDINAL american_scientific_journal-707 109 9 1 CARDINAL american_scientific_journal-707 109 10 ∈ 𝐴 Γ— 𝐴 ∢ πœ‹(π‘Ž ORG american_scientific_journal-707 111 1 π‘Ž ∈ 𝐴 ORG american_scientific_journal-707 111 2 β„±. GPE american_scientific_journal-707 111 3 𝛼(π‘Ž ORG american_scientific_journal-707 111 4 𝑔:𝐴 ORG american_scientific_journal-707 111 5 ∘ β„Ž(π‘Ž PERSON american_scientific_journal-707 111 6 πœ‹(π‘Ž PERSON american_scientific_journal-707 112 1 2 CARDINAL american_scientific_journal-707 113 1 two CARDINAL american_scientific_journal-707 113 2 πœ‹ PERSON american_scientific_journal-707 113 3 𝑔:𝑋 ORG american_scientific_journal-707 114 1 1:β„Ž~β„Žβ€² CARDINAL american_scientific_journal-707 114 2 𝑔~𝑔′ ORG american_scientific_journal-707 114 3 3 CARDINAL american_scientific_journal-707 114 4 two CARDINAL american_scientific_journal-707 114 5 πœ‹ ORG american_scientific_journal-707 115 1 π‘Œ ORG american_scientific_journal-707 116 1 67 CARDINAL american_scientific_journal-707 116 2 american scientific research journal ORG american_scientific_journal-707 116 3 2014 DATE american_scientific_journal-707 116 4 10 CARDINAL american_scientific_journal-707 116 5 61-73 CARDINAL american_scientific_journal-707 116 6 that𝑦1 DATE american_scientific_journal-707 118 1 ∘ DATE american_scientific_journal-707 120 1 andπ‘Ž2 ∈ 𝐴. ORG american_scientific_journal-707 120 2 𝛼(𝑦1 ORG american_scientific_journal-707 121 1 𝑔′ PRODUCT american_scientific_journal-707 123 1 68 CARDINAL american_scientific_journal-707 123 2 american scientific research journal ORG american_scientific_journal-707 123 3 2014 DATE american_scientific_journal-707 123 4 10 CARDINAL american_scientific_journal-707 123 5 61-73 CARDINAL american_scientific_journal-707 123 6 ∈ ORG american_scientific_journal-707 128 1 ∈ π‘Œ ORG american_scientific_journal-707 128 2 ∈ 𝐴. ORG american_scientific_journal-707 131 1 symbol,𝑛 ORG american_scientific_journal-707 133 1 ∈ 𝐴 ORG american_scientific_journal-707 147 1 π‘Ž ∈ 𝐴 ORG american_scientific_journal-707 147 2 69 CARDINAL american_scientific_journal-707 147 3 american scientific research journal ORG american_scientific_journal-707 147 4 2014 DATE american_scientific_journal-707 147 5 10 CARDINAL american_scientific_journal-707 147 6 61-73 CARDINAL american_scientific_journal-707 151 1 2 CARDINAL american_scientific_journal-707 151 2 4 CARDINAL american_scientific_journal-707 151 3 two CARDINAL american_scientific_journal-707 151 4 πœ‹ PRODUCT american_scientific_journal-707 151 5 1 CARDINAL american_scientific_journal-707 151 6 𝐴and 𝛼:π‘Œ β†’ 𝑋 ORG american_scientific_journal-707 151 7 π‘Ž ∈ 𝐴 ORG american_scientific_journal-707 153 1 π‘Ž ∈ 𝐴 ORG american_scientific_journal-707 154 1 𝐼𝐡 ∘ 𝑔′(𝑦 ORG american_scientific_journal-707 156 1 70 CARDINAL american_scientific_journal-707 156 2 american scientific research journal ORG american_scientific_journal-707 156 3 2014 DATE american_scientific_journal-707 156 4 10 CARDINAL american_scientific_journal-707 156 5 61-73 CARDINAL american_scientific_journal-707 157 1 ∘ CARDINAL american_scientific_journal-707 157 2 ∘ CARDINAL american_scientific_journal-707 158 1 two CARDINAL american_scientific_journal-707 158 2 𝐼𝐡 ORG american_scientific_journal-707 158 3 5 CARDINAL american_scientific_journal-707 158 4 homomorphisms PERSON american_scientific_journal-707 158 5 𝑔′ PRODUCT american_scientific_journal-707 158 6 ∘ 𝛼. therefore PERSON american_scientific_journal-707 158 7 𝑔~𝑔′ ORG american_scientific_journal-707 160 1 𝑔:𝑋 β†’ π΅π‘–π‘ π‘Ž ORG american_scientific_journal-707 160 2 β„±. PERSON american_scientific_journal-707 161 1 ∈ 𝐴 Γ— 𝐴 ∢ πœ‹(π‘Ž ORG american_scientific_journal-707 163 1 π‘Ž ∈ 𝐴 ORG american_scientific_journal-707 163 2 𝐡defined PERSON american_scientific_journal-707 163 3 algebras ORG american_scientific_journal-707 163 4 𝛼:𝑋 β†’ 𝐴 πœƒοΏ½ 71 american scientific research journal ORG american_scientific_journal-707 163 5 2014 DATE american_scientific_journal-707 163 6 10 CARDINAL american_scientific_journal-707 163 7 61-73 CARDINAL american_scientific_journal-707 163 8 6 CARDINAL american_scientific_journal-707 163 9 two CARDINAL american_scientific_journal-707 164 1 5 CARDINAL american_scientific_journal-707 164 2 algebras ORG american_scientific_journal-707 164 3 πœ‹ PERSON american_scientific_journal-707 164 4 algebras ORG american_scientific_journal-707 164 5 one CARDINAL american_scientific_journal-707 164 6 πœ‹ PERSON american_scientific_journal-707 164 7 𝑔′ PRODUCT american_scientific_journal-707 164 8 𝑔~𝑔′ ORG american_scientific_journal-707 164 9 β„Ž~β„Žβ€² ORG american_scientific_journal-707 165 1 two CARDINAL american_scientific_journal-707 165 2 two CARDINAL american_scientific_journal-707 165 3 ∘ CARDINAL american_scientific_journal-707 165 4 two CARDINAL american_scientific_journal-707 165 5 ⇔ ORG american_scientific_journal-707 165 6 ∘ DATE american_scientific_journal-707 165 7 ⇔ PERSON american_scientific_journal-707 165 8 ∘ DATE american_scientific_journal-707 165 9 two CARDINAL american_scientific_journal-707 165 10 one CARDINAL american_scientific_journal-707 165 11 two CARDINAL american_scientific_journal-707 167 1 algebras of ORG american_scientific_journal-707 170 1 u. m. swamy PERSON american_scientific_journal-707 173 1 72 CARDINAL american_scientific_journal-707 173 2 american scientific research journal ORG american_scientific_journal-707 173 3 2014 DATE american_scientific_journal-707 173 4 10 CARDINAL american_scientific_journal-707 173 5 61-73 CARDINAL american_scientific_journal-707 177 1 1 CARDINAL american_scientific_journal-707 177 2 burris PERSON american_scientific_journal-707 177 3 h.p. GPE american_scientific_journal-707 177 4 1981 DATE american_scientific_journal-707 177 5 2 CARDINAL american_scientific_journal-707 177 6 clifford bergman PERSON american_scientific_journal-707 177 7 crc press ORG american_scientific_journal-707 177 8 2011 DATE american_scientific_journal-707 177 9 3 CARDINAL american_scientific_journal-707 177 10 2013 DATE american_scientific_journal-707 177 11 4 CARDINAL american_scientific_journal-707 177 12 w. h. freeman PERSON american_scientific_journal-707 177 13 san francisco GPE american_scientific_journal-707 177 14 1980 DATE american_scientific_journal-707 178 1 5 CARDINAL american_scientific_journal-707 178 2 jonathan k. hodge PERSON american_scientific_journal-707 178 3 steven schlicker PERSON american_scientific_journal-707 178 4 crc press ORG american_scientific_journal-707 178 5 2013 DATE american_scientific_journal-707 178 6 6 CARDINAL american_scientific_journal-707 178 7 shelly l. wismath PERSON american_scientific_journal-707 178 8 2009 DATE american_scientific_journal-707 178 9 7 CARDINAL american_scientific_journal-707 178 10 shelly l. wismath PERSON american_scientific_journal-707 178 11 crc press ORG american_scientific_journal-707 178 12 2002 DATE american_scientific_journal-707 179 1 47-63 CARDINAL american_scientific_journal-707 179 2 8 CARDINAL american_scientific_journal-707 179 3 l. szabΓ³ PERSON american_scientific_journal-707 179 4 a. szendreilectures PERSON american_scientific_journal-707 179 5 1 CARDINAL american_scientific_journal-707 179 6 janos bolyai PERSON american_scientific_journal-707 179 7 2014 DATE american_scientific_journal-707 180 1 9 CARDINAL american_scientific_journal-707 180 2 paul b. garrett PERSON american_scientific_journal-707 180 3 crc press ORG american_scientific_journal-707 180 4 2007 DATE american_scientific_journal-707 180 5 page25-56 ORG american_scientific_journal-707 181 1 10 CARDINAL american_scientific_journal-707 181 2 peter hilton PERSON american_scientific_journal-707 181 3 yel-chiang wu PERSON american_scientific_journal-707 181 4 12 CARDINAL american_scientific_journal-707 181 5 wiley ORG american_scientific_journal-707 181 6 john wiley & sons ORG american_scientific_journal-707 181 7 1989 DATE american_scientific_journal-707 181 8 11 CARDINAL american_scientific_journal-707 181 9 pierce ORG american_scientific_journal-707 181 10 r.s. PERSON american_scientific_journal-707 181 11 rinehart GPE american_scientific_journal-707 181 12 winston GPE american_scientific_journal-707 181 13 new york GPE american_scientific_journal-707 181 14 1968 73 DATE american_scientific_journal-707 181 15 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22clifford+bergman%22&source=gbs_metadata_r&cad=5 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22klaus+denecke%22&source=gbs_metadata_r&cad=5 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22shelly+l.+wismath%22&source=gbs_metadata_r&cad=5 ORG american_scientific_journal-707 181 16 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22l.+szab%c3%b3%22&source=gbs_metadata_r&cad=5 GPE american_scientific_journal-707 181 17 https://www.google.com.et/search?tbo=p&tbm=bks&q=bibliogroup:%22colloquia+mathematica+societatis+janos+bolyai%22&source=gbs_metadata_r&cad=5 GPE american_scientific_journal-707 181 18 https://www.google.com.et/search?tbo=p&tbm=bks&q=bibliogroup:%22colloquia+mathematica+societatis+janos+bolyai%22&source=gbs_metadata_r&cad=5 GPE american_scientific_journal-707 181 19 https://www.google.com.au/search?tbo=p&tbm=bks&q=inauthor:%22paul+b.+garrett%22&source=gbs_metadata_r&cad=4 NORP