id sid eid entity type easat-3854 1 1 2576-8484 DATE easat-3854 1 2 8 CARDINAL easat-3854 1 3 6 CARDINAL easat-3854 1 4 8658-8666 2024 DATE easat-3854 1 5 10.55214/25768484.v8i6.3854 CARDINAL easat-3854 1 6 2024 CARDINAL easat-3854 1 7 2024 CARDINAL easat-3854 1 8 a_∞-algebras alaa hassan noreldeen1 PERSON easat-3854 1 9 fatma ahmed mohammed2* PERSON easat-3854 1 10 karar3 1,2,3department FAC easat-3854 1 11 science ORG easat-3854 1 12 aswan university ORG easat-3854 1 13 aswan GPE easat-3854 1 14 egypt GPE easat-3854 10 1 keywords: a_∞algebras ORG easat-3854 10 2 adams operations PERSON easat-3854 11 1 jel classification: ORG easat-3854 11 2 55n 91 DATE easat-3854 11 3 55n20 CARDINAL easat-3854 11 4 16e40 CARDINAL easat-3854 12 1 1 CARDINAL easat-3854 12 2 first ORDINAL easat-3854 12 3 the late 1930s DATE easat-3854 15 1 sullivan PERSON easat-3854 17 1 steenrod operations modulo𝛼 EVENT easat-3854 19 1 two CARDINAL easat-3854 20 1 1 CARDINAL easat-3854 20 2 2 CARDINAL easat-3854 21 1 steenrod GPE easat-3854 23 1 rost nilpotence ORG easat-3854 25 1 one CARDINAL easat-3854 26 1 8659 DATE easat-3854 26 2 2576-8484 DATE easat-3854 26 3 8 CARDINAL easat-3854 27 1 6 CARDINAL easat-3854 27 2 8658-8666 DATE easat-3854 27 3 2024 CARDINAL easat-3854 27 4 10.55214/25768484.v8i6.3854 CARDINAL easat-3854 27 5 2024 CARDINAL easat-3854 27 6 one CARDINAL easat-3854 27 7 𝐹𝛼 PERSON easat-3854 28 1 𝐹𝛼 PERSON easat-3854 28 2 π»π‘š(βˆ’ PERSON easat-3854 28 3 𝐹𝛼 PERSON easat-3854 31 1 2 CARDINAL easat-3854 31 2 multiplicative ORG easat-3854 32 1 j.m. lopez PERSON easat-3854 33 1 3 CARDINAL easat-3854 34 1 burghelea PERSON easat-3854 34 2 4 CARDINAL easat-3854 34 3 de-rham algebras ORG easat-3854 36 1 adams PERSON easat-3854 36 2 algebras GPE easat-3854 39 1 2 CARDINAL easat-3854 39 2 π“βˆž-algebras ORG easat-3854 39 3 algebras ORG easat-3854 41 1 1960 DATE easat-3854 42 1 2013 DATE easat-3854 42 2 alaa. PERSON easat-3854 43 1 h. PERSON easat-3854 45 1 5 CARDINAL easat-3854 45 2 6 CARDINAL easat-3854 45 3 7 CARDINAL easat-3854 46 1 2.1 CARDINAL easat-3854 46 2 8 CARDINAL easat-3854 46 3 0 ≀ MONEY easat-3854 49 1 πš€π“€)Μ‚ ORG easat-3854 49 2 1 ≀ QUANTITY easat-3854 49 3 𝛾(𝜎 GPE easat-3854 50 1 0 ≀ 𝚀1 MONEY easat-3854 50 2 ∈ β„€ ORG easat-3854 52 1 1 CARDINAL easat-3854 54 1 1 CARDINAL easat-3854 55 1 0 CARDINAL easat-3854 55 2 2 CARDINAL easat-3854 56 1 8660 CARDINAL easat-3854 56 2 2576-8484 DATE easat-3854 56 3 8 CARDINAL easat-3854 57 1 6 CARDINAL easat-3854 57 2 8658-8666 DATE easat-3854 57 3 2024 CARDINAL easat-3854 57 4 10.55214/25768484.v8i6.3854 CARDINAL easat-3854 57 5 2024 CARDINAL easat-3854 57 6 3 CARDINAL easat-3854 58 1 βˆ’ πœ•(𝚀3βˆ’2)πœ•(𝚀1,𝚀2 PERSON easat-3854 58 2 βˆ’ πœ•(𝚀2βˆ’1,𝚀3βˆ’1)πœ•(𝚀1 GPE easat-3854 58 3 π’žπ“… β†’ π’žπ“…} PERSON easat-3854 59 1 0 CARDINAL easat-3854 59 2 2 CARDINAL easat-3854 59 3 𝛿𝓇𝓅 = 𝓇𝓅𝛿. PERSON easat-3854 59 4 0 ≀ MONEY easat-3854 59 5 𝟎 ( CARDINAL easat-3854 59 6 𝟎 CARDINAL easat-3854 59 7 2 CARDINAL easat-3854 59 8 βˆ’1 DATE easat-3854 59 9 2 π“‡π“…βˆ’π“†πœ•(π“…βˆ’πš€π“† DATE easat-3854 59 10 π“…βˆ’πš€1 ORG easat-3854 59 11 0 CARDINAL easat-3854 60 1 3 CARDINAL easat-3854 60 2 2.2 CARDINAL easat-3854 60 3 6 CARDINAL easat-3854 60 4 ∈ β„€ PERSON easat-3854 60 5 0 CARDINAL easat-3854 60 6 1 CARDINAL easat-3854 61 1 π›Ώπœ‘π“…βˆ’1 DATE easat-3854 61 2 𝛿(πœ‘π“…βˆ’1 GPE easat-3854 61 3 4 CARDINAL easat-3854 61 4 4 CARDINAL easat-3854 61 5 β€’ CARDINAL easat-3854 61 6 1 CARDINAL easat-3854 62 1 0 CARDINAL easat-3854 62 2 β€’ DATE easat-3854 62 3 2 CARDINAL easat-3854 62 4 β€’ CARDINAL easat-3854 62 5 3 CARDINAL easat-3854 63 1 1βŠ—πœ‘0βŠ—1+ 1βŠ—2βŠ—πœ‘0 CARDINAL easat-3854 64 1 π’œβˆž-algebra ORG easat-3854 64 2 βˆ€π“‚ CARDINAL easat-3854 65 1 𝛿(𝓂)βˆ— NORP easat-3854 65 2 πœ‘π‘›(𝓂0βŠ—π“‚1βŠ— NORP easat-3854 67 1 2576-8484 DATE easat-3854 67 2 8 CARDINAL easat-3854 68 1 6 CARDINAL easat-3854 68 2 8658-8666 DATE easat-3854 68 3 2024 CARDINAL easat-3854 68 4 10.55214/25768484.v8i6.3854 CARDINAL easat-3854 68 5 2024 CARDINAL easat-3854 68 6 πœ‰ = 𝓅(π“…βˆ’1 ORG easat-3854 68 7 2 +βˆ‘ |π“‚πš€||𝓂𝑗|0β‰€πš€<𝑗≀𝓅 CARDINAL easat-3854 68 8 0 CARDINAL easat-3854 69 1 Β±1 PERSON easat-3854 71 1 2.3 CARDINAL easat-3854 72 1 5 CARDINAL easat-3854 73 1 6 CARDINAL easat-3854 73 2 one CARDINAL easat-3854 73 3 1 CARDINAL easat-3854 76 1 2.5 CARDINAL easat-3854 79 1 𝑓(𝓂)⋆. 2.6 PERSON easat-3854 80 1 8 CARDINAL easat-3854 81 1 5 CARDINAL easat-3854 81 2 2.7 CARDINAL easat-3854 82 1 5 CARDINAL easat-3854 84 1 ∞ 𝚀=1 PERSON easat-3854 85 1 6 CARDINAL easat-3854 85 2 2.8 CARDINAL easat-3854 86 1 9 CARDINAL easat-3854 86 2 π’ž(β„³)𝑠,𝑛 CARDINAL easat-3854 86 3 π’ž(β„³)𝑠,𝑛 CARDINAL easat-3854 87 1 βˆ€π‘› NORP easat-3854 87 2 0 CARDINAL easat-3854 87 3 𝓉𝑛(𝓂0βŠ— PRODUCT easat-3854 87 4 βˆ’1 DATE easat-3854 87 5 βŠ—π“‚π‘ βˆ’1 PERSON easat-3854 88 1 8662 DATE easat-3854 88 2 2576-8484 DATE easat-3854 88 3 8 CARDINAL easat-3854 89 1 6 CARDINAL easat-3854 89 2 8658-8666 DATE easat-3854 89 3 2024 CARDINAL easat-3854 89 4 10.55214/25768484.v8i6.3854 CARDINAL easat-3854 89 5 2024 CARDINAL easat-3854 89 6 βˆ— CARDINAL easat-3854 89 7 π’ž(β„³)𝑛,𝑝 CARDINAL easat-3854 89 8 0 ≀ MONEY easat-3854 89 9 βˆ’1)𝑠(π‘βˆ’1)1βŠ—π‘—βŠ—πœ‘π‘ βˆ’1βŠ—1βŠ—(π‘›βˆ’π‘ βˆ’π‘— PERSON easat-3854 89 10 0 ≀ QUANTITY easat-3854 90 1 + 1 DATE easat-3854 91 1 1 ≀ QUANTITY easat-3854 91 2 0,1 CARDINAL easat-3854 91 3 0 CARDINAL easat-3854 93 1 β„‹π’ž(β„³ ORG easat-3854 94 1 β„‹π’ž(β„³ ORG easat-3854 95 1 7 CARDINAL easat-3854 95 2 β„‹π’ž(β„³ ORG easat-3854 95 3 β„‹π’ž(β„³ ORG easat-3854 96 1 𝐷2 ORG easat-3854 97 1 0 CARDINAL easat-3854 97 2 𝐷2 ORG easat-3854 98 1 2.9 CARDINAL easat-3854 98 2 10 CARDINAL easat-3854 99 1 2𝓅 CARDINAL easat-3854 99 2 π’Ÿπ“… PERSON easat-3854 100 1 1 CARDINAL easat-3854 101 1 two CARDINAL easat-3854 102 1 ∈ ORG easat-3854 102 2 π‘Ÿ(𝓂1βŠ—π“‚2βŠ— NORP easat-3854 104 1 π‘Ÿ(𝓂1βŠ—π“‚2βŠ— NORP easat-3854 105 1 3 CARDINAL easat-3854 105 2 π“βˆž-algebras ORG easat-3854 105 3 11 CARDINAL easat-3854 105 4 12 CARDINAL easat-3854 105 5 π“š ORG easat-3854 105 6 zero CARDINAL easat-3854 105 7 1 CARDINAL easat-3854 105 8 4 CARDINAL easat-3854 106 1 8 CARDINAL easat-3854 107 1 8663 CARDINAL easat-3854 107 2 2576-8484 DATE easat-3854 108 1 8 CARDINAL easat-3854 109 1 6 CARDINAL easat-3854 109 2 8658-8666 DATE easat-3854 109 3 2024 CARDINAL easat-3854 109 4 10.55214/25768484.v8i6.3854 CARDINAL easat-3854 109 5 2024 CARDINAL easat-3854 110 1 𝓂0βŠ—π“‚1βŠ— GPE easat-3854 110 2 𝓃0βŠ—π“ƒ1βŠ— ORG easat-3854 111 1 β„³π’Ÿ ORG easat-3854 112 1 ∈ ORG easat-3854 113 1 β„³π’Ÿ ORG easat-3854 113 2 β„³π’Ÿ ORG easat-3854 113 3 β„³π’Ÿ ORG easat-3854 113 4 βŠ— CARDINAL easat-3854 113 5 β„³π’Ÿ ORG easat-3854 115 1 βŠ— πœ…(𝓃0βŠ—π“ƒ1βŠ— PERSON easat-3854 115 2 𝓃0βŠ—π“ƒ1βŠ— ORG easat-3854 117 1 ∈ ORG easat-3854 117 2 9 CARDINAL easat-3854 118 1 β„³π’Ÿ ORG easat-3854 118 2 5 CARDINAL easat-3854 119 1 β„³π’Ÿ ORG easat-3854 120 1 structure 𝛽(β„’ DATE easat-3854 120 2 the 𝓀-module: 𝛽(β„’ DATE easat-3854 120 3 β„³π’Ÿ ORG easat-3854 120 4 𝒯(π’₯𝒦[π’Ÿπ‘ ORG easat-3854 120 5 𝒯(π’₯𝒦[π’Ÿπ‘ ORG easat-3854 121 1 the 𝒦-module 𝛽(β„’ DATE easat-3854 122 1 ∈ ORG easat-3854 122 2 ∈ ORG easat-3854 122 3 ∈ ORG easat-3854 123 1 𝛽(β„’)π‘ βˆ’1 PERSON easat-3854 123 2 𝑓: 𝛽(β„’ DATE easat-3854 123 3 β„³π’Ÿ ORG easat-3854 124 1 𝑠𝓂[𝜐1| ORG easat-3854 124 2 10 CARDINAL easat-3854 125 1 0 CARDINAL easat-3854 128 1 1 CARDINAL easat-3854 128 2 𝜎 ORG easat-3854 129 1 𝓀 βŠ— PERSON easat-3854 129 2 𝑆(𝜐[𝜐1| ORG easat-3854 130 1 β„³π’Ÿ ORG easat-3854 130 2 βŠ— 𝒦[π’Ÿπ‘] 𝛽(𝜁 ORG easat-3854 131 1 11 CARDINAL easat-3854 131 2 π»π‘œπ‘šπ’¦[π’Ÿπ‘](𝛽(𝜁 ORG easat-3854 131 3 β„³π’Ÿ ORG easat-3854 132 1 β„³π’Ÿ ORG easat-3854 132 2 β„³π’Ÿ ORG easat-3854 133 1 11 CARDINAL easat-3854 133 2 8664 DATE easat-3854 133 3 2576-8484 DATE easat-3854 133 4 8 CARDINAL easat-3854 134 1 6 CARDINAL easat-3854 134 2 8658-8666 DATE easat-3854 134 3 2024 CARDINAL easat-3854 134 4 10.55214/25768484.v8i6.3854 CARDINAL easat-3854 134 5 2024 CARDINAL easat-3854 134 6 β„³π’Ÿ ORG easat-3854 134 7 β„³Μ‚π’Ÿ DATE easat-3854 134 8 𝛽(β„’ βŠ— DATE easat-3854 134 9 𝛽(β„’)βŠ— 𝛽(𝜁 ORG easat-3854 135 1 βˆ†π’¦π’Ÿ GPE easat-3854 135 2 βŠ— 𝛽(β„’ ORG easat-3854 135 3 𝛽(β„’))⋆ ORG easat-3854 135 4 𝛽(β„’))⋆ ORG easat-3854 135 5 𝛽(β„’)βŠ— 𝛽(β„’ ORG easat-3854 135 6 12 CARDINAL easat-3854 135 7 lemma ORG easat-3854 136 1 3.1 CARDINAL easat-3854 137 1 𝒦[πœ‚]-free GPE easat-3854 138 1 β„° βŠ— ORG easat-3854 138 2 β„° βŠ— 𝛽(β„’ ORG easat-3854 139 1 β„° βŠ— 𝛽(β„’ ORG easat-3854 140 1 βŠ— PERSON easat-3854 140 2 0 CARDINAL easat-3854 140 3 ∈ ℰ𝑖 ORG easat-3854 140 4 0 CARDINAL easat-3854 141 1 β„° βŠ— 𝛽(β„’ ORG easat-3854 141 2 βŠ— 𝒷)β„― ORG easat-3854 145 1 Ο‰(β„― βŠ— 𝓍)(β„“ PERSON easat-3854 146 1 βŠ— β„“ PERSON easat-3854 146 2 ∈ ORG easat-3854 147 1 𝛽(β„’)βŠ—π“‡)⋆ GPE easat-3854 149 1 lemma ORG easat-3854 149 2 3.1 CARDINAL easat-3854 151 1 ω⋆(𝓋 CARDINAL easat-3854 151 2 βˆ’π‘–βŠ—π’Άπ“… NORP easat-3854 151 3 0 CARDINAL easat-3854 152 1 𝒫𝑖 PERSON easat-3854 152 2 1 CARDINAL easat-3854 152 3 2 CARDINAL easat-3854 152 4 ∈ ORG easat-3854 152 5 ℋ𝓆+𝑠(𝛽(β„’)⋆ PRODUCT easat-3854 153 1 2 CARDINAL easat-3854 153 2 2 CARDINAL easat-3854 153 3 βˆ’1 DATE easat-3854 153 4 ∈ ORG easat-3854 154 1 2 CARDINAL easat-3854 154 2 2ℐ CARDINAL easat-3854 154 3 3.2 CARDINAL easat-3854 155 1 8665 CARDINAL easat-3854 155 2 2576-8484 DATE easat-3854 156 1 8 CARDINAL easat-3854 157 1 6 CARDINAL easat-3854 157 2 8658-8666 DATE easat-3854 157 3 2024 CARDINAL easat-3854 157 4 10.55214/25768484.v8i6.3854 CARDINAL easat-3854 157 5 2024 CARDINAL easat-3854 157 6 𝒫𝑖 PERSON easat-3854 157 7 β„‹π’Ÿπ‘ +𝑖(β„³)πœ€ LOC easat-3854 157 8 2 CARDINAL easat-3854 157 9 𝒫𝑖 PERSON easat-3854 157 10 2 CARDINAL easat-3854 158 1 𝒫𝑖 PERSON easat-3854 159 1 1 CARDINAL easat-3854 159 2 𝒫𝑖| DATE easat-3854 159 3 2 CARDINAL easat-3854 159 4 𝒫𝑖| DATE easat-3854 160 1 2 CARDINAL easat-3854 160 2 2𝑖 CARDINAL easat-3854 161 1 2 CARDINAL easat-3854 161 2 2𝑖 CARDINAL easat-3854 161 3 2 CARDINAL easat-3854 162 1 2 CARDINAL easat-3854 162 2 2 CARDINAL easat-3854 162 3 2𝑖 CARDINAL easat-3854 162 4 3 CARDINAL easat-3854 164 1 4 CARDINAL easat-3854 164 2 𝒫𝑖 PERSON easat-3854 164 3 2 CARDINAL easat-3854 164 4 1).π”…π›Ύπ’«π“Ž+π’·βˆ’π‘–π’«π‘– CARDINAL easat-3854 164 5 0 CARDINAL easat-3854 164 6 1 π‘“π‘œπ‘Ÿ QUANTITY easat-3854 164 7 2 CARDINAL easat-3854 164 8 2 CARDINAL easat-3854 164 9 two CARDINAL easat-3854 167 1 0 CARDINAL easat-3854 167 2 0 CARDINAL easat-3854 168 1 2 CARDINAL easat-3854 168 2 0 CARDINAL easat-3854 168 3 1 CARDINAL easat-3854 168 4 π”…π›Ύπ’«π“Žπ’«π’· DATE easat-3854 168 5 1 CARDINAL easat-3854 170 1 1).π”…π’«π“Ž+π’·βˆ’π‘–π’«π‘– CARDINAL easat-3854 170 2 βˆ’βˆ‘(βˆ’1)π“Ž+𝑖(π“Ž βˆ’ PERSON easat-3854 170 3 1 CARDINAL easat-3854 171 1 𝔅1𝒫𝑠 PERSON easat-3854 171 2 β„³-algebras ORG easat-3854 172 1 β„³π’Ÿ ORG easat-3854 172 2 𝒡/𝒫 ORDINAL easat-3854 172 3 β„³π’Ÿ ORG easat-3854 172 4 𝒦[π’Ÿπ‘]-algebras NORP easat-3854 174 1 3.1 CARDINAL easat-3854 177 1 2024 CARDINAL easat-3854 178 1 https://creativecommons.org/licenses/by/4.0/ WORK_OF_ART easat-3854 179 1 1 CARDINAL easat-3854 179 2 r. e. mosher PERSON easat-3854 179 3 m. c. tangora PERSON easat-3854 179 4 harper PERSON easat-3854 179 5 new york GPE easat-3854 179 6 evanston GPE easat-3854 179 7 london GPE easat-3854 179 8 dc GPE easat-3854 179 9 1968 DATE easat-3854 180 1 45-85 CARDINAL easat-3854 181 1 2 CARDINAL easat-3854 181 2 s. v. lapin PERSON easat-3854 181 3 𝐷-differential 𝐸-algebra WORK_OF_ART easat-3854 181 4 j. math. WORK_OF_ART easat-3854 182 1 152 CARDINAL easat-3854 182 2 3 CARDINAL easat-3854 182 3 8666 DATE easat-3854 182 4 2576-8484 DATE easat-3854 182 5 8 CARDINAL easat-3854 183 1 6 CARDINAL easat-3854 183 2 8658-8666 DATE easat-3854 183 3 2024 CARDINAL easat-3854 183 4 10.55214/25768484.v8i6.3854 CARDINAL easat-3854 183 5 2024 CARDINAL easat-3854 184 1 372 CARDINAL easat-3854 186 1 3 CARDINAL easat-3854 186 2 m. lopez PERSON easat-3854 186 3 213 CARDINAL easat-3854 186 4 4 CARDINAL easat-3854 187 1 409 CARDINAL easat-3854 187 2 2009 DATE easat-3854 188 1 4 CARDINAL easat-3854 188 2 d. burghelea PERSON easat-3854 189 1 z. fiedorwicz PERSON easat-3854 189 2 w. gojda PERSON easat-3854 189 3 de rham ORG easat-3854 189 4 4 CARDINAL easat-3854 189 5 3 CARDINAL easat-3854 190 1 269 CARDINAL easat-3854 192 1 5 CARDINAL easat-3854 192 2 a. h. noreldeen PERSON easat-3854 194 1 african NORP easat-3854 194 2 5 CARDINAL easat-3854 194 3 e00115 DATE easat-3854 194 4 2019 DATE easat-3854 195 1 6 CARDINAL easat-3854 195 2 a. h. noreldeen PERSON easat-3854 196 1 m. saad PERSON easat-3854 196 2 5 CARDINAL easat-3854 196 3 1 CARDINAL easat-3854 197 1 23-31 DATE easat-3854 197 2 2017 DATE easat-3854 199 1 7 CARDINAL easat-3854 199 2 s. v. lapin PERSON easat-3854 199 3 1, 2018 DATE easat-3854 201 1 8 CARDINAL easat-3854 201 2 f. r. karar PERSON easat-3854 201 3 o. h. fathy PERSON easat-3854 201 4 w. m. mahmoud PERSON easat-3854 201 5 6 CARDINAL easat-3854 201 6 874-882 CARDINAL easat-3854 201 7 2023 CARDINAL easat-3854 202 1 10.13189 CARDINAL easat-3854 204 1 9 CARDINAL easat-3854 205 1 r. l. krasauskas PERSON easat-3854 208 1 ussr GPE easat-3854 208 2 spornic GPE easat-3854 208 3 61 DATE easat-3854 208 4 1 CARDINAL easat-3854 209 1 23 CARDINAL easat-3854 210 1 10.1070 CARDINAL easat-3854 210 2 10 CARDINAL easat-3854 210 3 w. m. mahmoud PERSON easat-3854 210 4 a. h. noreldeen PERSON easat-3854 210 5 h. fathy PERSON easat-3854 210 6 s. a. a. PERSON easat-3854 210 7 algebras ORG easat-3854 210 8 african NORP easat-3854 210 9 25 DATE easat-3854 210 10 e02288 NORP easat-3854 210 11 september 2024 DATE easat-3854 212 1 11 CARDINAL easat-3854 212 2 m. elhamdadi PERSON easat-3854 215 1 72 DATE easat-3854 216 1 4539-4545, 2003 DATE easat-3854 217 1 12 CARDINAL easat-3854 218 1 21 CARDINAL easat-3854 218 2 2 CARDINAL easat-3854 219 1 341 CARDINAL easat-3854 219 2 1998 DATE