Knowledge (XXG)

Flexible algebra

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122: 57: 345: 139:(that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a 154:
operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the
340: 313: 240: 277: 174: 140: 151: 147: 197: 190: 178: 163: 305: 170: 309: 223: 40: 319: 281: 36: 32: 323: 257: 136: 298: 207: 155: 334: 213: 276:
Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”,
117:{\displaystyle a\bullet \left(b\bullet a\right)=\left(a\bullet b\right)\bullet a} 17: 202: 28: 252: 228: 236: 159: 219:
Similarly, the following classes of nonassociative magmas are flexible:
285: 193:, the following classes of nonassociative algebras are flexible: 231:(which are associative magmas, and which are also alternative) 239:, and all algebras constructed from these by iterating the 143:
is flexible if its multiplication operator is flexible.
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and showed that they satisfy the flexible identity.
297: 116: 8: 300:An introduction to non-associative algebras 59: 269: 173:examined the algebras generated by the 7: 25: 346:Properties of binary operations 278:American Journal of Mathematics 1: 296:Schafer, Richard D. (1995) . 241:Cayley–Dickson construction 362: 341:Non-associative algebras 210:(which are commutative) 175:Cayley–Dickson process 141:nonassociative algebra 118: 243:, are also flexible. 162:, which are not even 127:for any two elements 119: 198:Alternative algebras 191:associative algebras 58: 47:if it satisfies the 306:Dover Publications 224:Alternative magmas 171:Richard D. Schafer 114: 49:flexible identity 18:Flexible identity 16:(Redirected from 353: 327: 303: 288: 274: 123: 121: 120: 115: 107: 103: 85: 81: 37:binary operation 33:abstract algebra 21: 361: 360: 356: 355: 354: 352: 351: 350: 331: 330: 316: 295: 292: 291: 286:10.2307/2372583 275: 271: 266: 258:Maltsev algebra 249: 208:Jordan algebras 187: 135:of the set. A 93: 89: 71: 67: 56: 55: 31:, particularly 23: 22: 15: 12: 11: 5: 359: 357: 349: 348: 343: 333: 332: 329: 328: 314: 290: 289: 268: 267: 265: 262: 261: 260: 255: 248: 245: 233: 232: 226: 217: 216: 214:Okubo algebras 211: 205: 200: 186: 183: 156:multiplication 125: 124: 113: 110: 106: 102: 99: 96: 92: 88: 84: 80: 77: 74: 70: 66: 63: 24: 14: 13: 10: 9: 6: 4: 3: 2: 358: 347: 344: 342: 339: 338: 336: 325: 321: 317: 315:0-486-68813-5 311: 307: 302: 301: 294: 293: 287: 283: 279: 273: 270: 263: 259: 256: 254: 251: 250: 246: 244: 242: 238: 230: 227: 225: 222: 221: 220: 215: 212: 209: 206: 204: 201: 199: 196: 195: 194: 192: 184: 182: 180: 176: 172: 167: 165: 161: 157: 153: 149: 144: 142: 138: 134: 130: 111: 108: 104: 100: 97: 94: 90: 86: 82: 78: 75: 72: 68: 64: 61: 54: 53: 52: 50: 46: 42: 38: 34: 30: 19: 299: 272: 234: 218: 203:Lie algebras 188: 168: 145: 132: 128: 126: 48: 44: 26: 280:76: 435–46 164:alternative 152:associative 148:commutative 29:mathematics 335:Categories 324:0145.25601 264:References 229:Semigroups 253:Zorn ring 237:sedenions 169:In 1954, 160:sedenions 109:∙ 98:∙ 76:∙ 65:∙ 247:See also 189:Besides 185:Examples 45:flexible 177:over a 39:• on a 322:  312:  146:Every 179:field 137:magma 310:ISBN 235:The 131:and 35:, a 320:Zbl 282:doi 158:of 150:or 43:is 41:set 27:In 337:: 318:. 308:. 304:. 166:. 51:: 326:. 284:: 133:b 129:a 112:a 105:) 101:b 95:a 91:( 87:= 83:) 79:a 73:b 69:( 62:a 20:)

Index

Flexible identity
mathematics
abstract algebra
binary operation
set
magma
nonassociative algebra
commutative
associative
multiplication
sedenions
alternative
Richard D. Schafer
Cayley–Dickson process
field
associative algebras
Alternative algebras
Lie algebras
Jordan algebras
Okubo algebras
Alternative magmas
Semigroups
sedenions
Cayley–Dickson construction
Zorn ring
Maltsev algebra
American Journal of Mathematics
doi
10.2307/2372583
An introduction to non-associative algebras

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