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Alternativity

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595: 87: 32: 50: 341: 251: 503: 376: 456: 283: 97: 655: 636: 405:) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be alternative. The 68: 112: 155: 127: 134: 629: 141: 660: 123: 622: 519: 410: 296: 206: 461: 346: 420: 414: 256: 606: 571: 555: 514: 406: 185: 177: 567: 148: 575: 563: 189: 20: 540: 649: 398: 594: 86: 402: 559: 602: 104: 378:
A magma that is both left and right alternative is said to be
80: 25: 541:"Automated theorem proving in quasigroup and loop theory" 458:
cannot be proven to be identical to expressions such as
409:, however, is not true, in contrast to the situation in 610: 108: 46: 464: 423: 349: 299: 259: 209: 41:
may be too technical for most readers to understand
497: 450: 370: 335: 277: 245: 413:. In fact, an alternative magma need not even be 630: 8: 113:introducing citations to additional sources 637: 623: 539:Phillips, J. D.; Stanovský, David (2010), 463: 422: 348: 298: 258: 208: 69:Learn how and when to remove this message 53:, without removing the technical details. 103:Relevant discussion may be found on the 531: 51:make it understandable to non-experts 7: 591: 589: 609:. You can help Knowledge (XXG) by 14: 593: 96:relies largely or entirely on a 85: 30: 656:Properties of binary operations 492: 489: 486: 477: 471: 465: 445: 436: 433: 424: 327: 318: 312: 303: 240: 231: 219: 210: 16:Property of a binary operation 1: 336:{\displaystyle y(xx)=(yx)x} 246:{\displaystyle (xx)y=x(xy)} 677: 588: 498:{\displaystyle (x(x(xx)))} 18: 505:purely by alternativity. 417:: already the expression 371:{\displaystyle x,y\in G.} 451:{\displaystyle (xx)(xx)} 278:{\displaystyle x,y\in G} 19:Not to be confused with 605:-related article is a 499: 452: 372: 337: 279: 247: 560:10.3233/AIC-2010-0460 500: 453: 373: 338: 280: 248: 462: 421: 411:alternative algebras 347: 297: 257: 207: 109:improve this article 520:Power associativity 184:is a property of a 495: 448: 401:magma (that is, a 368: 333: 275: 243: 618: 617: 548:AI Communications 415:power-associative 289:right alternative 174: 173: 159: 79: 78: 71: 668: 639: 632: 625: 597: 590: 580: 578: 554:(2–3): 267–283, 545: 536: 515:Flexible algebra 504: 502: 501: 496: 457: 455: 454: 449: 392: 391: 384: 383: 377: 375: 374: 369: 342: 340: 339: 334: 291: 290: 284: 282: 281: 276: 252: 250: 249: 244: 201: 200: 199:left alternative 186:binary operation 178:abstract algebra 169: 166: 160: 158: 117: 89: 81: 74: 67: 63: 60: 54: 34: 33: 26: 676: 675: 671: 670: 669: 667: 666: 665: 646: 645: 644: 643: 586: 584: 583: 543: 538: 537: 533: 528: 511: 460: 459: 419: 418: 389: 388: 381: 380: 345: 344: 295: 294: 288: 287: 255: 254: 205: 204: 198: 197: 170: 164: 161: 124:"Alternativity" 118: 116: 102: 90: 75: 64: 58: 55: 47:help improve it 44: 35: 31: 24: 21:Alternatization 17: 12: 11: 5: 674: 672: 664: 663: 658: 648: 647: 642: 641: 634: 627: 619: 616: 615: 598: 582: 581: 530: 529: 527: 524: 523: 522: 517: 510: 507: 494: 491: 488: 485: 482: 479: 476: 473: 470: 467: 447: 444: 441: 438: 435: 432: 429: 426: 367: 364: 361: 358: 355: 352: 332: 329: 326: 323: 320: 317: 314: 311: 308: 305: 302: 274: 271: 268: 265: 262: 242: 239: 236: 233: 230: 227: 224: 221: 218: 215: 212: 195:is said to be 172: 171: 107:. Please help 93: 91: 84: 77: 76: 38: 36: 29: 15: 13: 10: 9: 6: 4: 3: 2: 673: 662: 661:Algebra stubs 659: 657: 654: 653: 651: 640: 635: 633: 628: 626: 621: 620: 614: 612: 608: 604: 599: 596: 592: 587: 577: 573: 569: 565: 561: 557: 553: 549: 542: 535: 532: 525: 521: 518: 516: 513: 512: 508: 506: 483: 480: 474: 468: 442: 439: 430: 427: 416: 412: 408: 404: 400: 395: 393: 385: 365: 362: 359: 356: 353: 350: 330: 324: 321: 315: 309: 306: 300: 292: 272: 269: 266: 263: 260: 237: 234: 228: 225: 222: 216: 213: 202: 194: 191: 187: 183: 182:alternativity 179: 168: 157: 154: 150: 147: 143: 140: 136: 133: 129: 126: –  125: 121: 120:Find sources: 114: 110: 106: 100: 99: 98:single source 94:This article 92: 88: 83: 82: 73: 70: 62: 59:November 2021 52: 48: 42: 39:This article 37: 28: 27: 22: 611:expanding it 600: 585: 551: 547: 534: 396: 387: 379: 286: 196: 192: 181: 175: 162: 152: 145: 138: 131: 119: 95: 65: 56: 40: 399:associative 382:alternative 650:Categories 576:1204.68181 526:References 135:newspapers 403:semigroup 360:∈ 270:∈ 105:talk page 509:See also 407:converse 390:flexible 343:for all 253:for all 165:May 2024 603:algebra 568:2647941 149:scholar 45:Please 574:  566:  151:  144:  137:  130:  122:  601:This 544:(PDF) 190:magma 188:. A 156:JSTOR 142:books 607:stub 397:Any 285:and 128:news 572:Zbl 556:doi 394:). 293:if 203:if 176:In 111:by 49:to 652:: 570:, 564:MR 562:, 552:23 550:, 546:, 180:, 638:e 631:t 624:v 613:. 579:. 558:: 493:) 490:) 487:) 484:x 481:x 478:( 475:x 472:( 469:x 466:( 446:) 443:x 440:x 437:( 434:) 431:x 428:x 425:( 386:( 366:. 363:G 357:y 354:, 351:x 331:x 328:) 325:x 322:y 319:( 316:= 313:) 310:x 307:x 304:( 301:y 273:G 267:y 264:, 261:x 241:) 238:y 235:x 232:( 229:x 226:= 223:y 220:) 217:x 214:x 211:( 193:G 167:) 163:( 153:· 146:· 139:· 132:· 115:. 101:. 72:) 66:( 61:) 57:( 43:. 23:.

Index

Alternatization
help improve it
make it understandable to non-experts
Learn how and when to remove this message

single source
talk page
improve this article
introducing citations to additional sources
"Alternativity"
news
newspapers
books
scholar
JSTOR
abstract algebra
binary operation
magma
associative
semigroup
converse
alternative algebras
power-associative
Flexible algebra
Power associativity
"Automated theorem proving in quasigroup and loop theory"
doi
10.3233/AIC-2010-0460
MR
2647941

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