122:
57:
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
345:
340:
313:
17:
277:
240:
174:
140:
151:
147:
323:
8:
197:
190:
178:
163:
305:
298:
170:
309:
223:
40:
319:
281:
36:
32:
276:
Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”,
257:
136:
117:{\displaystyle a\bullet \left(b\bullet a\right)=\left(a\bullet b\right)\bullet a}
207:
155:
334:
213:
202:
28:
219:
Similarly, the following classes of nonassociative magmas are flexible:
252:
228:
285:
236:
159:
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.
60:
181:
and showed that they satisfy the flexible identity.
297:
116:
332:
300:An introduction to non-associative algebras
295:
173:examined the algebras generated by the
14:
333:
24:
25:
357:
346:Properties of binary operations
278:American Journal of Mathematics
270:
13:
1:
296:Schafer, Richard D. (1995) .
263:
7:
246:
241:Cayley–Dickson construction
184:
10:
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:
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:
59:
56:
55:
31:, particularly
23:
22:
15:
12:
11:
5:
359:
349:
348:
343:
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:
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:
269:
259:
256:
254:
251:
250:
244:
242:
238:
230:
227:
225:
222:
221:
220:
215:
212:
209:
206:
204:
201:
199:
196:
195:
194:
192:
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:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.