35:. A ternary equivalence relation is symmetric, reflexive, and transitive, where those terms are meant in the sense defined below. The classic example is the relation of
173:
Karzel, Helmut; Pianta, Silvia (2008), "Binary operations derived from symmetric permutation sets and applications to absolute geometry",
126:
Araújo, João; Konieczny, Janusz (2007), "A method of finding automorphism groups of endomorphism monoids of relational systems",
319:
128:
270:
248:
329:
324:
195:
Karzel, Helmut; Marchi, Mario; Pianta, Silvia (December 2010), "The defect in an invariant reflection structure",
217:
50:
43:. In an abstract set, a ternary equivalence relation determines a collection of equivalence classes or
45:
32:
58:
54:
215:
Karzel, Helmut; Taherian, Sayed-Ghahreman (2018), "Groups with a ternary equivalence relation",
279:
257:
226:
204:
182:
137:
25:
153:
40:
29:
292:
313:
36:
17:
187:
141:
230:
208:
284:
57:. In the same way, a binary equivalence relation on a set determines a
246:
Pickett, H.E. (1966), "A note on generalized equivalence relations",
261:
268:
Rainich, G.Y. (1952), "Ternary relations in geometry and algebra",
160:
Karzel, Helmut (2007), "Loops related to geometric structures",
296:
90:
Reflexivity: . Equivalently, in the presence of symmetry, if
302:-ary equivalence relations and their application to geometry
305:, Warsaw: Instytut Matematyczny Polskiej Akademi Nauk
152:, Die Grundlehren der mathematischen Wissenschaften,
87:
Symmetry: If then and . (Therefore also , , and .)
83:, written , that satisfies the following axioms:
150:Aufbau der Geometrie aus dem Spiegelungsbegriff
8:
283:
186:
69:A ternary equivalence relation on a set
239:Metric planes and metric vector spaces
7:
115:and and then . (Therefore also .)
14:
162:Quasigroups and Related Systems
1:
271:Michigan Mathematical Journal
249:American Mathematical Monthly
148:Bachmann, Friedrich (1959),
102:are not all distinct, then .
22:ternary equivalence relation
346:
188:10.1016/j.disc.2006.11.058
142:10.1016/j.disc.2006.09.029
237:Lingenberg, Rolf (1979),
231:10.1007/s00010-018-0543-x
209:10.1007/s00022-010-0058-7
218:Aequationes Mathematicae
320:Mathematical relations
285:10.1307/mmj/1028988890
39:among three points in
175:Discrete Mathematics
129:Discrete Mathematics
33:equivalence relation
330:Projective geometry
197:Journal of Geometry
325:Incidence geometry
105:Transitivity: If
55:incidence geometry
337:
306:
288:
287:
264:
242:
233:
211:
191:
190:
169:
156:
144:
114:
101:
97:
93:
82:
72:
53:in the sense of
26:ternary relation
345:
344:
340:
339:
338:
336:
335:
334:
310:
309:
293:Szmielew, Wanda
291:
267:
262:10.2307/2314183
245:
236:
214:
194:
172:
159:
154:Springer-Verlag
147:
125:
122:
106:
99:
95:
91:
74:
70:
67:
41:Euclidean space
28:analogous to a
12:
11:
5:
343:
341:
333:
332:
327:
322:
312:
311:
308:
307:
289:
265:
243:
234:
212:
203:(1–2): 67–87,
192:
170:
157:
145:
121:
118:
117:
116:
103:
88:
73:is a relation
66:
63:
13:
10:
9:
6:
4:
3:
2:
342:
331:
328:
326:
323:
321:
318:
317:
315:
304:
303:
299:
294:
290:
286:
281:
278:(2): 97–111,
277:
273:
272:
266:
263:
259:
255:
251:
250:
244:
240:
235:
232:
228:
224:
220:
219:
213:
210:
206:
202:
198:
193:
189:
184:
180:
176:
171:
167:
163:
158:
155:
151:
146:
143:
139:
136:: 1609–1620,
135:
131:
130:
124:
123:
119:
113:
109:
104:
89:
86:
85:
84:
81:
77:
64:
62:
60:
56:
52:
48:
47:
42:
38:
34:
31:
27:
24:is a kind of
23:
19:
301:
297:
275:
269:
253:
247:
238:
222:
216:
200:
196:
178:
174:
165:
161:
149:
133:
127:
111:
107:
79:
75:
68:
51:linear space
49:that form a
44:
37:collinearity
21:
15:
256:: 860–861,
225:: 415–423,
181:: 415–421,
18:mathematics
314:Categories
120:References
65:Definition
59:partition
295:(1981),
241:, Wiley
168:: 47–76
46:pencils
98:, and
30:binary
20:, a
298:On
280:doi
258:doi
227:doi
205:doi
183:doi
179:308
138:doi
134:307
16:In
316::
274:,
254:73
252:,
223:92
221:,
201:99
199:,
177:,
166:15
164:,
132:,
110:≠
94:,
78:⊂
61:.
300:n
282::
276:1
260::
229::
207::
185::
140::
112:b
108:a
100:c
96:b
92:a
80:X
76:E
71:X
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.