17:
319:
77:
221:
is also a partial order. The properties required of critical pairs ensure that, when the relationship
232:
16:
243:
84:
but that could be made comparable without requiring any other changes to the partial order.
283:
313:
304:, Johns Hopkins Series in Mathematical Sciences, Baltimore: Johns Hopkins Univ. Press
81:
290:
of linear extensions is a realizer if and only if it reverses every critical pair.
69:
98:
be a partially ordered set. Then a critical pair is an ordered pair
231:
is added, the addition does not cause any violations of the
207:
is a critical pair, then the binary relation obtained from
302:
Combinatorics and partially ordered sets: Dimension theory
20:
Hasse diagram of a partial order with a critical pair ⟨
40:without requiring any other changes. Conversely, ⟨
8:
282:. This property may be used to characterize
286:of finite partial orders: A nonempty set
15:
270:if there exists a linear extension in
114:with the following three properties:
72:, a discipline within mathematics, a
7:
211:by adding the single relationship
14:
48:⟩ is not a critical pair, since
1:
76:is a pair of elements in a
336:
278:occurs earlier than
300:Trotter, W. T. (1992),
65:
78:partially ordered set
19:
125:are incomparable in
233:transitive property
66:
244:linear extensions
327:
305:
289:
281:
277:
273:
269:
265:
254:a critical pair
249:
241:
230:
220:
210:
206:
190:
180:
170:
166:
159:
149:
139:
135:
128:
124:
120:
113:
109:
97:
32:line would make
31:
335:
334:
330:
329:
328:
326:
325:
324:
310:
309:
299:
296:
287:
279:
275:
271:
267:
255:
247:
239:
222:
212:
208:
196:
182:
172:
168:
164:
151:
141:
137:
133:
126:
122:
118:
111:
110:of elements of
99:
88:
29:
12:
11:
5:
333:
331:
323:
322:
312:
311:
308:
307:
295:
292:
193:
192:
161:
130:
87:Formally, let
28:⟩. Adding the
13:
10:
9:
6:
4:
3:
2:
332:
321:
318:
317:
315:
303:
298:
297:
293:
291:
285:
263:
259:
253:
245:
236:
234:
229:
225:
219:
215:
204:
200:
189:
185:
179:
175:
162:
158:
154:
148:
144:
131:
117:
116:
115:
107:
103:
95:
91:
85:
83:
79:
75:
74:critical pair
71:
63:
59:
55:
51:
47:
43:
39:
35:
27:
23:
18:
320:Order theory
301:
261:
257:
251:
237:
227:
223:
217:
213:
202:
198:
194:
187:
183:
177:
173:
156:
152:
146:
142:
105:
101:
93:
89:
86:
82:incomparable
73:
70:order theory
67:
61:
57:
53:
49:
45:
41:
37:
33:
25:
21:
250:is said to
294:References
274:for which
163:for every
132:for every
56:, but not
284:realizers
80:that are
314:Category
252:reverse
238:A set
186:<
181:then
176:<
171:, if
160:, and
155:<
150:then
145:<
140:, if
121:and
96:, ≤)
60:<
52:<
36:<
30:grey
266:in
246:of
242:of
195:If
167:in
136:in
92:= (
68:In
316::
260:,
235:.
226:≤
216:≤
201:,
104:,
306:.
288:R
280:x
276:y
272:R
268:P
264:)
262:y
258:x
256:(
248:P
240:R
228:y
224:x
218:y
214:x
209:P
205:)
203:y
199:x
197:(
191:.
188:z
184:x
178:z
174:y
169:S
165:z
157:y
153:z
147:x
143:z
138:S
134:z
129:,
127:P
123:y
119:x
112:S
108:)
106:y
102:x
100:(
94:S
90:P
64:.
62:b
58:d
54:c
50:d
46:b
44:,
42:c
38:c
34:b
26:c
24:,
22:b
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.