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Critical pair (order theory)

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is also a partial order. The properties required of critical pairs ensure that, when the relationship
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but that could be made comparable without requiring any other changes to the partial order.
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of linear extensions is a realizer if and only if it reverses every critical pair.
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be a partially ordered set. Then a critical pair is an ordered pair
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is added, the addition does not cause any violations of the
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is a critical pair, then the binary relation obtained from
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Combinatorics and partially ordered sets: Dimension theory
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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

Index


order theory
partially ordered set
incomparable
transitive property
linear extensions
realizers
Category
Order theory

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