A Mathematical Bridge between Discretized Gauge Theories in Quantum Physics and Approximate Reasoning in Pairwise Comparisons

<table class="fixed-width table-group" id="tab1"><tr><td><table class="table"><colgroup><col style="width:18.00em"/><col style="width:23.57em"/><col style="width:18.00em"/></colgroup><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left">PC matrices</td><td class="align_center">Discretized gauge theories</td><td class="align_center">Continuum gauge theories</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left">Consistency</td><td class="align_center">0-holonomy</td><td class="align_center">0-curvature</td></tr><tr><td class="align_left">Consistency in <svg height="8.69875pt" id="M141" style="vertical-align:-0.3499298pt" version="1.1" viewbox="-0.0498162 -8.34882 26.097 8.69875" width="26.097pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 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376V378Z" id="g113-52"></path></g></svg> matrices</td><td class="align_center">0-holonomy on the border of a 2-simplex <svg height="11.8174pt" id="M142" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -8.6359 15.0414 11.8174" width="15.0414pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M600 0V24L339 665L311 657L43 24V0H600ZM497 50H107L303 539L497 50Z" id="g113-133"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.958,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg></td><td class="align_center"><svg height="11.5564pt" id="M143" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 63.3163 11.5564" 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11.8174" width="15.0414pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M600 0V24L339 665L311 657L43 24V0H600ZM497 50H107L303 539L497 50Z" id="g113-133"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.958,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg></td></tr><tr><td class="align_left">Koczkodaj’s inconsistency indicator ii</td><td class="align_center"><svg height="12.5794pt" id="M146" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 105.081 12.5794" width="105.081pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M319 325C317 349 306 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81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg> is any 2-simplex of the triangulation</td><td class="align_center"><span style="width: 50.8028ptpx;"><svg height="12.6603pt" id="M148" style="vertical-align:-3.61338pt" version="1.1" viewbox="-0.0498162 -9.04692 50.8028 12.6603" width="50.8028pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M319 325C317 349 306 409 297 431C277 440 250 449 209 449C117 449 57 389 57 319C57 243 122 209 182 182C232 159 261 135 261 91C261 48 227 21 190 21C130 21 85 79 68 145L41 140C41 104 51 36 58 22C75 7 121 -12 172 -12C252 -12 337 35 337 126C337 195 286 231 210 262C166 281 126 304 126 348C126 388 152 417 191 417C240 417 274 378 294 318L319 325Z" id="g121-113"></path></g><g transform="matrix(.013,0,0,-0.013,4.706,0)"><path d="M518 50L491 51C452 54 444 60 444 110V444C429 441 405 437 377 434C348 430 314 427 287 426V403L323 397C356 392 365 386 365 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colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Advances in Mathematical Physics

tab1

Table 1

Table 1: A Mathematical Bridge between Discretized Gauge Theories in Quantum Physics and Approximate Reasoning in Pairwise Comparisons 