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The optimal version of Hua's fundamental theorem of geometry of rectangular matrices
Šemrl, Peter
Huajev fundamentalni izrek geometrije matrik opiše splošno obliko bijektivnih preslikav na prostoru ▫$m \times n$▫ matrik nad (ne nujno komutativnim) obsegom ▫$\mathbb{D}$▫, ki ohranjajo sosednost v ...obe smeri. Motivirani s številnimi uporabami se ukvarjamo z možnimi izboljšavami tega izreka. Le te so možne v treh smereh. Najprej se vprašamo, ali je mogoče nadomestiti predpostavko ohranjanja sosednosti v obe smeri s šibkejšo predpostavko, da se sosednost ohranja zgolj v eno smer in pri tem še vedno dobiti enak zaključek. Ali lahko omilimo predpostavko bijektivnosti? In končno, ali je mogoče dobiti podobne strukturne rezultate za ohranjevalce sosednosti med matričnimi prostori različnih dimenzij? EAS obseg je tak obseg, ki ni izomorfen nobenemu svojemu pravemu podobsegu. Za matrike nad takimi obsegi hkrati rešimo vse tri zgoraj naštete probleme in s tem dobimo optimalno verzijo Huajevega izreka. Pri splošnih obsegih dobimo tak optimalni rezultat le v primeru, ko je domena prostor kvadratnih matrik. S primeri pokažemo, da se kvadratni primer ne da posplošiti na poljubne pravokotne matrike.
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