TY  - BOOK
AU  - Semrl,Peter
TI  - The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices
T2  - Memoirs of the American Mathematical Society Series
SN  - 9781470418922
AV  - QA188 .S45 2014 
U1  - 512.9/434 
PY  - 2014///
CY  - Providence
PB  - American Mathematical Society
KW  - Geometry, Algebraic
KW  - Matrices
KW  - Electronic books
N1  - Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Notation and basic definitions -- Chapter 3. Examples -- Chapter 4. Statement of main results -- Chapter 5. Proofs -- 5.1. Preliminary results -- 5.2. Splitting the proof of main results into subcases -- 5.3. Square case -- 5.4. Degenerate case -- 5.5. Non-square case -- 5.6. Proofs of corollaries -- Acknowledgments -- Bibliography -- Back Cover
N2  - Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all m\times n matrices over a division ring \mathbb{D} which preserve adjacency in both directions. Motivated by several applications the author studies a long standing open problem of possible improvements. There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes? A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings the author solves all three problems simultaneously, thus obtaining the optimal version of Hua's theorem. In the case of general division rings he gets such an optimal result only for square matrices and gives examples showing that it cannot be extended to the non-square case
UR  - https://ebookcentral.proquest.com/lib/orpp/detail.action?docID=5295322
ER  -