| 词条 | Moufang plane |
| 释义 |
In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically it is a special type of translation plane. A translation plane is a projective plane that has a translation line, that is, a line with the property that the group of automorphisms that fixes every point of the line acts transitively on the points of the plane not on the line.[1] A translation plane is Moufang if every line of the plane is a translation line.[2] CharacterizationsA Moufang plane can also be described as a projective plane in which the little Desargues Theorem holds.[3] This theorem states that a restricted form of Desargues' theorem holds for every line in the plane.[4] Every Desarguesian plane is a Moufang plane.[5] In algebraic terms, a projective plane over any alternative division ring is a Moufang plane,[6] and this gives a 1:1 correspondence between isomorphism classes of alternative division rings and Moufang planes. As a consequence of the algebraic Artin–Zorn theorem, that every finite alternative division ring is a field, every finite Moufang plane is Desarguesian, but some infinite Moufang planes are non-Desarguesian planes. In particular, the Cayley plane, an infinite Moufang projective plane over the octonions, is one of these because the octonions do not form a division ring.[7] PropertiesThe following conditions on a projective plane P are equivalent:[8]
Also, in a Moufang plane:
Notes1. ^That is, the group acts transitively on the affine plane formed by removing this line and all its points from the projective plane. 2. ^{{harvnb|Hughes|Piper|1973|loc=p. 101}} 3. ^{{harvnb|Pickert|1975|loc=p. 186}} 4. ^This restricted version states that if two triangles are perspective from a point on a given line, and two pairs of corresponding sides also meet on this line, then the third pair of corresponding sides meet on the line as well. 5. ^{{harvnb|Hughes|Piper|1973|loc=p. 153}} 6. ^{{harvnb|Hughes|Piper|1973|loc=p. 139}} 7. ^{{Citation | last1=Weibel | first1=Charles | title=Survey of Non-Desarguesian Planes | url=http://www.ams.org/notices/200710/ | year=2007 | journal= Notices of the AMS | volume= 54 | issue=10 | pages=1294–1303}} 8. ^H. Klein Moufang planes 9. ^{{harvnb|Stevenson|1972|loc=p. 392}} Stevenson refers to Moufang planes as alternative planes. 10. ^If transitive is replaced by sharply transitive, the plane is pappian. References
Further reading
1 : Projective geometry |
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