Properties
F1 is believed to have the following properties.
- Finite sets are both affine spaces and projective spaces over F1.
- Pointed sets are vector spaces over F1.
- The finite fields Fq are quantum deformations of F1, where q is the deformation.
- Weyl groups are simple algebraic groups over F1:
- Given a Dynkin diagram for a semisimple algebraic group, its Weyl group is the semisimple algebraic group over F1.
- Spec Z is a curve over F1.
- Groups are Hopf algebras over F1. More generally, anything defined purely in terms of diagrams of algebraic objects should have an F1-analog in the category of sets.
- Group actions on sets are projective representations of G over F1, and in this way, G is the group Hopf algebra F1.
- Toric varieties determine F1-varieties. In some descriptions of F1-geometry the converse is also true, in the sense that the extension of scalars of F1-varieties to Z are toric Whilst other approaches to F1-geometry admit wider classes of examples, toric varieties appear to lie at the very heart of the theory.
- The zeta function of PN over F1 should be ζ(s) = s(s − 1)···(s − N).
- The mth K-group of F1 should be the mth stable homotopy group of the sphere spectrum.
Read more about this topic: Field With One Element
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)