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QUASIFIELD

Quasifield

Division ring with weakened conditions

In mathematics, a quasifield is an algebraic structure ( Q , + , ⋅ ) {\displaystyle (Q,+,\cdot )} where + {\displaystyle +} and ⋅ {\displaystyle \cdot

Quasifield

Hall plane

based on the Hall quasifield (also called a Hall system), H of order p2n for p a prime. The creation of the plane from the quasifield follows the standard

Hall plane

Hall_plane

Associative property

Property of a mathematical operation

nature of infinitesimal transformations. Other examples are quasigroup, quasifield, non-associative ring, and commutative non-associative magmas. In mathematics

Associative property

Associative_property

Translation plane

Special type of projective plane

translation planes, it is always possible to coordinatize with a quasifield. However, some quasifields satisfy additional algebraic properties, and the corresponding

Translation plane

Translation_plane

Field (mathematics)

Algebraic structure with addition, multiplication, and division

various other weaker algebraic structures related to fields such as quasifields, near-fields and semifields. There are also proper classes with field

Field (mathematics)

Field_(mathematics)

Planar ternary ring

Construction in projective geometry

\forall a,b,c\in R} . For example, the planar ternary ring associated to a quasifield is (by construction) linear. Given a planar ternary ring ( R , T ) {\displaystyle

Planar ternary ring

Planar_ternary_ring

Near-field (mathematics)

Algebraic structure

convention. A (right) near-field is called "planar" if it is also a right quasifield. Every finite near-field is planar, but infinite near-fields need not

Near-field (mathematics)

Near-field_(mathematics)

Semifield

Algebraic structure

commutative is a field. A semifield by this definition is a special case of a quasifield. If S is finite, the last axiom in the definition above can be replaced

Semifield

Non-Desarguesian plane

Projective plane not satisfying Desargues' theorem

division rings, semifields, nearfields, right nearfields, quasifields and right quasifields. In a Desarguesian projective plane a conic can be defined

Non-Desarguesian plane

Non-Desarguesian_plane

André plane

norm function from K {\displaystyle K} to F {\displaystyle F} . Define a quasifield Q {\displaystyle Q} with the same elements and addition as K, but with

André plane

André_plane