In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
consisting of a non-empty set
and a ternary mapping
may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by
Marshall Hall to construct
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
s by means of coordinates. A planar ternary ring is not a
ring
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
in the traditional sense, but any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
gives a planar ternary ring where the operation
is defined by
. Thus, we can think of a planar ternary ring as a generalization of a field where the ternary operation takes the place of both addition and multiplication.
There is wide variation in the terminology. Planar ternary rings or ternary fields as defined here have been called by other names in the literature, and the term "planar ternary ring" can mean a variant of the system defined here. The term "ternary ring" often means a planar ternary ring, but it can also simply mean a ternary system.
Definition
A planar ternary ring is a structure
where
is a set containing at least two distinct elements, called 0 and 1, and
is a mapping which satisfies these five axioms:
#
;
#
;
#
, there is a unique
such that :
;
#
, there is a unique
, such that
; and
#
, the equations
have a unique solution
.
When
is finite, the third and fifth axioms are equivalent in the presence of the fourth.
No other pair (0', 1') in
can be found such that
still satisfies the first two axioms.
Binary operations
Addition
Define
. The structure
is a
loop with
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
0.
Multiplication
Define
. The set
is closed under this multiplication. The structure
is also a loop, with identity element 1.
Linear PTR
A planar ternary ring
is said to be ''linear'' if
.
For example, the planar ternary ring associated to a
quasifield
In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
Definition
A ...
is (by construction) linear.
Connection with projective planes
Given a planar ternary ring
, one can construct a
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
with point set ''P'' and line set ''L'' as follows: (Note that
is an extra symbol not in
.)
Let
*
, and
*
.
Then define,
, the
incidence relation In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point ''lies on'' a line" or "a line is ''contained in'' a plane" are used. The most basic incidence relation is that betw ...
in this way:
:
:
:
:
:
:
:
:
:
Every projective plane can be constructed in this way, starting with an appropriate planar ternary ring. However, two nonisomorphic planar ternary rings can lead to the construction of isomorphic projective planes.
Conversely, given any projective plane π, by choosing four points, labelled ''o'', ''e'', ''u'', and ''v'', no three of which lie on the same line, coordinates can be introduced in π so that these special points are given the coordinates: ''o'' = (0,0), ''e'' = (1,1), ''v'' = (
) and ''u'' = (0). The ternary operation is now defined on the coordinate symbols (except
) by ''y'' = T(''x'',''a'',''b'') if and only if the point (''x'',''y'') lies on the line which joins (''a'') with (0,''b''). The axioms defining a projective plane are used to show that this gives a planar ternary ring.
Linearity of the PTR is equivalent to a geometric condition holding in the associated projective plane.
Intuition
The connection between planar ternary rings (PTRs) and two-dimensional geometries, specifically projective and
affine geometries, is best described by examining the affine case first. In affine geometry, points on a plane are described using
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
, a method that is applicable even in
non-Desarguesian geometries — there, coordinate-components can always be shown to obey the structure of a PTR. By contrast,
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
, typically used in projective geometry, are unavailable in
non-Desarguesian contexts. Thus, the simplest
analytic
Analytic or analytical may refer to:
Chemistry
* Analytical chemistry, the analysis of material samples to learn their chemical composition and structure
* Analytical technique, a method that is used to determine the concentration of a chemical ...
way to construct a projective plane is to start with an affine plane and extend it by adding a "line at infinity"; this bypasses homogeneous coordinates.
In an affine plane, when the plane is Desarguesian, lines can be represented in
slope-intercept form
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficie ...
. This representation extends to non-Desarguesian planes through the ternary operation of a PTR, allowing a line to be expressed as
. Lines parallel to the y-axis are expressed by
.
We now show how to derive the
analytic
Analytic or analytical may refer to:
Chemistry
* Analytical chemistry, the analysis of material samples to learn their chemical composition and structure
* Analytical technique, a method that is used to determine the concentration of a chemical ...
representation of a general projective plane given at the start of this section. To do so, we pass from the affine plane, represented as
, to a representation of the projective plane
, by adding a line at infinity. Formally, the projective plane is described as
, where
represents an affine plane in Cartesian coordinates and includes all finite points, while
denotes the line at infinity. Similarly,
is expressed as
. Here,
is an affine line which we give its own Cartesian coordinate system, and
consists of a single point not lying on that affine line, which we represent using the symbol
.
Related algebraic structures
PTR's which satisfy additional algebraic conditions are given other names. These names are not uniformly applied in the literature. The following listing of names and properties is taken from .
A linear PTR whose additive loop is
associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
(and thus a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
), is called a cartesian group. In a cartesian group, the mappings
, and
must be permutations whenever
. Since cartesian groups are groups under addition, we revert to using a simple "+" for the additive operation.
A
quasifield
In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
Definition
A ...
is a cartesian group satisfying the right distributive law:
.
Addition in any quasifield is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
.
A
semifield
In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed.
Overview
The term semifield has two conflicting meanings, both of which inc ...
is a quasifield which also satisfies the left distributive law:
A planar
nearfield is a quasifield whose multiplicative loop is associative (and hence a group). Not all nearfields are planar nearfields.
Notes
References
*
*
*
*
*
*
*
*
*
*
*
* {{citation, last= Stevenson, first=Frederick, title=Projective Planes, year=1972, publisher=W.H. Freeman and Company, place=San Francisco, isbn=071670443-9
Algebraic structures
Projective geometry