Planar ternary ring

In mathematics, an algebraic structure ${\displaystyle (R,T)}$ consisting of a non-empty set ${\displaystyle R}$ and a ternary mapping ${\displaystyle T\colon R^{3}\to R\,}$ may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall Hall^[1] to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field gives a planar ternary ring where the operation ${\displaystyle T}$ is defined by ${\displaystyle T(a,b,c)=ab+c}$. 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 ${\displaystyle (R,T)}$  where ${\displaystyle R}$  is a set containing at least two distinct elements, called 0 and 1, and ${\displaystyle T\colon R^{3}\to R\,}$  is a mapping which satisfies these five axioms:^[2]

1. ${\displaystyle T(a,0,b)=T(0,a,b)=b,\quad \forall a,b\in R}$ ;
2. ${\displaystyle T(1,a,0)=T(a,1,0)=a,\quad \forall a\in R}$ ;
3. ${\displaystyle \forall a,b,c,d\in R,a\neq c}$ , there is a unique ${\displaystyle x\in R}$  such that : ${\displaystyle T(x,a,b)=T(x,c,d)\,}$ ;
4. ${\displaystyle \forall a,b,c\in R}$ , there is a unique ${\displaystyle x\in R}$ , such that ${\displaystyle T(a,b,x)=c\,}$ ; and
5. ${\displaystyle \forall a,b,c,d\in R,a\neq c}$ , the equations ${\displaystyle T(a,x,y)=b,T(c,x,y)=d\,}$  have a unique solution ${\displaystyle (x,y)\in R^{2}}$ .

When ${\displaystyle R}$  is finite, the third and fifth axioms are equivalent in the presence of the fourth.^[3]

No other pair (0', 1') in ${\displaystyle R^{2}}$  can be found such that ${\displaystyle T}$  still satisfies the first two axioms.

Binary operations

Addition

Define ${\displaystyle a\oplus b=T(a,1,b)}$ .^[4] The structure ${\displaystyle (R,\oplus )}$  is a loop with identity element 0.

Multiplication

Define ${\displaystyle a\otimes b=T(a,b,0)}$ . The set ${\displaystyle R_{0}=R\setminus \{0\}\,}$  is closed under this multiplication. The structure ${\displaystyle (R_{0},\otimes )}$  is also a loop, with identity element 1.

Linear PTR

A planar ternary ring ${\displaystyle (R,T)}$  is said to be linear if ${\displaystyle T(a,b,c)=(a\otimes b)\oplus c,\quad \forall a,b,c\in R}$ . For example, the planar ternary ring associated to a quasifield is (by construction) linear.

Connection with projective planes

 

Coordinates of a projective plane to establish a planar ternary ring

Given a planar ternary ring ${\displaystyle (R,T)}$ , one can construct a projective plane with point set P and line set L as follows:^[5][6] (Note that ${\displaystyle \infty }$  is an extra symbol not in ${\displaystyle R}$ .)

Let

• ${\displaystyle P=\{(a,b)|a,b\in R\}\cup \{(a)|a\in R\}\cup \{(\infty )\}}$ , and
• ${\displaystyle L=\{[a,b]|a,b\in R\}\cup \{[a]|a\in R\}\cup \{[\infty ]\}}$ .

Then define, ${\displaystyle \forall a,b,c,d\in R}$ , the incidence relation ${\displaystyle I}$  in this way:

${\displaystyle ((a,b),[c,d])\in I\Longleftrightarrow T(a,c,d)=b}$ 

${\displaystyle ((a,b),[c])\in I\Longleftrightarrow a=c}$ 

${\displaystyle ((a,b),[\infty ])\notin I}$ 

${\displaystyle ((a),[c,d])\in I\Longleftrightarrow a=c}$ 

${\displaystyle ((a),[c])\notin I}$ 

${\displaystyle ((a),[\infty ])\in I}$ 

${\displaystyle ((\infty ),[c,d])\notin I}$ 

${\displaystyle ((\infty ),[a])\in I}$ 

${\displaystyle ((\infty ),[\infty ])\in I}$ 

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 = (${\displaystyle \infty }$ ) and u = (0).^[7] The ternary operation is now defined on the coordinate symbols (except ${\displaystyle \infty }$ ) 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.^[8]

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, 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, typically used in projective geometry, are unavailable in non-Desarguesian contexts. Thus, the simplest analytic 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 ${\displaystyle y=mx+c}$ . This representation extends to non-Desarguesian planes through the ternary operation of a PTR, allowing a line to be expressed as ${\displaystyle y=T(x,m,c)}$ . Lines parallel to the y-axis are expressed by ${\displaystyle x=c}$ .

We now show how to derive the analytic representation of a general projective plane given at the start of this section. To do so, we pass from the affine plane, represented as ${\displaystyle R^{2}}$ , to a representation of the projective plane ${\displaystyle R\mathbb {P} ^{2}}$ , by adding a line at infinity. Formally, the projective plane is described as ${\displaystyle R\mathbb {P} ^{2}:=R^{2}\cup R\mathbb {P} ^{1}}$ , where ${\displaystyle R^{2}}$  represents an affine plane in Cartesian coordinates and includes all finite points, while ${\displaystyle R\mathbb {P} ^{1}}$  denotes the line at infinity. Similarly, ${\displaystyle R\mathbb {P} ^{1}}$  is expressed as ${\displaystyle R\mathbb {P} ^{1}:=R^{1}\cup R\mathbb {P} ^{0}}$ . Here, ${\displaystyle R^{1}}$  is an affine line which we give its own Cartesian coordinate system, and ${\displaystyle R\mathbb {P} ^{0}}$  consists of a single point not lying on that affine line, which we represent using the symbol ${\displaystyle \infty }$ .

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 Dembowski (1968, p. 129).

A linear PTR whose additive loop is associative (and thus a group ), is called a cartesian group. In a cartesian group, the mappings

${\displaystyle x\longrightarrow -x\otimes a+x\otimes b}$ , and ${\displaystyle x\longrightarrow a\otimes x-b\otimes x}$ 

must be permutations whenever ${\displaystyle a\neq b}$ . Since cartesian groups are groups under addition, we revert to using a simple "+" for the additive operation.

A quasifield is a cartesian group satisfying the right distributive law: ${\displaystyle (x+y)\otimes z=x\otimes z+y\otimes z}$ . Addition in any quasifield is commutative.

A semifield is a quasifield which also satisfies the left distributive law: ${\displaystyle x\otimes (y+z)=x\otimes y+x\otimes z.}$ 

A planar nearfield is a quasifield whose multiplicative loop is associative (and hence a group). Not all nearfields are planar nearfields.

Notes

1. Hall 1943
2. Hughes & Piper 1973, p. 113, Thm. 5.1.
3. Hughes & Piper 1973, p. 118, Theorem 5.4
4. In the literature there are two versions of this definition. This is the form used by Hall (1959, p. 355), Albert & Sandler (1968, p. 50), and Dembowski (1968, p. 128), while ${\displaystyle a\oplus b=T(1,a,b)}$  is used by Hughes & Piper (1973, p. 117), Pickert (1975, p. 38), and Stevenson (1972, p. 274). The difference comes from the alternative ways these authors coordinatize the plane.
5. R. H. Bruck, Recent Advances in the Foundations of Euclidean Plane Geometry, The American Mathematical Monthly, vol. 66, pp. 2-17 (1955) Appendix I.
6. Hall 1943, p.247 Theorem 5.4
7. This can be done in several ways. A short description of the method used by Hall (1943) can be found in Dembowski (1968, p. 127).
8. Dembowski 1968, p. 129

References

• Albert, A. Adrian; Sandler, Reuben (1968). An Introduction to Finite Projective Planes. New York: Holt, Rinehart and Winston.
• Artzy, Rafael (2008) [1965], "Chapter 4 Axiomatic Plane Geometry", Linear Geometry, Dover, ISBN 978-0-486-46627-9
• Benz, Walter; Ghalieh, Khuloud (1998), "Groupoids associated with the ternary ring of a projective plane", Journal of Geometry, 61 (1–2): 17–31, doi:10.1007/bf01237490, S2CID 123135402
• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275
• Grari, A. (2004), "A necessary and sufficient condition so that two planar ternary rings induce isomorphic projective planes", Arch. Math. (Basel), 83 (2): 183–192, doi:10.1007/s00013-003-4580-9, S2CID 122203312
• Hall, Marshall, Jr. (1943), "Projective planes", Transactions of the American Mathematical Society, 54 (2), American Mathematical Society: 229–277, doi:10.2307/1990331, ISSN 0002-9947, JSTOR 1990331, MR 0008892{{citation}}: CS1 maint: multiple names: authors list (link)
• Hall, Marshall Jr. (1959), The Theory of Groups, New York: The MacMillan Company, MR 0103215
• Hughes, D.R. (1955), "Additive and multiplicative loops of planar ternary rings", Proceedings of the American Mathematical Society, 6 (6): 973–980, doi:10.1090/s0002-9939-1955-0073568-8, MR 0073568
• Hughes, Daniel R.; Piper, Fred C. (1973), Projective Planes, Graduate Texts in Mathematics (6), New York: Springer-Verlag, ISBN 0387900446, MR 0333959
• Martin, G.E. (1967), "Projective planes and isotopic ternary rings", The American Mathematical Monthly, 74 (10): 1185–1195, doi:10.2307/2315659, hdl:10338.dmlcz/101204, JSTOR 2315659, MR 0223972
• Pickert, Günter (1975), Projektive Ebenen, Berlin: Springer-Verlag, ISBN 3540072802
• Stevenson, Frederick (1972), Projective Planes, San Francisco: W.H. Freeman and Company, ISBN 071670443-9