A Modern View of Geometry by Leonard M. Blumenthal

Author:Leonard M. Blumenthal
Language: eng
Format: epub
Publisher: INscribe Digital
Published: 2017-09-01T04:00:00+00:00

Figure 18

Since x · y = T(x, y, 0), t + z = T(t, 1, z) (x, y, z, t ∈ Γ), it follows that T(x, y, z)=xy + z= T(xy, 1, z) = T[T(x, y, 0), 1, z].

We recall that a coordinate system was established in the affine plane Π by selecting an arbitrary point O of Π for the origin and by choosing three pairwise distinct lines on O and designating them the x-line, the y-line, and the unit line. On the unit line any point distinct from O was chosen and labeled I, and an arbitrary abstract set Γ was selected (subject to the sole requirement that there is a one-to-one correspondence γ between the points of the line OI and the elements of Γ), and γ(O), γ(I) were labeled 0, 1, respectively. Then a procedure was defined that established a one-to-one correspondence between the points of Π and the elements of the set of all ordered pairs of elements of Γ. The ordered pair of elements of Γ corresponding to a point P of Π are the coordinates of P. A ternary operation T(a, m, b) was defined in Γ, and the system [Γ, T] was called a planar ternary ring.

Now if a different point O′ of Π were chosen for the origin, and three lines on O′ selected as the x′-line, y′-line, and unit line (with point I′ selected on it), the same set Γ could serve as the coordinate set, since there exists a one-to-one correspondence between the elements of Γ and the points of the new unit line [each line of Π contains the same number (finite or transfinite) of points]. But the ternary operation T′, defined by the new lines and new origin, would, in general, be quite different from the ternary operation T defined by the original coordinate system (though the domain of the two operators is the same set Γ), so a different planar ternary ring [Γ, T′] would result. If the first Desargues property is assumed for Π, both operators T, T′ are linear; that is T(x, y, z) = xy + z, T′(x, y, z) = [x y] ⊕ z, where the addition ⊕ and the multiplication are, in general, different from the addition + and multiplication · defined by using the original coordinate system. We have established the following result.

THEOREM V.3.1. If the first Desargues property is valid in the affine plane Π, every (planar) ternary ring defined in Π is linear.

Is the converse of this theorem valid; that is, if every (planar) ternary ring defined in the affine plane Π is linear, does it follow that the first Desargues property is valid in Π? The answer is affirmative, as we shall now show.

THEOREM V.3.2. If every (planar) ternary ring defined in an affine plane Π is linear, Π has the first Desargues property.

Proof. Let A*, A′, A″ and B* B′, B″ be any two triples of points of Π (each two points distinct) such that lines

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