# Betweenness plane geometry and its relationship with convex, linear, and projective plane geometries/Vahelsustasandi geomeetria ja selle seos kumer-, lineaar- ning projektiivtasandi geomeetriaga.

1. INTRODUCTIONIn the foundation of geometry the betweenness relation has fascinated the investigators for a long time. Already C. F. Gauss, in his letter to F. Bolyai (6 March 1832, see [1], p. 222), pointed to the absence of betweenness postulates in Euclid's treatment. Elimination of this defect was started 50 years later by Pasch [2]. Further development in the 19th century (through the works of G. Peano, F. Amodeo, G. Veronese, G. Fano, F. Enriques, and M. Pieri) led to Hilbert's fundamental Grundlagen der Geometrie [3], where the betweenness relation is subject to the axioms of connection and of order (I 1-7, II 1-5 of Hilbert's list), called by Schur [4] the projective axioms of geometry. Here the axioms of order II 1-5 are presented as dependent on the axioms I 1-7.

In the first decade of the 20th century Hilbert's projective axioms were investigated by Moore [5] and Veblen [6]. Moore indicated some redundancy in Hilbert's axioms of order, which Hilbert took into consideration in the following editions (e.g. in the seventh edition of [3]). But in these editions there was not considered the question asked by Henri Poincare in 1902: "Ne setait-il pas preferable de donner aux axiomes du deuxieme groupe une forme qui les affranchit de cette dependance et les separat completement du premier groupe?" (see [7], Appendice. Ch. 1: Les foundaments de la geometrie, page 112, first published in Journal des savants, Mai 1902). (1)

Veblen [6] was the first to respond to this question in 1904. He gave independent axioms of betweenness relation and showed that the lines and planes can then be defined as special sets of points.

Huntington [8-11] gave an elaborated system of axioms for the betweenness relation, but only in dimension one, i.e. for the case of a line.

This standpoint was developed further in Estonia, first by Nuut [12] in 1929 (for dimension one, as a geometrical foundation of real numbers). Then Sarv [13] proposed in 1931 an axiomatics for the betweenness relation for an arbitrary dimension n, extending the Moore-Veblen approach so that all axioms of connection, including also those concerning lines, planes, etc., became consequences. This self-dependent axiomatics was simplified and then perfected by Nuut [14] and Tudeberg (from 1936 Humal) [15]. As a result, an extremely simple axiomatics was worked out for n-dimensional geometry using only two basic concepts: "point" and "between".

The author of the present paper developed a comprehensive theory of the models of betweenness, based on this axiomatics [16]. At the same time he established [17] that in dimension > 2 this model reduces to a convex domain in an n-dimensional linear space over an ordered skew field. (2) As a whole, the theory of these models, including also the Huntington-Nuut theory for dimension 1, can be called betweenness geometry. The same term was introduced independently in a similar situation by Hashimoto [19] in 1958.

In 1970-80 Rubinshtein [20-22] developed (together with Rutkovskij) a theory of axial structures. This theory is tightly connected with betweenness geometry and uses some of its results (with exact references to [16,17]).

Independently there evolved also another approach, independent of the axioms of connection. In 1909, Schur [23] tried to work out a part of geometry based on the basic concepts "point" and "line segment" (Ger. Strecke). This approach was elaborated in 1961 by Prenowitz [24] (see also [25,26]). The segment was considered as the "join" of its endpoints, and so the join operation was introduced in the set of points. This approach was then developed as the theory of convexity spaces in the 1970s by Bryant and Webster [27], Doignon [28] and others (summarized in papers [29-31], and in monographs [32-33]).

In a recent paper [34] the author studied the relationship of betweenness geometries with join geometries, treated in [32]. He proved there Theorems 14 and 15, according to which betweenness geometry coincides with the Pasch-Peano geometry of [28]. Due to Theorems 16 and 18 of [34], betweenness geometry coincides with convex geometry, according to the Theorem in Sec. 1.3 of [28].

Betweenness and convex geometries were developed by several authors to so-called linear geometry (in the sense of [32,33,35,36]).

Betweenness geometry is tightly connected also with projective geometry (and also with absolute geometry, see [37]). Already in [16], [section] 20 and [17] it was established that in the 3-dimensional betweenness space every bundle of lines through a fixed point has the structure of a Desarguesian projective plane (see also [34], Sec. 8). In the further study of the betweenness planes (which can also be non-Desarguesian), which follows below, several constructions of the theory of projective planes will be useful. First, however, betweenness geometry must be recapitulated.

2. TOWARDS BETWEENNESS GEOMETRY

Recall that the author worked out betweenness geometry, considered here, more than 40 years ago in [16,17], being guided by [13,15]. (Independently it was initiated by Hashimoto [19].) Recently this geometry was recapitulated in [34] as follows.

Let S be a set, and let B be a subset in S x S x S (i.e. a ternary relation for S). Further (abc) will mean that (a, b, c) [member of] B, then b is said to be between (or inter) a and c. Moreover, let us denote

<abc> = (abc) [disjunction] (bca) [disjunction] (cab), [abc] = <abc> [disjunction] (a = b) [disjunction] (b = c) [disjunction] (c = a): (1)

The triplet (a, b, c) is said to be correct if <abc>, and collinear if [abc].

Definition 1. The pair (S, B) is called an interimity model and B the interimity relation if

B1: (a [not equal to] b) [??] [there exists] c, (abc), B2: (abc) = (cba), B3: (abc) [??] [logical not] (acb), B4: <abc> [disjunction] [abd] [??] [cda], B5: (a [not equal to] b) [??] [there exists] c, [logical not] [abc].

For any two different a, b, a [not equal to] b the subset ab = {x|(axb)} is called an interval with ends a and b, and the subset [L.sub.ab] = {x[parallel]xab]} is said to be a line through a and b.

For any non-collinear a, b, c the subset [P.sub.abc] = [Q.sub.a] [union] [Q.sub.b] [union] [Q.sub.c], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is called a plane through a, b, c. Here [Q.sub.b] and [Q.sub.c] are obtained by reordering a, b, c in the definition of [Q.sub.a], hence [P.sub.abc] does not depend on this reordering.

A one-to-one map f : S [right arrow] S of an interimity model onto itself is said to be a cointerrelation if (abc) [??] (f(a)f(b)f(c)), i.e. if the interimity relation remains valid by f. Then this f is also a collineation, because due to (1), [abc] [??] [f(a)f(b)f(c)], i.e. every collinear point-triplet maps into a collinear point-triplet, hence every line maps into a line.

The basic concept will be introduced by the following

Definition 2. If in an interimity model, in addition,

B6: [logical not] [abc] [conjunction] (abd) [conjunction] (bec) [??] [there exists] f, ((afc) [conjunction] (def)),

then this model is called a betweenness model and its relation is said to be the betweenness relation (see [16,17]).

A subsidiary concept gives now the following

Definition 3. If in an interimity model B6 is replaced by

B6': [logical not][abc] [conjunction] (abd) [conjunction] (aec) [??] [there exists] f, ((bfc) [conjunction] (dfe)),

then this model is called a betwixtness (3) model and its relation is said to be the betwixtness relation.

The connecting instrument for the betweenness and betwixtness models is the so-called Pasch postulate

P: [logical not] [abc] [conjunction] (bec) [conjunction] (d [member of] [P.sub.abc]) [conjunction] (d [not member of] [L.sub.bc]) [conjunction] (a [not member of] [L.sub.de]) [??] [there exists] f, (f [member of] [L.sub.de]) [conjunction] [(afb) [conjunction] (afc)].

The above postulates B6 and B6' can be considered as forms of this Pasch postulate in terms of the initial betweenness and betwixtness relation, respectively. (4)

It is natural to start with the interimity model.

Lemma 4. In an interimity model (abc) implies that a, b, c are three distinct points.

Proof. Indeed, B3 excludes b = c, and together with B2 excludes also b = a. Finally, a = c is impossible as well, because if c = a, then b [not equal to] a and due to B5 [there exists] d, [logical not] [abd], but on the other hand (abc) [??] <acb> and (a = c) [??] [acd], and these together imply due to B4 that [dba] = [abd], but this contradicts [logical not] [abd] and finishes the proof.

If a triplet a, b, c is correct, i.e. <abc>, then due to Lemma 4 here a, b, c are three different points, and due to B2, B3 only one of them is between the two others. Recall that if [abc], then a, b, c are said to be collinear. It is obvious that correctness and collinearity of any three a, b, c does not depend on their order, i.e.

<abc> = <bca> = <cab>, [abc] = [bca] = [cab]. (2)

Lemma 5. In an interimity model let a, b, c be collinear, i.e. [abc] and so (1) holds. Here only the following four possibilities occur:

1) (a = b) [disjunction] (b = c) [disjunction] (c = a), 2) (abc), 3) (bca), 4) (cab).

Each of them excludes the three others.

Proof. The first possibility follows from Lemma 4. Due to B2, B3, (abc) = (cba) [??] [logical not] (cab), (abc) [??] [logical not](acb) = [logical not] (bca). Due to the same Lemma 4, (abc) [??] [logical not] [(a = b) [disjunction] (b = c) [disjunction] (c = a)].

Lemma 6. In an interimity model there hold

[logical not] [abc] [conjunction] <abd> [??] [logical not] [acd], (3)

[logical not] [abc] [conjunction] [abd] [conjunction] [adc] [??] (a = d), (4)

(abc) [conjunction] (bcd) [??] <abd>, (5)

[logical not] [abc] [conjunction] (adb) [conjunction] (aec) [??] d [not equal to] e, (6)

Proof. Let us suppose for (3), by reductio ad absurdum, that [acd]. Then due to (1) and B4, <abd> [conjunction] [acd] = <adb> [conjunction] [adc] [??] [bca] = [abc], but this is impossible.

For (4), [logical not] [abc] [??] (a [not equal to] b), [logical not] [abc] [conjunction] [adc] [??] (b [not equal to] d), now by reductio ad absurdum,

[abd] [conjunction] (a [not equal to] b) [conjunction] (b [not equal to] d) [conjunction] (a [not equal to] d) [??] <abd> = <adb>

and then due to B4 <adb> [conjunction] [adc] [??] [bca] = [abc], but this is impossible.

For (5), due to Lemma 4, (abc) [conjunction] (bcd) [??] (a [not equal to] b) [conjunction] (b [not equal to] d). Also d [not equal to] a, because otherwise, due to B2, (bcd) = (bca) = (acb) and, now due to B3, [logical not] (abc), which is impossible. Further, due to (1), (abc) [conjunction] (bcd) = <bca> [conjunction] [bcd], and now due to B4, [adb], which is, due to (1), equivalent with [abd], but this together with (a [not equal to] b) [conjunction] (b [not equal to] d) [conjunction] (d [not equal to] d) implies <abd>, as needed.

For (6), (adb) [??] [abd], and now, by reductio ad absurdum, if one supposes d = e, then (aec) = (adc) [??] [adc], and (4) would yield a = d. On the other hand, due to (1), (adb) [??] a [not equal to] d, which gives a contradiction.

This finishes the proof.

For a line the following assertions can be proved, which show that in an interimity model the points a, b are not some specific points of a line [L.sub.ab], but can be exchanged by every two of its different points c, d. Indeed, there holds

Lemma 7. If c [member of] [L.sub.ab] and c [not equal to] a, then [L.sub.ac] = [L.sub.ab].

Proof. This is obvious if c = b. Otherwise [abc] [conjunction] (a [not equal to] b) [conjunction] (b [not equal to] c) [conjunction] (c [not equal to] a) [??] <abc> and, due to B4, <abc> [conjunction] [abx] [??] [cxa], thus x [member of] [L.sub.ab] [??] x [member of] [L.sub.ac]. But also <acb> [conjunction] [acy] [??] [bya], thus y [member of] [L.sub.ac] [??] y [member of] [L.sub.ab].

Using this Lemma two times, one obtains

Theorem 8. If in an interimity model two different points c, d belong to a line [L.sub.ab], then [L.sub.cd] = [L.sub.ab].

Hence, a line is uniquely determined by any two of its different points.

Recall that in the definition of a line [L.sub.ab] due to (1) [xab] = (xab) [conjunction] (abx) [conjunction] (bxa) [conjunction] (x = a) [conjunction] (x = b) (note that here a = b is excluded). Hence a and b divide the remaining part of [L.sub.ab] into three subsets: 1) ab = {x|(axb)} (note that, due to B2, ab = ba), 2) a/b = {x|(xab)}, and 3) b/a = {x|(abx) = (xba)}.

Recall that ab is the interval with ends a and b, further, a/b will be called its extension over an end a.

It follows that [L.sub.ab] = ab [union] (a/b) [union] (b/a) [union] a [union] b, i.e. a line [L.sub.ab] is a union of an interval, its ends, and its extensions over both ends.

Note that up to now only B1-B4 are used and, in an extreme case, S can consist only of the points of one and the same line [L.sub.ab].

Further let also B5 be taken along. Here [logical not] [abc] means that a, b, c are three non-collinear points, i.e. three different points, not belonging to one line.

If a, b, c are non-collinear, then they are said to be vertices; the intervals bc, ca, ab are the sides (opposite to a, b, c, respectively) of the triangle [DELTA]abc, which is considered as the union of all of them.

Here a/b and b/a are the extensions of the side ab, and ab [union] a [union] b is the closed side.

Note that the subset [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the definition of a plane [P.sub.abc] can now be interpreted as the union of points on the lines, which are determined by a vertex a of the triangle [DELTA]abc and the points of its opposite closed side.

The plane [P.sub.abc] itself can be interpreted as the union of the points on the lines, which are determined by any of the vertices and the points of its opposite closed side of a triangle [DELTA]abc.

An interimity model will turn into a betweenness model if one adds B6 to B1-B5. The premise [logical not] [abc] of B6 means that there exists a triangle [DELTA]abc. The other premises (abd) [conjunction] (bec) mean that there are d [member of] b/a and e [member of] bc, where the side bc and extension b/a have a common endpoint b.

Note that here the premises of B6' differ from those of B6 only by the fact that e [member of] bc is replaced by e [member of] ac and so bc is changed by the side ac, which does not have a common endpoint with the extension b/a.

The theory of the betweenness model, called betweenness geometry, is developed in [34], where the following theorems have been proved:

(1) Every betweenness geometry is also a betwixtness geometry, i.e. if B1-B6 hold, then also B6' holds (Theorem 11 in [34]).

(2) The interval ab is not empty but is an infinite subset (Theorem 12 in [34]).

(3) For a triangle [DELTA]abc, the subset {x|[there exists] y, (byc) [conjunction] (axy)} does not depend on the reordering of vertices a, b, c (Theorem 13 in [34]).

It is natural to call the subset considered in the last theorem the interior of the triangle [DELTA]abc. Here any permutation of a, b, c is admissible.

The interpretation of the Pasch postulate P can be detailed as follows. Its premises mean that there is a line [L.sub.ed], which is determined by a point e of a side bc of the triangle [DELTA]abc and a point d of the plane [P.sub.abc] of this triangle, and does not contain any of its vertices. The assertion is that this line must intersect at least one of the other two sides in a point f. (Both of these cannot intersect, because this is excluded by Lemma 6 in [34].) Briefly: If a line in a plane of a triangle intersects one side and does not contain any of the vertices, then it intersects one of the other two sides (but not both of them).

Theorems 14 and 15 of [34] state the following:

In betweenness geometry the Pasch postulate is valid. In an interimity geometry, the Pasch postulate P yields B6, i.e. one can obtain a betweenness geometry adding P, instead of B6, to B1-B5.

3. COORDINATES AND TERNARY OPERATION

Coordinates and the ternary operation, introduced for the projective plane by Hall [38] (see also [39,40]), can also be defined for a betweenness plane.

Let O, X, Y be some three non-collinear points of the betweenness plane and E be a point in the interior of the triangle [DELTA]OXY, i.e. [there exists] F, (OEF) [conjunction] (XFY). The line [L.sub.OX] with the point X excluded will be called the x-axis, the line [L.sub.OY] with Y excluded the y-axis, and the line [L.sub.XY] with Y excluded the [[xi]]-axis.

To the points of the line [L.sub.OE] the coordinates (a, a) will be assigned, where these a are symbols, different for different points. In particular, to O and E the symbols 0 and 1 will be assigned, respectively, thus O = (0, 0) and E = (1, 1). If for a point p the line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] intersects [L.sub.OE] in (a, a) and the line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] intersects [L.sub.OE] in (b, b), then to p the coordinates (a, b) will be assigned. Then the intersection points of these lines with the x-axis and y-axis, respectively, are (a, 0) and (0, b).

On the [[xi]]-axis the symbol [m] will be assigned to the point, which is collinear with (0, 0) and (1,m). For instance, [1] is collinear with O = (0, 0) and E = (1, 1) (and also with every (a, a)).

The ternary operation, which was originally defined for the projective plane [38-40], can be applied in the case of the betweenness plane. Namely, for the sides OX and OY of the triangle [DELTA]OXY let this operation be defined by the following construction.

Let (x, 0), [m], (0, b) be points of the appropriate sides. Due to B6' there exists a point q such that ((x, 0)qY) [conjunction] ((0, b)qX), and then p such that ((0, b)p[m]).

Further, [logical not] [(0, 0)(x, 0)Y] [conjunction] ((0, 0)(x, 0)X) [conjunction] ((x, 0)pY); hence, due to B6, there exists such a point (0, y) that ((0, 0)(0, y)Y). Here this y will be defined as the result of the ternary operation:

y = x o m / b; (7)

this definition can be extended naturally also to x = 0, m = 0, or b = 0.

This ternary operation has the following properties:

0 o m / b = x o 0 / b = b, (8)

1 o m / 0 = m o 1 / 0 = m, (9)

x o m / z = y can have only unique solution with respect to z, (10)

x o [m.sub.1] / [b.sub.1] = x o [m.sub.2] / [b.sub.2] (with [m.sub.1] [not equal to] [m.sub.2]) can have only unique solution with respect to x, (11)

system [a.sub.1] o m / b = [c.sub.1], [a.sub.2] o m / b = [c.sub.2] (with ([a.sub.1], [c.sub.1]) [not equal to] ([a.sub.2], [c.sub.2])) can have only unique solution with respect to the pair m, b. (12)

The properties (8), (9), and (10) follow immediately from the definition (7). For (11) the construction of x can be seen if one makes a suitable figure, uniqueness can be proved easily. (If [m.sub.1] = [m.sub.2], then no solution exists, in general, unless also [b.sub.1] = [b.sub.2], when the solution x becomes arbitrary.) To verify (12), let (7) be considered as the equation of the line [L.sub.[m](0,b)], and let us state that only one line can pass through two different points.

As a consequence, the following properties can be obtained:

x o m/b = c (m [not equal to] 0) can have only a unique solution with respect to x, (13)

a o [xi] / b = c (a [not equal to] 0) can have only a unique solution with respect to [xi]. (14)

Indeed, due to (11), the equation x o m / b = x o 0 / c (with m [not equal to] 0) can have only a unique solution x. But due to (8), x o 0 / c = c. This verifies (13). Further, due to (12), the system a o [xi] / y = c, 0 o [xi] / y = b (with a [not equal to] 0) can have only a unique solution with respect to the pair [xi], y, but due to (8), here y = b. This verifies (14).

Following Hall [38,40], natural operations + and x can be defined by this ternary operation, now for the betweenness plane:

a + b = (a o 1) / b, (15)

a x m = (a o m) / 0. (16)

These operations have the following properties:

a + 0 = 0 + a = a, (17)

m x 0 = 0 x m = 0, (18)

m x 1 = 1 x m = m, (19)

a + x = c can have only a unique solution with respect to x, (20)

x x m = c (m [not equal to] 0) can have only a unique solution with respect to x, (21)

a x [xi] = c (a [not equal to] 0) can have only a unique solution with respect to [xi]. (22)

The properties (17)-(19) follow immediately from (8) and (9). The properties (20)-(22) follow, respectively, from (12)-(14).

4. ALGEBRAIC EXTENSION OF A GIVEN BETWEENNESS PLANE

For a betweenness plane there exists a linear order on every line. This order on the lines through O = (0, 0) can be chosen so that E follows O, then on the x-axis (1, 0) follows O = (0, 0), and on the y-axis (0, 1) follows O = (0, 0). For all points (a, 0) of the side OX on the x-axis then a > 0, similarly, for all points (0, a) of the side OY on the y-axis a > 0.

Let us consider Eq. (20) with c = 0 and a > 0; i.e. the equation a + x = 0 with a > 0. It is natural to denote the unique solution of this equation, if it exists, by x = -a, so that a + (-a) = 0; here naturally -a < 0. The point (0, -a), if it exists, lies in the remaining part of the y-axis, except Y .

Considering the tetragon with vertices (a, 0), (a, a), [1], X, one can see that one of its diagonals intersects the y-axis [L.sub.OY] at the point (0, a), the other at the point (0, -a). Hence the pair of points (0, a), (0, -a) is harmonic with respect to the pair (0, 0), Y. But there is the possibility that this intersection point (0, -a) does not really exist in the considered betweenness plane. Then it will be added as an ideal point, following the procedure which is given for projective planes by Moufang [41] and Sperner [42] to expand a limited plane part by adding the ideal points.

For the points (0, -y) of the y-axis, including the ideal ones, each of which forms with (0, y) [member of] OY, y > 0, a harmonic pair with respect to the pair O, Y, as above, there exists -y < 0. The same procedure can be performed on the x-axis: the points (0, -x), harmonic with respect to O, X, including the ideal ones, can be added to the points (x, 0) [member of] OX.

The ternary operation can be extended as well, carrying the ideal points not only on the x- and y-axes, but also on the [[xi]]-axis. After that this operation works on a linearly ordered set (called in [39] the ternar).

Introducing the ideal points (x, y) as the intersection points of [L.sub.Y] (x,0) and [L.sub.X] (0, y), where one of (x, 0), (0, y) is ideal (or both of them are ideal), one extends the given betweenness plane to an ordered projective plane with a ternary operation. In the properties (10)-(14), (20)-(22) the words can have only may be then replaced by has, and one has the so-called complete linearly ordered ternar.

Theorem 9. By means of a complete linearly ordered ternar with elements a, b, ..., x, y, ..., a betweenness plane can be constructed, taking the pairs (a, b) as the points and considering ([a.sub.2], [b.sub.2]) being between ([a.sub.1], [b.sub.1]) and ([a.sub.3], [b.sub.3]) if either

(1) there exist m, b so that [b.sub.i] = [a.sub.i] o m / b for i [member of] {1, 2, 3} and [a.sub.2] is between [a.sub.1], [a.sub.3], like [b.sub.2] is between [b.sub.1], [b.sub.3] in this ordered ternar, or

(2) [a.sub.2] is between [a.sub.1], [a.sub.3], and [b.sub.2] = [b.sub.1] = [b.sub.3], or

(3) [a.sub.2] = [a.sub.1] = [a.sub.3] and [b.sub.2] is between [b.sub.1], [b.sub.3].

Proof. For the set of these points (a, b), with this relation "between", the axioms B1-B4 are satisfied, because they hold for the set of the ternar's elements, as the linearly ordered set. Here the lines are the subsets {(x, x o m / b)}, {(x, b)}, and {(a, y)}. For each of them there exist points not belonging to this line. Hence B5 is also satisfied.

For a line {(x, y)|y = x o m / c} the two complementary half-planes, whose edge is this line, are {(x, y)|x o m / c < y} and {(x, y)|x o m / c > y}. The validity of the Pasch postulate can be established in the following way.

On the betweenness plane just constructed, let the triangle with vertices ([a.sub.i], [b.sub.i]), i [member of] {1, 2, 3} be such that its vertices ([a.sub.1], [b.sub.1]) and ([a.sub.2], [b.sub.2]) belong to the two different half-planes above and the third vertex ([a.sub.3], [b.sub.3]) is not on the line above. Then [a.sub.1] o m / c < [b.sub.1] and [a.sub.2] o m / c > [b.sub.2], but [a.sub.3] o m / c [not equal to] [b.sub.3]. Here [a.sub.3] o m / c < [b.sub.3] or [a.sub.3] o m / c > [b.sub.3], but not both together. In the first case the vertices ([a.sub.2], [b.sub.2]) and ([a.sub.3], [b.sub.3]) are in the two different half-planes, i.e. the line above intersects the side between them. In the first case the same can be said for the vertices ([a.sub.1], [b.sub.1]) and ([a.sub.3], [b.sub.3]). So the Pasch postulate is valid, and thus the proof is finished.

If the complete linearly ordered ternar in this theorem is obtained by the above extension of the ternar of a given betweenness plane, then the new betweenness plane, constructed according to this theorem, can be seen as an algebraic extension of the given plane. The latter is then a convex subset of the extended betweenness plane. Here convex means that with any two different points a, b it contains also the interval ab.

5. COLLINEATIONS AND LENZ-BARLOTTI CLASSIFICATION

A one-to-one map f : S [right arrow] S of a betweenness plane onto itself is said to be a collineation if (abc) [??] (f(a)f(b)f(c)), i.e. if the betweenness relation remains valid under f. Then, due to (1), also [abc] [??] [f(a)f(b)f(c)], i.e. every collinear point-triplet maps into a collinear point-triplet, hence every line maps into a line, from this stems the term "collineation". If considering the projective plane obtained by algebraic extension, one has here a collineation in the sense of the theory of projective geometry. The only difference is now that the linear order on every line of the betweenness plane remains invariable.

If the collineation has a fixed point z and a fixed line A, so that all lines through z and all points of A also remain fixed, then this collineation is called (z, A)-collineation, with centre z and axis A.

If there is a fixed point z and a fixed line A on the plane, so that the group of all (z, A)-collineations acts transitively on this plane, then the plane is said to be (z, A)-transitive. There exist the algebraic properties of the betweenness plane, which are equivalent to the (z, A)-transitivity.

Let the coordinates on the (z, A)-transitive betweenness plane be taken so that [L.sub.XY] = A and Y = z, and let us consider the (Y, [L.sub.XY])-collineation f such that f(O) = (0, b). Then all lines through Y remain invariant by f, like all points of the line [L.sub.XY], in particular X, [1], and [m].

According to the construction, which led above to the result (1), there is a point p with coordinates (x, x o m / b), and a point u which has, due to (10), the coordinates (x, x x m). There is a point w which belongs to the line [L.sub.O[1]] and thus has the coordinates (x x m, x x m). Due to (9), there is a point r with coordinates (x x m, x x m + b).

Since the points [m] and [1] are fixed by the f under consideration, like the lines through Y, it holds that f : u [right arrow] p, f : w [right arrow] r. Also the point X is fixed. Since u, w are collinear, lying on a line through X, the corresponding p, r must lie on a line through X. Therefore their second coordinates must be

x o m / b = x x m + b. (23)

Hence for a (Y, [L.sub.XY])-transitive betweenness plane the ternary operation is composed of two binary natural operations.

Further, it can be proved that, with respect to the natural operation of addition, the considered linearly ordered ternar forms a group. It is sufficient to show that this addition is associative, having in mind the properties (17) and (20).

Let us consider the (Y, [L.sub.XY])-collineation f above. Here y = x transforms to y = x + b; x = c to x = c; (c, c) to (c, c + b); y = c to y = c + b; and also x = a to x = a.

Hence f((a, c)) = (a, c + b). Let now the same be done with (Y, [L.sub.XY])-collineation g, defined by g(O) = (0, d). The result is that g((u, v)) = (u, v + d). For the product gf there exist

(gf)(O) = g(f(O)) = g((0, b)) = (0, b + d).

Consequently, in general, (gf)((a, c)) = (a, c + (b + d)). But g(f(a, c)) = = g((a, c + d)) = (a, (c + b) + d). Hence

c + (b + d) = (c + b) + d, (24)

i.e. addition has, indeed, the associativity property.

Conversely, if the ternar of a betweenness plane has the properties (23) and (24), i.e. if a) the ternary operation is composed of two natural binary operations: addition and multiplication, and b) with respect to the addition one has a linearly ordered group, then this plane is (Y, [L.sub.XY])-transitive.

Indeed, let us consider for an arbitrary b a map f, which leaves Y and [m] invariant and transforms (a, c) to (a, c + b), and also leaves the lines [L.sub.XY] and x = a invariant and transforms the line y = x x m + t to the line y = x x m + (t + b). This f is a collineation, because if a point (a, c) lies on the line y = x x m + t, then c = a x m + t, and thus

c + b = (a x m + t) + b = a x m + (t + b),

i.e. the point (a, c + b) lies on the line y = x x m + (t + b). Moreover, this collineation f is a (Y, [L.sub.XY])-collineation. Since b is arbitrary, the plane is (Y, [L.sub.XY])-transitive. For the projective planes Lenz [43] gave in 1954 a classification considering the set L of all pairs (z, A) with z [member of] A such that the plane is (z, A)-transitive. He obtained seven classes: I (this set is empty)--VII (this set contains all such pairs), where IV and VI are both subdivided into two subclasses: a and b.

Three years later Barlotti [44] perfected this classification by abandoning the condition z [member of] A and excluding class VI, which turned out to be empty, according to a result by H. F. Gingerich in 1945 (see [45], p. 103).

In [46] the Lenz-Barlotti classification is presented as follows (the classes of [43], which turned out to be empty, are excluded).

Let B be the set of all pairs (z, A) with z 62 [not member of] such that the plane is (z, A)-transitive. The classes are now the following.

[I.sub.1]. L = 0 = B.

[I.sub.2]. L = 0, [absolute value of B] = 1.

[I.sub.3]. L = 0, [absolute value of B] = 2, where [absolute value of B] is the power of the set B (i.e. B = {([z.sub.1], [A.sub.1]), ([z.sub.2], [A.sub.2])}) and [z.sub.1] [member of] [A.sub.2] [A.sub.1], [z.sub.2] [member of] [A.sub.1] \ [A.sub.2].

[I.sub.4]. L = 0, [absolute value of B] = 3, and B= {(a, [L.sub.bc]), (b, [L.sub.ca]), (c, [L.sub.ab])} with non-collinear a, b, c.

[I.sub.5]. L = 0. There exists a point a and a line A with a [member of] A, and a bijection [beta] of A \ {a} onto the set of lines X with a [member of] X,X [not equal to] A such that B = {(x, [beta](x))|x [member of] A \ {a}}.

[II.sub.1]. [absolute value of L] = 1 and B = 0.

[II.sub.2]. [absolute value of L] = 1 = [absolute value of B] and for B = {(b, B)} there is b [member of] A and a [member of] B.

[II.sub.3]. [absolute value of L] = 1 and B is such as in [I.sub.6], while (a, A) [member of] L.

[III.sub.1]. There exists only one point a and one line Z with a [member of] Z so that L = {(z, [L.sub.za])|z [member of] Z} and B = 0.

[III.sub.2]. There exists only one point a and one line Z with a [member of] Z so that L = {(z, [L.sub.za])|z [member of] Z} and B = {(a, Z)}.

[IV.sub.a1]. There exists only one line A with L = {(z, A)|z [member of] A} and B = 0.

[IV.sub.a2]. There exists only one line A with L = {(z, A)|z [member of] A}. There exist a, a' [member of] A, a [not equal to] a' and the lines X, Y so that B = {(a, X)|a' [member of] X} [union] {(a', Y)|a [member of] Y}.

[IV.sub.a3]. There exists only one line A with L = {(z, A)|z [member of] A}. There exists an involution [beta] : A [right arrow] A without fix points so that B = {(x, X)|x [member of] A, X [not equal to] A, [beta](x) [member of] X}.

[IV.sub.b1], dual to [IV.sub.a1].

V. There exists only one point a and one line Z with a [member of] Z so that L = {(z, Z)|z [member of] Z} [union] {(a, A)|a [member of] A} and B = 0.

[VII.sub.1]. L = {(z, A)|z [member of] A} and B = 0.

[VII.sub.2]. L = {(z, A)|z [member of] A} and B [not equal to] L.

The same classification is presented in [47] as a table, where also the corresponding ternar of each class is characterized. Moreover, the special features of this classification for the ordered projective planes (and thus for betweenness planes) are discussed in [47] (see also [48]).

6. TRANSLATION PLANE AND ORDERED QUASIFIELDS

If for a betweenness plane its algebraic extension is (z, A)-transitive for an arbitrary centre z [member of] A, then it is called the translation plane with respect to the axis A.

For this, it is sufficient that this plane is ([z.sub.1], A)-transitive and ([z.sub.2], A)-transitive for two different centres [z.sub.1], [z.sub.2], on the axis A, i.e. [z.sub.1], [z.sub.2] [member of] A; [z.sub.1] [not equal to] [z.sub.2].

Indeed, let then z be an arbitrary point of A, z [not equal to] [z.sub.1], [z.sub.2], and L a line through z, L [not equal to] A. Further, let u, v be two arbitrary points of L such that the lines [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] intersect at a point w. There exists a ([z.sub.1], A)-collineation [f.sub.1] such that [f.sub.1](u) = w, and a ([z.sub.2], A)-collineation [f.sub.2] such that [f.sub.2](w) = v. Then f = [f.sub.2][f.sub.1] is a desirable (z, A)-collineation, which shows that one has the (z, A)-transitivity.

Let us consider now a betweenness plane which is a translation plane with respect to the axis L. The coordinates on such a plane can be taken so that [L.sub.XY] = A. This plane is (Y, [L.sub.XY])-transitive and also (X, [L.sub.XY])-transitive. Due to the result of Sec. 5, the ternary operation is composed of two binary operations: addition and multiplication, and with respect to the first of them, one has a linearly ordered group. In the proof of this result there is a (Y, [L.sub.XY])-collineation f with the property f((a, c)) = (a, c + b). Similarly there exists an (X, [L.sub.XY])-collineation g with the property g((a, c)) = (a, c + d).

It appears that fg = gf. Indeed, for an arbitrary point u, u [not member of] [L.sub.XY], one has then collinears Y, u, f(u), X, u, g(u), Y, g(u), f(g(u)), and X, f(u), g(f(u)). Here X, f(u), g(f(u)) = (gf)(u) are also collinear, like Y, g(u), f(g(u)) = (fg)(u). Hence (gf)(u) and (fg)(u) are both the intersection points of two different lines [L.sub.Xf](u) and [L.sub.Yg](u), thus gf = fg, as asserted.

Since (gf)(a, c) = g(a, c + b) = (a, c + b + d) and (fg)(a, c) = f(a, c + d) = (a, c + d + b), one has

b + d = d + b, (25)

i.e. the linearly ordered group is an Abelian group with respect to addition.

The complete linearly ordered ternar has the properties (19), (21), and (22) with respect to multiplication, which shows that one has here a quasigroup with unit, i.e. a loop.

There are some properties which connect these two binary natural operations: addition and multiplication of the complete linearly ordered ternar.

Let us consider an (X, [L.sub.XY])-collineation h such that h(0, 0) = (b, 0). Then y = x transforms to y = x - b; y = a to y = a; (a, a) to (a + b, a); x = a to x = a + b; y = a x m to y = a x m; (a, a x m) to (a + b, a x m); and y = x x m to y = x x m - b x m.

Since the point (a, a x m) lies on the line y = x x m, and the point (a + b, a x m) lies on the line y = x x m - b x m, there holds a x m = (a + b) x m - b [cents] m, thus

(a + b) x m = a x m + b x m, (26)

i.e. the multiplication is right-distributive with respect to the addition.

The properties (11) and (23) give together the following:

x x [m.sub.1] = x x [m.sub.2] (with [m.sub.1] [not equal to] [m.sub.2]) has a unique solution with respect to x. (27)

The ternar, 1) whose ternary operation is composed of its two binary natural operations, 2) which is an Abelian group with respect to the addition and a loop with respect to the multiplication, and 3) which has the properties (26) and (27), has been considered in [49], therefore it is called a Veblen-Wedderburn system (also said to be a quasifield).

For the betweenness plane this is, of course, the linearly ordered quasifield (considered, e.g., by Jousson [50]).

7. ALTERNATIVE QUASIFIELD AND MOUFANG-TYPE PLANE

Let us consider the betweenness plane, whose algebraic extension gives the ordered projective plane (with the excluded line [L.sub.XY]), which is a translation plane with respect to every line through the point Y as the axis. It is known that for this it is sufficient that this plane is the translation plane with respect to two lines intersecting in Y (see [40], Theorem 20.5.1).

Theorem 10. If the betweenness plane is such that its algebraic extension is a translation plane with respect to every line going through the point Y, then

(1) all its lines, different from [L.sub.XY], are given by the linear equations x = c and y = x x m + b, and

(2) the coordinates are subjected to the following requirements:

2.1) according to addition, they form a linearly ordered Abelian group,

2.2) according to multiplication, they form a loop (quasigroup with unit),

2.3) (a + b) x m = a x m + b x m,

2.4) a x (s + t) = a x s + a x t,

2.5) every element a [not equal to] 0 has an inverse [a.sup.-1] such that a x [a.sup.-1] = 1 = [a.sup.-1] x a,

2.6) [a.sup.-1] x (a x b) = b.

Proof. Among the lines through the point Y there is the line [L.sub.XY]. Hence, due to the results of the previous Sec. 6, here (1) and 2.1), 2.2), and 2.3) are valid.

To prove the remaining three requirements, let a (Y, A)-collineation be considered, whose axis A is the line x = 0, going through the centre Y (in such a case this collineation is called an elation, according to [40], [sections] 20.2), and which maps the point X = [0] into the point [m]. Then all points (0, b) are invariant, like the lines [L.sub.XY] and x = c. Here successively [0] [??] [m], (0, b) [??] (0, b), y = b [??] y = x x m + b, x = a [??] x = a, (a, b) [??] (a, a x m + b). In particular, (1, t) [??] (1, m + t) and (0, 0) [??] (0, 0), hence y = x x t [??] y = x x (m + t). But (a, a x t) [??] (a, a x m + a x t), where (a, a x t) belongs to the line y = x x t. Therefore (a, a x m + a x t) belongs to the line y = x x (m + t), hence the left distributivity requirement 2.4) is satisfied:

a x m + a x t = a x (m + t). (28)

Now let us consider another elation, whose centre is (0, 0), axis x = 0, and which maps X = [0] into (-1 -a, 0). For this (0, 1 + a) [??] (0, 1 + a), thus y = 1 + a [??] y = x + 1 + a. Further, [0] [??] (-1-a, 0), (0, b + a x b) [??] (0, b + a x b), hence y = b + a x b [??] y = x x b + b + a x b. Now y = 1 + a [??] y = x + 1 + a, y = x x (1 + a) [??] y = x x (1 + a), hence (1, 1 + a) [??] (d, d + 1 + a), if a [not equal to] 0, therefore

d x (1 + a) = d + 1 + a.

Further, Y [??] Y, (1, 1 + a) [??] (d, d + 1 + a), thus x = 1 [??] x = d. Consequently,

y = x x (b + a x b) [??] y = x x (b + a x b),

y = b + a x b [??] y = x x b + b + a x b,

hence (1, b + a [cents] b) [??] (d, d x (b + a x b)), where d x (b + a x b) = d x b + b + a x b.

Let now not only a [not equal to] 0 but also (-1-a, 0) [not equal to] (0, 0), i.e. a [not equal to] -1. For such an element there exists a d, so that d x (1+a) = d+1+a and d x (b + a x b) = d x b + b + a x b for an arbitrary b. Taking d = u + 1 and using the distributive property, one finds that ua = 1 and u x (a x b) = b. For a = -1 these relations remain valid if u = -1. Since for u [not equal to] 0 there exists such an element v that vu = 1 and v x (u x a) = a, it follows that v = a. Consequently, u = [a.sup.-1], and the result will be 2.5): a x [a.sup.-1] = [a.sup.-1] x a = 1, and also 2.6): [a.sup.-1] x (a x b) = b. This finishes the proof.

One more identity may be deduced following Bruck [51]. Let there be

[[y.sup.-1] - [(y + [z.sup.-1])sup.-1] x [y x (z x y) + y] = t,

where y [not equal to] 0, y [not equal to] -[z.sup.-1]. Multiplying here by y + [z.sup.-1], one obtains

(y + [z.sup.-1]) x t = (y + [z.sup.-1])[zy + 1 - [(y + [z.sup.-1]).sup.-1](y x (z x y)) - [(y + [z.sup.-1]).sup.-1]y] = y x (z x y) + y + y + [z.sup.-1] - y x (z x y) - y = y + [z.sup.-1].

Hence, t = 1, and the elements [y.sup.-1] - [(y + [z.sup.-1]).sup.-1] and y x (z x y) + y are inverse to each other. Then for an arbitrary x there exists

[[y.sup.-1] - [(y + [z.sup.-1]).sup.-1]] x [(y x (z x y)) x x + y x x] = x.

If one denotes

[[y.sup.-1] - [(y + [z.sup.-1]).sup.-1]] x [y x z x (y x x)) + y x x] = w,

then

(y + [z.sup.-1]) x w = (y + [z.sup.-1])[z x (y x x) + x] - y x [z x (y x x)] - y x x = y x x + [z.sup.-1] x x = (y + [z.sup.-1]) x x.

Consequently, w = x. Comparing the expressions for w and x, one obtains

[y x (z x y)] x x = y x [z x (y x x)]: (29)

This identity is called the Moufang identity, it is valid also for the excluded values y = 0, y = -[z.sup.-1], thus for all elements.

In particular, for z = 1 there follows the left alternativity:

(y x y) x x = y x (y x x): (30)

A ternar, satisfying the conditions (2.1)-(2.6), (29), and (30), is called an alternative quasifield. A projective plane with such a ternar is said to be a Moufang plane.

A betweenness plane whose algebraic extension gives a Moufang projective plane is said to be the Moufang-type betweenness plane. For such a plane the following algebraic result (Skornyakov [52], Bruck and Kleinfeld [53]) is important:

Every ordered alternative quasifield is a skewfield.

For betweenness planes this means:

Every Moufang-type betweenness plane is Desarguesian.

Note that for the ordered projective planes this statement is formulated in [47], p. 264, and in [46], p. 135.

ACKNOWLEDGEMENTS

The author would like to thank the referee Jaak Peetre and an anonymous referee whose suggestions helped to improve the exposition. He also thanks his grandson Imre for technical assistance.

Received 5 April 2007, in revised form 8 June 2007

REFERENCES

[1.] Gauss, C. F. Werke, Bd. 8. Grundlagen der Geometrie. Teubner, Gottingen, 1900.

[2.] Pasch, M. Vorlesungen uber neuere Geometrie. Teubner, Leipzig, 1882 (2nd ed. by Pasch, M. and Dehn, M., Springer, Berlin, 1926).

[3.] Hilbert, D. Grundlagen der Geometrie, Festschrift zur Feier der Enthullung des Gauss-Weber-Denkmals in Gottingen. Teubner, Leipzig, 1899 (Siebente umgearbeitete und vermehrte Auflage, Teubner, Leipzig/Berlin, 1930).

[4.] Schur, F. Ueber die Grundlagen der Geometrie. Math. Ann., 1901, 55, 265-292.

[5.] Moore, E. H. On the projective axioms of geometry. Trans. Amer. Math. Soc., 1902, 3, 142-158.

[6.] Veblen, O. A system of axioms for geometry. Trans. Amer. Math. Soc., 1904, 5, 343-384.

[7.] Poincare, H. Dernieres pensees. Ed. Ernest Flammarion, Paris, 1913.

[8.] Huntington, E. V. and Kline, J. R. Sets of independent postulates for betweenness. Trans. Amer. Math. Soc., 1917, 18, 301-325.

[9.] Huntington, E. V. Set of completely independent postulates for cyclic order. Proc. Nat. Acad. Sci. USA, 1924, 10, 73-78.

[10.] Huntington, E. V. A new set of postulates for betweenness with proof of complete independence. Trans. Amer. Math. Soc., 1926, 26, 6-24.

[11.] Huntington, E. V. The four principal types of order. Trans. Amer. Math. Soc., 1935, 38, 1-9.

[12.] Nuut, J. Topologische Grundlagen des Zahlbegriffs. Acta Comment. Univ. Tartuensis (Dorpatensis), 1929, A15, No. 5.

[13.] Sarv, J. Geomeetria alused. Acta Comment. Univ. Tartuensis (Dorpatensis), 1931, A19, No. 4.

[14.] Nuut, J. Einige Bemerkungen uber Vierpunktaxiome. Acta Comment. Univ. Tartuensis (Dorpatensis), 1932, A23, No. 4.

[15.] Tudeberg, A. Uber die Beweisbarkeit einiger Anordnungsaussagen in geometrischen Axiomensystemen. Acta Comment. Univ. Tartuensis (Dorpatensis), 1934, A26, No. 6.

[16.] Lumiste, U. Geomeetria alused. Tartu Riiklik Ulikool, Tartu, 1964.

[17.] Lumiste, U. On models of betweenness. Eesti NSV Tead. Akad. Toim. Fuus.-Mat.- ja Tehn.-tead., 1964, 13, 200-209 (in Russian, summaries in Estonian and English, Math. Rev., 1965, 29.6360).

[18.] Pimenov, R. I. Spaces of kinematic type (a mathematical theory of space-time). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 1968, 6 (in Russian, English transl.: Kinematic Spaces. Consultation Bureau, New York/London, 1970).

[19.] Hashimoto, J. Betweenness geometry. Osaka Math. J., 1958, 10, 147-158.

[20.] Rubinshtein, G. Sh. On an intrinsic characteristic of relatively open convex sets. Dokl. Akad. Nauk SSSR, 1970, 193, 5, 1004-1007 (in Russian).

[21.] Rubinshtein, G. Sh. Duality in mathematical programming and some problems of convex analysis. Uspekhi Mat. Nauk, 1970, 25, 5(155), 171-201 (in Russian).

[22.] Rubinshtein, G. Sh. and Rutkovskij, N. V. Axial structures and foundations of geometry. Dokl. Akad. Nauk SSSR, 1980, 254, 6, 1342-1345 (in Russian).

[23.] Schur, F. Grundlagen der Geometrie. Teubner, Leipzig, 1909.

[24.] Prenowitz,W. A contemporary approach to classical geometry. Amer. Math. Month., 1961, 68 (No. 1, Part II), 1-67.

[25.] Prenowitz,W. and Jantosciak, J. Geometries and join spaces. J. reine angew. Math., 1972, 257, 100-128.

[26.] Prenowitz, W. and Jordan, M. Basic Concepts of Geometry. Blaisdell, New York, 1965.

[27.] Bryant, V. W. and Webster, R. J. Generalizations of the theorems of Radon, Helly and Caratheodory. Monatsh. Math., 1969, 73, 309-315.

[28.] Doignon, J.-P. Caracterisations d'espaces de Pasch-Peano. Acad. Roy. Belg. Bull. Cl. Sci., 1976, 62, 679-699.

[29.] Bryant, V. W. and Webster, R. J. Convexity spaces. I. J. Math. Anal. Appl., 1972, 37, 206-213.

[30.] Bryant, V. W. and Webster, R. J. Convexity spaces. III. J. Math. Anal. Appl., 1977, 57, 382-392.

[31.] Cantwell, J. and Kay, D. C. Geometric Convexity. III. Trans. Amer. Math. Soc., 1978, 246, 211-230.

[32.] Prenowitz, W. and Jantosciak, J. Join Geometries. A Theory of Convex Sets and Linear Geometry. Springer, New York, 1979.

[33.] Coppel, W. A. Foundations of Convex Geometry. Cambridge University Press, Cambridge, 1998.

[34.] Lumiste, U. Relationship between join and betweenness geometries. Proc. Estonian Acad. Sci. Phys. Math., 2005, 53, 131-153.

[35.] Pambuccian, V. A methodologically pure proof of a convex geometry problem. Beitrage zur Algebra und Geometry, 2001, 42, 401-406.

[36.] Pambuccian, V. Forms of the Pasch axiom in ordered geometry. J. Geometry (submitted).

[37.] Sorensen, K. Eine Bemerkung zu absoluten Ebenen. Beitr. Algebra Geom., 1993, 24, 23-30.

[38.] Hall, M. Projective planes. Trans. Amer. Math. Soc., 1943, 54, 229-277.

[39.] Skornyakov, L. A. Projective planes. Uspekhi Mat. Nauk, 1951, 6, 6(46), 112-154 (in Russian).

[40.] Hall, M. The Theory of Groups. MacMillan, 1959 (Russian translation: Moskva, 1962).

[41.] Moufang, R. Die Einfuhrung der idealen Elemente in die ebene Geometrie mit Hilfe des Satzes vom vollstandigen Vierseit. Math. Ann., 1931, 105, 759-778.

[42.] Sperner, E. Zur Begrundung der Geometrie im begrenzten Ebenenstuck. Schriften der Konigsberger Gelehrten Ges. (Math. Naturwiss. Kl.), 1938, 14. Jahr Heft 6, 121-143, Gesammelte Werke. Heldermann Verlag, Lemgo, 2005, 135-157.

[43.] Lenz, H. Kleiner Desarguesscher Satz und Dualitat in projectiven Ebenen. Jahresb. Deutsch. Math.-Verein., 1954, 57, 381-387.

[44.] Barlotti, A. Le possibili configurazioni del sistema delle coppie punto-retta (A,a) per cui un piano grafico risulta (A,a)-transitivo. Boll. Un. Mat. Ital., 1957, 12, 212-226.

[45.] Pickert, G. Projektive Ebenen. Springer, Berlin, 1955 (2nd ed. 1975).

[46.] Karzel, H. und Kroll, H.-J. Geschichte der Geometrie seit Hilbert. Wissenschaftliche Buchgesellschaft, Darmstadt, 1988.

[47.] Priess-Crampe, S. Angeordnete Strukturen: Gruppen, Korper, projektive Ebenen. Springer, Berlin, 1983.

[48.] Jagannathan, T. V. S. Ordered projective planes. J. Indian Math. Soc., 1972, 36, 65-78.

[49.] Veblen, O. and Wedderburn, J. H. M. Non-Desarguesian and non-Pascalian geometries. Trans. Amer. Math. Soc., 1907, 8, 379-388.

[50.] Jousson, J. Konstruktion archimedischer Anordnungen von freien Ebenen. Result. Math., 1981, 4, 55-74.

[51.] Bruck, R. H. Difference sets in a finite group. Trans. Amer. Math. Soc., 1955, 78, 464-481.

[52.] Skornyakov, L. A. Alternativkorper. Ukrain. Mat. Zhurn., 1950, 2, 70-85 (in Russian).

[53.] Bruck, R. H. and Kleinfeld, E. The structure of alternative division rings. Proc. Amer. Math. Soc., 1951, 2, 878-890.

(1) "Wouldn't it be better to give the axioms in the second group a form which makes them free of this dependence and separates them from the first group?"

(2) Later Pimenov [18] (in Appendix: Local betweenness relation) called this perfected axiomatics the Humal-Lumiste axiomatics and its model in dimension 2, when the above result cannot be used, the Lumiste plane.

(3) Here the word "betwixt" has been in mind (which, according to dictionaries, is now archaic except in the expression betwixt and between), as well as the word "interim".

(4) They are called the outer and inner Pasch axioms, respectively, and denoted by OP and IP (see [36], where it is proved that IP does not imply OP, while OP does imply IP, according to [6], see also [16], Theorem 13, and [34], Theorem 11, this means that every betweenness model is also a betwixtness model, but not vice versa).

Ulo Lumiste

Institute of Pure Mathematics, Faculty of Mathematics and Computer Science, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia, lumiste@math.ut.ee