It is well known that every Euclidean plane (E, L, α, ≡) is isomorphic to an affine plane AG(2,K) over a Pythagorean ordered commutative field K .There is a corrisponding theorem for hyperbolic panes (cf. [2]). For both proofs one first considers the group of motions, in particular the line reflections

Kreuzer, Alexander

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LIX (2004) – Fasc. I–II, pp. 263–270SPACES OF DIMENSION THREE WITH CONGRUENCEALEXANDER KREUZER1. Introduction.It is well known that every Euclidean plane (E,L, α,≡) is isomorphicto an affine plane AG(2,K) over a Pythagorean ordered commutative field K .There is a corrisponding theorem for hyperbolic panes (cf. [2]). For both proofsone first considers the group of motions, in particular the line reflections.Reflections at all are a powerful tool in order to prove geometric theorems.For example they have been used by J. Hjelmslev, G. Thomsen, A. Schmidt, F.Bachmann, E. Sperner and many others as a basic concept for an axiomatizationof plane absolute geometry. For characterizations of absolute planes one has toconsider products of reflections. One central theorem of absolute planes is thewell known three reflections theorem. Let A, B,C be lines, either through acommon point or perpendicular to a common line. Then the product of the linereflections about A, B,C is a line reflection again.In the famous paper [9], K. So¨rensen introduces planes with congruenceusing only three clear axioms and shows that for the definition of a line reflectionan order relation is not necessary. One needs only linear structure of (E,L)and the congruence relation ≡, but for the proof that the line reflection is amotion it seems that additional assumptions on the geometry are necessary. Forexample K. So¨rensen assumes that for given lines G1,G2, there exist distinctlines H1, H2 through a common point z which intersect G1,G2. But it sufficesto assume an exchange plane with congruence not using this property.In the plane with congruence (E,L, α,≡) two perpendicular lines may264 ALEXANDER KREUZERhave an empty intersection. In the case that in a plane any two perpendicularlines have an empty intersection, then the relation “perpendicular” together withthe “identity relation” is transitive (cf. 3.1).If there are two perpendicular lines with a non empty intersection, one canshow that for any line G and any point x ∈ G there exists a perpendicular linethrough x . This is true for finite or affine planes, but not known in general. Butfor spaces with congruence of dimension at least three an answer is possible.We assume that there are two points with a midpoint, all known examples inwhich the line reflections are motions have dimension two.We use the property that if two points have midpoint, then they have amidline in every plane. For a space of dimension greater than two one canshow that for any line G of a plane E and any point x ∈ G there is a uniqueperpendicular line through x in E . First we have to prove that any line reflectionis a motion. Therefore we need some steps. We show that the restriction of a linereflection to a plane is a motion (section 2). Then we consider a subspace U ofdimension three and construct some special lines in U for which the reflectionsare motions. Finally we generalize this result to all lines ofU . It follows that forany two points b, z there exists a unique point b′ ∈ z, b \{b} with (z, b) ≡ (z, b′)and also point reflections are motions. Hence one can introduce a loop operation+ in a more general context then in [1].This is the preparation of a talk given at Conference Combinatorics 2004 inCatania–Capomulini. For complete proofs of all propositions mentioned here,we refer to the papers [9], [4], [5].2. Planes with congruence.Let (P,L) denote a linear space or incidence space with the point set P ,the line set L and at least three points on every line, i.e., for any two points thereis exactly one line containing them and for any line L ∈L we have |L| ≥ 3.A subspace is a subset U ⊂ P such that for all distinct points x , y ∈ Uthe unique line passing through x and y , denoted by x , y, is contained in U .Let U denote the set of all subspaces. For every subset X ⊂ P we define thefollowing closure operation(1) :  (P)→ , X → X by X :=⋂U∈ X⊂UUFor U ∈ we call dimU := inf{|X | − 1 : X ⊂ U and X = U } the dimensionof U. A subspace of dimension two is a plane. For a set {a, b, c, . . .} we writea, b, c, . . . instead of {a, b, c, . . .}.SPACES OF DIMENSION THREE WITH CONGRUENCE 265We introduce the concept of a space (P,,≡) with congruence (cf. [9]).We assume in this section a linear space (P,) which satisfies the followingexchange condition.(EC) Let S ⊂ P and let x , y ∈ P with x ∈ S ∪ {y} \ S. Then y ∈ S ∪ {x}Let ≡ be a congruence relation on P × P , i.e. ≡ is a equivalence relationwith (a, b) ≡ (b, a) and (a, a) ≡ (b, c) if and only if b = c.We use the notation (x1, x2, x3) ≡ (y1, y2, y3) if and only if (xi , xj ) ≡(yi , yj ) for i, j ∈ {1, 2, 3}. (P,,≡) is a space with congruence if the axioms(W1), (W2) and (W3) are satisfied.(W1) Let a, b, c ∈ P be distinct and collinear, and let a′, b′ ∈ P with (a, b) ≡(a′, b′). Then there exists exactly one c′ ∈ a′, b′ with (a, b, c) ≡(a′, b′, c′).(W2) Let a, b, x ∈ P be non-collinear and let a′, b′, x ′ ∈ P with (a, b, x ) ≡(a′, b′, x ′). For any c ∈ a, b and c′ ∈ a′, b′ with (a, b, c) ≡ (a′, b′, c′) itholds (x , c) ≡ (x ′, c′).(W3) For a, b, x ∈ P non-collinear there exists exactly one x ′ ∈ {a, b, x} \ {x}with (a, b, x ) ≡ (a, b, x ′).We call a bijective mapping φ : P → P a motion, if (x , y) ≡ (φ(x ), φ(y))for all x , y ∈ P . We quote from [9]:Lemma 2.1.(i) If a, b, c are collinear points and a′, b′, c′ ∈ P with (a, b, c) ≡ (a′, b′, c′),then a′, b′, c′ are collinear.(ii) Any motion φ is a collineation.(iii) Two distinct points a, b have at most one point m ∈ a, b with (a,m) ≡(b,m).An important proposition is (1.5) of [9]. There implicitly the exchangecondition is used. We give here a proof using that in a plane the set of allmidpoints of two points is contained in a line. For a subspace U and pointsa, b ∈U we define MU (a, b) := {x ∈U : (a, x ) ≡ (b, x )}. We call MU (a, b) amidpoint, a midline, or a midplane of a, b, respectively, if it is a point, a line, ora plane, respectively.Lemma 2.2.(i) MU (a, b) is a subspace of U .(ii) Let a, b, x ∈ P be non-collinear points and let x ′ ∈ a, b, x \ {x} with(a, b, x ) ≡ (a, b, x ′). Then for any c ∈ a, b, x it holds (x , c) ≡ (x ′, c)if and only if c∈ a, b.266 ALEXANDER KREUZERProof. (i) Assume a = b and let x , y ∈ MU (a, b) be distinct points, hence(a, x , y) ≡ (b, x , y). By (W1) the points a, b, x , y are not collinear. Forw ∈ x , y by (W2) it follows (a, w) ≡ (b, w), i.e., w ∈ MU (a, b).(ii) For E := a, b, x we have a, b ∈ ME (x , x ′). Since x  ∈ ME (x , x ′) itfollows ME (x , x ′) = E . By (i), ME (x , x ′) is a subspace of E , and since E isan exchange plane and a, b∈ ME (x , x ′), it follows ME (x , x ′) = a, b. For a line L ∈L, x ∈ P \ L and a, b∈ L with a = b there exists by (W3)the unique point x ′ ∈ L ∪ {x} \ {x} with (a, b, x ) ≡ (a, b, x ′). By (W2), x ′ isindependent of the choice of a, b∈ L , hence we may denote x ′ = L(x ).The following mapping is called line reflectionL˜ : P → P; x →{x if x ∈ L,L(x ) if x  ∈ L.Clearly L˜ is an involutory bijection with z = L˜(z) if and only if z ∈ L .For the proof that the restriction of a line reflection to a plane is a motion,So¨rensen uses an additional property mentioned in the introduction. If twoperpendicular lines have an intersection, a distinct proof can be found in [4],which work also in general (cf. [5] (2.4), (2.5) and (2.6)). We obtain:Theorem 2.3. Let L be a line of a plane E . Then L˜|E is an involutory motionwith z = L˜(z) if and only if z ∈ L.We define for lines A, B ∈L:A ⊥ B ⇐⇒ A˜(B) = B and A = B.Lemma 2.4. If A ⊥ B, then B ⊥ A.Proof. Let a0, a1 ∈ A \ B be distinct points, b ∈ B \ A and b′ = A(b) ∈ B .Then (a0, a1, b) ≡ (a0, a1, b′). For a2 = B(a1) we have (b′, b, a1) ≡ (b′, b, a2),and by Lemma 2.2 (ii), (b, a2) ≡ (b, a1) ≡ (b′, a1) ≡ (b′, a2) impliesa2 ∈ A = a1, a0, hence by the involutoriety of B˜ we have B˜(A) = A. By [9], (1.15):Theorem 2.5. Let A, B,C be lines of a plane E through a point a. Then thereexists a line D ⊂ E through a with A˜B˜C˜|E = D˜|ELemma 2.6. Let E be a plane with distinct points a,m, c∈ E:(i) There exists at most one line L in E . with L(a) = c(ii) If (a,m) ≡ (c,m), there exists a unique line X ⊂ E with m ∈ X andX (a) = c, i.e., X = ME (a, c) (cf. [9] (1.7.2),(1.8),(4.2)).SPACES OF DIMENSION THREE WITH CONGRUENCE 267Lemma 2.7. The following propositions are equivalent (cf. [9] (4.3))(a) There are three collinear points a, b, b′ with (a, b) ≡ (a, b′).(b) For any points x , y there is a point y ′ ∈ x , y \ {y} with (x , y) ≡ (x , y ′).(c) There are two perpendicular lines A, B with A ∩ B = ∅.(d) For a line L of any plane E and x ∈ L, there exists a line G ⊂ E throughx with L ⊥ G.It is an open problem for a plane with congruence, if for x ∈ L there existsonly one perpendicular line G ⊥ L through x . For spaces an answer is given inCorollary 3.5 (i)).3. Spaces with congruence.In this section let (P,L,≡) be a linear space with congruence anddim P ≥ 3 satisfying the exchange condition .Let A ∈ L be any line. For any two points x , y ∈ P there is a threedimensional subspace U of P with A ∪ {x , y} ⊂ U . To show that A˜ is amotion it suffices to show that the restriction of A˜ to U is a motion. Thereforein the following we consider a three dimensional subspace U of P and a lineA ⊂ U .Remarks. We recall that if there are two points in P with a midpoint, thenby Lemma 2.7 for any distinct a,m ∈ A there exists a point c ∈ A with(a,m) ≡ (c,m).Further we remark that for affine spaces (P,L) with dim P ≥ 3 one can showthat perpendicular lines have a non empty intersection. With Lemma 2.7 itfollows that for affine spaces (P,L) with congruence and with dim P ≥ 3on every line there exist points a, c with a midpoint.By [9] (2.3) finite planes with congruence are affine planes. Further finiteaffine spaces with congruence have dimension two. Therefore finite spaces withcongruence are affine planes.One can also show that in the case that any perpendicular lines have anempty intersection, the relation “perpendicular” together with the “identity” istransitive:Theorem 3.1. Assume that for any two perpendicular lines A, B we haveA ∩ B = ∅. Then for distinct lines G, H, L of a plane E with L ⊥ G, Hwe have G ⊥ H .268 ALEXANDER KREUZERProof. Let a ∈ L . Since G˜|E , H˜ |E , are motions by Theorem 2.3,(G(a), H (a)) ≡ (GHG(a),G(a)) .Because L ⊥ G, H , clearly a, H (a), GHG(a) ∈ L , and by assumption andLemma 2.7, there exits only one point x ∈ L with (x ,G(a)) ≡ (H (a),G(a)),i.e., GHG(a) = H (a).Now it is easy to show that G˜ H˜ G˜|E = ˜˜G(H )|E , and since a /∈ H ,G˜(H ) = H is by Lemma 2.6 (i) the unique midline of a and H (a) in E . HenceG ⊥ H . We know already that the restriction of a line reflection to a plane is amotion. To show that the line reflections are motion at all, we start with theproof for some special lines. One can show that the reflection on the intersectionline of two mid planes is a motion. Then one shows that this is the case for everyline. First we consider the intersection of mid planes:Lemma 3.2.(i) Let a, c ∈ U be distinct points, then dimMU (a, c) ≤ 2. If a, c have amidpoint m ∈ a, c, then MU (a, c) is a plane of U .(ii) Let A, B ⊂ U be perpendicular lines, let a ∈ A \ B and b ∈ B \ A. If thepoint m = A ∩ B exists, then X := MU (a, B(a)) ∩ MU (b, A(b)) is a linewith m ∈ X .Proof. (i). Since a, c /∈ MU (a, c), dimMU (a, c) ≤ 2. If m is the midpoint ofa, c, then by Lemma 2.6 (ii) in every plane E of U containing a, c there existsa line L ⊂ MU (a, c), hence dimMU (a, c) = 2.(ii). The point m is the midpoint of a, B(a) and b, A(b). By (i), MU (b, A(b))is a plane with A ⊂ MU (b, A(b)) and by Lemma 2.6 (ii) there is a lineX ⊂ MU (b, A(b)) with m ∈ X and X (a) = B(a), hence X = MU (a, B(a)) ∩MU (b, A(b)). Using the method of [4] one can show (cf. [5]):Lemma 3.3. Let A, B ⊂ U be lines with A ⊥ B and for a ∈ A\B and b∈ B\Alet X := MU (a, B(a)) ∩ MU (b, A(b)) be a line. Then:X˜ |U is a motion.A generalization of this Lemma to all lines shows (cf. [5] (3.4)):Theorem 3.4. Let (P,L,≡) be a space with congruence and dim P ≥ 3. Ifthere are two points in P with a midpoint, then any line reflection is a motion.SPACES OF DIMENSION THREE WITH CONGRUENCE 269Now using line reflections one proves that in a plane for every line we havea unique perpendicular line through every point of this line:Corollary 3.5. Let A, B ∈L be lines in the three-dimensional subspaceU withA ⊥ B and m = A ∩ B. Then:(i) B is the unique line through m in A ∪ B with A ⊥ B.(ii) For a ∈ A \ {m} there exists exactly one point b ∈ A \ {a} with (a,m) ≡(b,m).(iii) There exists only one line X ⊂ U with X ⊥ A, B and it holds X ⊥ C forany line C ⊂ A ∪ B with m ∈C.(iv) Let z, a, b be distinct points, let a′ ∈ z, a \ {a} and b′ ∈ z, b \ {b} with(z, a) ≡ (z, a′) and (z, b) ≡ (z, b′), then (a, b) ≡ (a′, b′).In the following we assume, that there are two perpendicular lines witha non-empty intersection. Then by Lemma 2.7 and Corollary 3.5 for any twopoints a, b there is a unique point b′ ∈ a, b \ {b} with (a, b) ≡ (a, b′).Now we consider point reflections. For distinct points a, x ∈ P we denote witha(x ) the unique point a(x ) ∈ a, x \ {x} with (a, x ) ≡ (a, a(x )). We call thefollowing mapping point reflection:a˜ : P → P; x →{x if x = a,a(x ) if x = a.Now Lemma 3.5 (iv) imply:Theoem 3.6. Every point reflection a˜ is an involutory motion with x = a˜(x ) ifand only if x = a.With this result one can introduce a binary operation + in a more generalcontext then in [1]. Assume that any two distinct points a, a′ have a midpointm ∈ a, a′ with (a,m) ≡ (a′,m). For a fixed point 0∈ P we denote for any pointa ∈ P \ {0} the unique midpoint of 0 and a by a/2. We denote 0 = 0/2. Thenfor the point reflection corresponding to a/2 we havea˜/2(0) = a and a˜/2(a) = 0 .We define for points a, b the addition+ on the point set P by (cf. [1], [4])a + b := a˜/2 ◦ 0˜(b) = a˜/2 0˜(b) .Theorem 3.7. ([3], [1], [4]) Let (P,L,≡) be a space with congruence ofdim P ≥ 3 such that any two distinct points have a midpoint. Then fixing a270 ALEXANDER KREUZERpoint 0 ∈ P, an addition + can be defined on P such that (P,+) is a Bruckloop (for a definition cf. [8] ) with the neutral element 0. The point −a := 0˜(a)is the inverse of a ∈ P.(P,+) is associative if and only if for three points a, b, c∈ P the product a˜ b˜ c˜is a point reflection, too.REFERENCES[1] H. Karzel, Recent developments on absolute geometries and algebraization by K-loops, Discrete Math., 208/209 (1999), pp. 387–409.[2] H. Karzel - K. So¨rensen - D. Windelberg, Einfu¨hrung in die Geometrie, UTBVandenhoeck, Go¨ttingen, 1973.[3] A. Konrad, Nichteuklidische Geometrie und K-loops, Ph.D Thesis, TechnischeUniversita¨t Mu¨nchen, 1995.[4] A. Kreuzer, Reflection loops of spaces with congruence and hyperbolic incidencestructure, Comment. Math. Univ. Carolinae, 45 (2004), pp. 303–320.[5] A. Kreuzer - K. So¨rensen, Spaces with Congruence, to appear in Disc. Math..[6] H.-J. Kroll - K. So¨rensen, Pseudo-Euklidische Ebenen und Euklidische Ra¨ume, J.Geom., 8 (1976), pp. 95–115.[7] H.-J. Kroll - K. So¨rensen, Hyperbolische Ra¨ume, J. Geom., 64 (1998), pp. 141 –149.[8] H.O. Pflugfelder, Quasigroups and loops: Introduction, Heldermann Verlag, Ber-lin, 1990.[9] K. So¨rensen, Ebenen mit Kongruenz, J. Geom., 22 (1984), pp. 15–30.[10] K. So¨rensen, Eine Bemerkung u¨ber absolute Ebenen, J. Geom., 64 (1999), pp.160–166.Fachbereich Mathematik,Universita¨t Hamburg,Bundesstr. 55, 20146 Hamburg (GERMANY)

Spaces of dimension three with congruence

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Spaces of dimension three with congruence

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Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)