DEMONSTRATE MATIIEMATJCA Vol. XXVII No 1 1994 Michal Muzalewski ON ORTHOGONALITY RELATION DETERMINED BY A GENERALIZED SEMI-INNER PRODUCT In the paper we arc dealing with structures QJD = (V, W; IL), where V and W are vector spaccs over a skew-field K. We assume that the relation IL C V X W satisfies some very weak axioms which are usually satisfied by the orthog- onality relations and wc prove that then the relation IL is induced by an appriopiate form f: V x W A', i.e IL = iL^, where xlL^y :<£> £(x,y) = 0, for x € V, y <E W. 1. Introductory definitions and facts DEFINITION 1. Let V, W be left vector spaccs over skew-fields K and L, respectively. Let us denote L(V) := {L: L is a subspace of V A dim L = 1} , H(V) := {H: H is a subspace of V A codim H = 1} and S(V) := {U: U is a subspace of V} . In the case of finite dimensional projective spaccs by a correlation in ^3(V) := (L(V), H(V); C) wc mean any isomorphism K:<P(V)-(H(V),L(V);D). which is usually identified with its natural extension K:S(V) —• S(V). If dim V = oo, then there is no such function K, therefore we must change and generalize our definition. 54 M. Mnzalcwski A map K: L(V) H(W)[7 : H(V) L(W)], is said to be correlation [dual correlation], if it satisfies the following condition: Corrn : L0 C Lt + ... + in kL0 D kLj n... n «Ln, [DCorrn : H0 D H[ n ...nHn » 7H0 C 7H1+ ... + tH„] for all n > 1. A correlation K is strong if it satisfies SlrCorrn: for all Hq G H(W), Li, ..., Ln G L(V) such that Hq 2 Li H ... n L£ there is L0 C Li + ... + Ln with = L£ THEOREM 1. Let V, W be left vector spaces over a skew-field K and dimW > 3. A transformation K: L(W) H(V), where dimW > 3, is a correlation [strong correlation] iff there exist anti-monomorphism [anti-auto- morphism] J : K K and a nondegcnerate J-sesquilinear form £ : V x W —> K such that KL = AL for L e L(W), where 1L = 11^. DEFINITION 2. A groupoid (5;©) is said to be a midpoint algebra if it satisfies the following axioms: Ml a © a = a (idemptotency), M 2 a © 6 = ¿> © a (commutativity), MS (a © b) © (c © d) = (a © c) © (b © d) (bi-commutativity), M 4 Vab3x x®a = b. Let 5 be a set and V be a group. A map u: S2 —V is said to be an atlas if it satisfies the following conditions: (1) •) : 5 —s- V is a bijcction for a £ S , (2) b) + ui(b, c) = u>(«, c) for a, 6, c € S . A midpoint algebra 9Jt = (5;©) and a group V are said LJ-associated iff u> : S2 —• V is an atlas and the following condition holds (©) c = a®b<&u(a, c) = w(c,6). A commutative group (V; 6, -f) is said to be a group with the operator \ if it satisfies the following conditions: (3) Va3x x + x = a , (4) a + a = 0=i>a = 0. THEOREM 2. A structure iXTi = (5;©), rvhere ©: S2 —s- S, is a midpoint algebra iff there exist a group V with operator | and an atlas OJ: S2 —> V, such that 231 and V are UJ-associated. Orthogonality relation 55 In the monograph [5] W. Szmiclcw introduced the parallclity planes (5; ||), where || C S'1. In the paper wc will omit upper bound axiom, which must be satisfied by the parallclity planes. Obtained in that manner struc- tures are called the parallclity spaces. DEFINITION 3. A pair IL = (5; ||) is said to be a parallelity space if S is a set, || C 54 and il satisfies the following axioms P 0 ab\\ba, P1 ab\\cc, P 2 a ^ b Aab\\pq A ab\\rs pq\\rs, P 3 ab\\ac ba\\bc, PA 3abc->ab\\ac, P 5 3q (p / q A ab\\pq). A structure il = (5; ||. ©) is referred to as a midpoint space if (1) (5; ||) is a parallclity space, (2) (5; ©) is a midpoint algebra, and 11 satisfies the following axioms: T o/flA oa||o& => 3d [oc||or/ A ac||&cZ] (trapezoid axiom), F a, 6||a © c, b © c (Fa.no axiom). A midpoint space (.S';||,©) and a vector space = (V, A'; •) over a skew-field K are said to be u>-associated if >15 0 u>: S2 —* V is an atlas, AS 1 the midpoint algebra (5; ©) and V are w-associated, -45 2 ab\\cd vectors u(ab), uj(cd) are linearly dependent in for all a, b, c, d € S. THEOREM 3. A midpoint space il = (S, ||, ©) is a Desarguean midpoint space iff there exist a vector space OT = (V, F\ •) over a skew-field K with characteristic different than 2u>-associated with space il. 2. Left orthogonality In Section 2 wc shall give an axiomatic description of the class of all structures 9JD = (V,W; IL^), where V x W —K is an arbitrary left-linear form. Wc begin this section with some basic defini- tions concerning alFine and metric afiine geometry. One can regard them as a motivation for adopting a axiom system (see Definition 1), characterizing a class of corresponding "metric-vector" spaces. 56 M. Muzalcwski DEFINITION 4. Let il = (S; ||,ffi) be a Desarguean midpoint space (see Section 1), V be a vector spacc over a skew-field 5" with characteristic ^ 2, and u: S2 V be an atlas such that: (1) ab\\cd vectors u(a, b),u:{c, d) are linearly dependent, (2) a@b = v(a, c) = w(c, b); consider arbitrary relations 1C 5"1 and IL C V2. The structures 1I'D := (il, 1) = (5; -L) and 235) = (V, JL) are said to be u>-associated, if (3) ab I cd& u;(«, b) IL w(c, d). Let ii = (5"; ||,©) be a Desarguean midpoint spacc and ICS4. The structure Hi) is said to be a Desarguean midpoint space with nondegenerate left orthogonality if Hi) satisfies the following axioms: ALO 1 oo ± op, ARO 1 op JL oo, ALO 2 oa -L op A ob A. op => o, a ® b ± o, p, ALO 3 oa 1 op A oa\\ob A o ^ a =>• ob I op, ALO 4 oa / op =>• 3b(xb ± op A ob\\oa), and ARO 5 o / a => 3p {op JL oa). DEFINITION 5. Let us consider a structure 235) = (V,W; 1L), where V, W are left vector spaces over a skew-field K and IL C V X W is an arbitrary binary relation. The structure 23® is said to be a vector space with left orthogonality, if it satisfies the following axioms: VLO 1 6 IL b, VLO 2 ai, a2 IL b aj + a2 IL b, and VLO 3 alLb =*> AalLb. Let us note that if HD = (ii; 1} and 935) = (V, V; IL ) are a»-a.ssociatcd, then ii5) is a Desarguean midpoint space satisfying ALO 1, ALO 2, and ALO 3 iff 235) is a vector spacc with left orthogonality. We will use the following symbols: ^B := {xe V:xJLLB}, := {y e W : AlLy}, and ^b := ^-{b}, a* := {a}^ . PROPOSITION 1. Let 235) = (V,W; IL), where V, W are left vector spaces over a skew-field K and J! C V x W be an arbitrary binary relation. (1) The structure 235) is a vector space with left-orthogonality iff -^b C V is a subspace for any b 6 W. (2) //2J5) is a vector space with left-orthogonality and B C W is a subset, then ^B C V is a subspace. Orthogonality relation 57 Proof. (1) follows directly by VLO 1, VLO 2 and VLO 3 • (2) By (1) and the evident equality: -^B = n{1b:beB}, we get the thesis • • Definition 6. A map V x W K is said to be a lefl-linear form if (1) £(xi + x2,y) = £(xi,y) + £(x2,y), (2) f(ax,y) = ai(x,y). Remark 1. Let C be the complex field and V be a vector space over C. A left-linear form £ : V x V —> C is called a semi-inner product if £ satisfies the following conditions: (3) £(x, x) > 0 when x ^ 0, (4) |£(x,y)|2<£(x,xK(y,y). The investigations on semi-inner products originated in the papers [3] and [2], and were motivated by Quantum Mechanics. Let us note that if £ is a semi-inner product, then the orthogonality relation need not be sym- metric. It seems that the notion of left-linear form may be considered as a natural and convenient basis for studying semi-inner products. Proposition 2. // V x W — K is a left-linear form, over left K- spaces V and W, then the structure (V, W; _LL^) is a vector space with left orthogonality m Definition 7. Let Q3D = (V, W; JL ) be a vector space with left orthog- onality; 03*1) is said to be a vector space with nondegenerate left orthogonality if it satisfies VLO 4 a-ji b =>• 3a(x — aa 1L b) (about projection, see fig. 1.5), VRO 5 b ± 6 =» 3a (a.j£b), VRO 1 aJLfl. Proposition 3. //HD, 935) are ui-associated, then: (1) HD 1= ALOl iff WD (= VLOl, (2) 115) 1= ARO1 iff WD 1= VRO 1, (3) 11® t= ALOl A AL02 A ALOZ iff WD 1= VLO 1 A VL02 A VLO?,, (4) WD N ALOA iffWB t= VLOA, and (5) UV N AR05 iff WD 1= VROh m Let V and W be arbitrary vector spaces; let !L C V x W.