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Pseudo Planes and Pseudo Ternaries* View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURhXL OF ALGEBRA 4, 300-316 (1966) Pseudo Planes and Pseudo Ternaries* REUBEN SANDLER The University of Chicugo, Chicago, Illinois Communicated by R. H. Bruck Received March 1, 1965 In working with projective planes and the planar ternary rings coordina- tizing them, it often seems as though the “essential” property of the planar ternary ring is that the naturally defined addition and multiplication both give rise to loop structures, while the further solvability conditions sometimes seem to be merely an afterthought. The purpose of this paper is to investigate in some detail the algebraic and geometric consequences of such a notion. For it turns out that the correct definition of “ternary loops” gives rise to a certain type of incidence structure which is a generalization of the idea of projective plane. In fact, this incidence structure (defined in Section 1) can be coordinatized (Section 2) in a manner completely analogous to what occurs in the study of projective planes, and many of the theorems valid for pro- jective planes and planar ternary rings are also valid for the new objects (pseudo planes and pseudo ternaries). In Section 3, the properties of finite pseudo planes are investigated along with certain existence questions, and Sections 4 and 5 explore the collinea- tions of pseudo planes and the relationships between “central collineations” of pseudo planes and algebraic properties of pseudo ternaries. The paper is concluded with an interpretation of one of the results on finite pseudo planes in the notation of zero-one matrices. The study of this particular generalization of the geometric axioms for projective planes seems to be interesting for several reasons. First, there is some hope that a study of some such objects will give some insight into existence and nonexistence questions about projective planes. Secondly, this seems to be the most general set of axioms which allows one to set up the unique correspondence between coordinatizing rings and the geometric structures in the manner of projective planes. Third, it gives rise to a set of * The research described in this paper was partially supported by the National Science Foundation. 300 PSEUDOPLANES AND PSEUDOTERNARIJ!S 301 algebraic systems which are quite natural generalizations of planar ternary rings, and tinally, there seem to be many interesting problems which arise from a consideration of the axioms and some of the examples which are discussed in Section 3. 1. PSEUDO PLANES A set, ?r, of points and lines, together with an incidence relation, I, will be called a pseudo plane if there are two distinct points PI , P, E r, and two distinct lines L, , L, E rr such that PI 1 L, , PI IL, , P, 1 L, , and such that (1) For any point P such that P 1 L, or P 1 L, , and any Q E X, there is a unique line L E rr with P 1 L and Q 1 L. (2) For any line L, such that PI IL or Pz IL, and any line M E W, there is a unique point P with P IL and P 1 M. (3) There are four points in ST,no three of which are incident with the same line. The set P, , Pz , L, , L, will be called a coordinutizing configuration of rr. The relevance of the notion of pseudo planes and coordinatizing configura- tions will be made more explicit when it is shown that every pseudo plane can be coordinatized, and the algebraic properties of the coordinatizing ring are explored in some detail. At this point, however, some facts about pseudo planes can be ascertained in a purely geometric manner. Observe first that every projective plane rr is a pseudo plane in which any set of two points P, , P, and two lines, L, , L, with PI IL,; Pi IL, , i = 1, 2, is a coordina- tizing configuration. Thus, the notion of pseudo plane is a generalization of the concept of projective plane, and we shall show later that it is a proper generalization. We note that axioms (1) and (2) are mutually dual, and wish to show that the principle of duality holds for pseudo planes. For this we need first observe that there must be at least one point # PI on L, , and then prove THEOREM 1. Let VT be a pseudo plane with coordinatizing configuration P,,P,,L,,L,. ThenifOisanypoint #P,,ofL,,thereisapointPofr such that no three of P, , P, , 0, P are incident with a common line of T. Proof. Assume not. Then every point of w must lie on one of the three lines L, , L, , L, , where L, is the unique line containing 0 and P, (written 0 * P, if the line must exist and is unique). In fact, by axiom (3) at least two of these lines must contain points other than P, , Pz , 0. Say Q IL, , R IL3 . But by axiom (1) there are unique lines Ml = P, * R, containing PI and R, 302 SANDLER and M, = Pz * Q. Then, by axiom (2), these lines must have a unique point of intersection, which cannot be on L, , L, , or L, , contradicting our hypoth- esis. The same argument holds if Q IL, , R 1 L, , or Q IL2 , R IL3 , completing the proof of the theorem. As an immediate consequence we have COROLLARY. If T is a pseudo plane, w contains four lines, no three of which contain a common point. Proof. Consider the points PI , Pz , 0, P, given by Theorem 1, and the lines L, , L, , L3 = 0 * P, L, = Pz * P. Note that L4 and Lb are unique by axiom (1). Then axioms (1) and (2) immediately imply that no three of these lines contain a common point. The axioms for pseudo planes, then, are self-dual, and by considering the duals of pseudo planes (obtained from pseudo planes by interchanging points and lines) we can demonstrate the validity of the principle of duality for pseudo planes: Given any theorem valid for pseudo planes, the statement obtained by interchanging the words “point” and “line” is also a valid theorem about pseudo planes. 2. COORDINATIZATION AND PSEUDOTERNARY RINGS Having defined the geometric notion of pseudo plane, we proceed to the task of introducing a coordinate system into an arbitrary pseudo plane 7rIT, with coordinatizing configuration PI , P, , L, , L, as described in the previous section. The process is analogous to the coordinatization of a projective plane and the methods are essentially the same, though care must be exer- cised to verify that the desired elements exist and are unique. Let 0 be any point # P, on L, , and J, any point such that no three of PI, Pz , J, 0 are collinear. Let L = 0 * J be the unique line determined by 0 and J, and associate with each point Q on L except L n L, (unique by 2) a distinct element a(Q) of a set R of symbols (including the symbols 0, 1) such that or(O) = 0, a(J) = 1. To each such point Q IL, Q # L n L, , we give the name (4Q), a(Q)). Now if P is any point of m - L1 , consider the unique lines Ml = PI * P, and M, = P, * P. By axiom (2), since PI 1 Ml , and P, 1 MS , we have L n M, = (a, a); i.e., L and M, intersect in a unique point, and L n M, = (b, b), a, b, E R. The point P is given the name (a, 6). Clearly, we have established a one-one correspondence between ordered pairs (x, y) for x, y E R, and the points of r - L, . The points of L, are named as follows: Let N,,, be the line determined by 0 and (1, m), and the point of intersection of N, and L, is called (m). Conversely, let P IL, be any point # PI . Then PSEUDO PLANES AND PSEUDO TERNARIES 303 if N is the line P -0, and N’ = PI * (1, l), then by axiom (2) Q = N n N is unique, and clearly Q = (1, m) for some m. Finally, let P, = (co). The lines of ?r are named as follows: L, = L, , if M = (co) . (a, a), then M is called [co, a] and if M is a line intersecting L, in a point (m), and inter- secting [co, 0] in a point (0, k), then M is called [m, k]. In this way, every ordered pair of elements in R is made to correspond to a unique line of ‘1~ not incident with (co), and conversely. Also, the lines # L, containing (co) are in one-one correspondence with the elements of R. So far, we have merely named each of the points and lines of r by a method almost exactly the same as is done in the coordinatization of a projective plane. Observe that precisely the axioms (1) and (2) were needed in order to establish this naming in a consistent fashion. Also the name “coordinatizing configura- tion” has been justified. The next step is to define the analogue of the Planar Ternary Ring and see if the geometric axioms force the corresponding alge- braic structure to have reasonable properties. Thus we define the following ternary function. DEFINITION. Let r be a pseudo plane which is coordinatized as above, and deJine Y =F(x,m, k)* (x,y)I Em,4 (4) Thus we have defined a “ternary” function (possibly many-valued) whose domain is a subset of R x R x R, and whose range is a subset of R.
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