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Bur. Standards Sect. B 69 (1965) 67-72. [4] J. Folkman and J. Lawrence, Oriented matroids, J . Combin. Theory 25 (B) (1978) 199-236. [5] B. Griinbaum, Convex Polytopes (Interscience, London, 1967). [6] V. W. Walkup, The d-step conjecture for polyhedra of dimension d < 6, Acta Math. 117 (1967) 53-78. [7] M. Las Vergnas, Extensions ponctuelles d’une geometric combinatoire orientee, in: Problemes combinatoires et thCorie des graphes, Proc. Colloq. S. (1978) 265-270. [8] M. Las Vergnas, Bases in oriented matroids, J.

If w is a d-tuple of X', let [ ( a )= +(u,u ) . Then if [ is not identically zero (XI, 5) is a (d - l)-ordered set. Since the proof is short, we give it to render the flavor of proofs in d-ordered sets. Clearly 5 is alternating. We verify (A2). Suppose, for a a ( d - 1)-tuple [(s, ( Y ) [ ( T ) = - 1. Then +(s, a,u ) ~ ( T v, ) = - 1, so by (A2) either there is an i with or else +(U> a,u)+(7, s) = - 1. Clearly the latter is not the case, since +(v, a,u ) = 0 by (Al). The expression on the left in the former is so (A2) is satisfied.

C,,,) be the ( k , + * * . + k,)-tuple obtained by their concatenation. In particular, u = (Lb,Eb, Rb) for any i with O < i G k. A d-ordered set is a pair (X, +), where X is a set and is a function on ( d + 1)-tuples (T = (x", . . , xd) of elements of X with values in { - 1 , 0 , l}, not identically zero, with the two properties: ( A l ) 4 is alternating; that is, if the (d+l)-tuple (T is obtained from T by interchanging two entries, then + ( a )= - $ ( T ) , and (A2) if s EX, u is a d-tuple of elements of X , and T is a (d + 1)-tuple from X such that (a) +(E*T,u ) ~ ( L ' T s, ,R ' T ) ~ O for each i with 0 s i s d, then (b) tp(s, c)+f7) 3 0.