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Includes sections on Algebraic $K$- and $L$-theory, surgical procedure and its functions, crew activities

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It is easily verified that this defines a convex geometry on X. Indeed the axiom (C) holds trivially if c e {a1,bi} or if de { c,b2 } and otherwise holds in X because it holds in X 1 and X2 • Moreover this convex geometry on X satisfies those of the axioms (L2),(L3),(L4),(P) which are satisfied by both the given convex geometries on x l,X2. 4 EXAMPLES (continued) It will now be shown that a real vector space, or indeed a vector space over any ordered division ring, is a convex geometry with the usual definition of segments.

Evidently Cs X1• IfC =X 1, there is nothing more to do. If C '# X 1, the argument can be repeated. The process must eventually terminate, yielding c = xk and Xk-1 =cu Xk e '16. D There is an interesting connection between this result and linguistics. Our language has as its alphabet the elements of the finite set X. The admissible words of our language are the finite sequences a = x 1 ••• xn,, where X; e E(X \ {x1, ••• ,x;_ 1}) for 1 < i < m. This language obviously has the properties (o) 0 is an admissible word; (i) if a= x 1 ••• Xm is an admissible word, then x; '# Xj for 1 < i

For a,b e X we define [a,b] = {a,b} if a e X 1, be X2 or if a e X2, be X1, and [a,b] = [a,bh if a,b e Xk (k = 1,2). It is easily verified that this defines a convex geometry on X. Indeed the axiom (C) holds trivially if c e {a1,bi} or if de { c,b2 } and otherwise holds in X because it holds in X 1 and X2 • Moreover this convex geometry on X satisfies those of the axioms (L2),(L3),(L4),(P) which are satisfied by both the given convex geometries on x l,X2. 4 EXAMPLES (continued) It will now be shown that a real vector space, or indeed a vector space over any ordered division ring, is a convex geometry with the usual definition of segments.