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A reader interested in a local Lie semigroup theory and Lie's Fundamental Theorems can go directly to Chapter IV without having to pass through the preceding parts of the book. The global theory described in Chapters V and VI may also be approached directly, but at least a perusal of the local theory of Chapter IV is advisable; special results introduced and proved in Chapter IV are definitely used in these later parts, but the full power of the Fundamental Theorem of the local theory is not used.

Geometrically, a support hyperplane of a wedge W at a point x c W is a closed hyperplane containing x and bounding a closed half-space containing W. I. The geometry of cones 16 Definition. Let W be a wedge in a topological vector space L and x E W. 15. functional w E W* which is non-zero on W - W and satisfies (w I x) = 0. According to this definition, a hyperplane T of L is a support hyperplane of W at x if and only if there is an w E x1 n W* = op x which is non-zero on W - W. i. 16. Proposition.

A Lie theory for semigroups was asked for. We have proposed a theory of local and global infinitesimally generated subsemigroups in finite dimensional real Lie groups. The reader was taken on an occasional foray into infinite dimensions. Chapter VII finally presents an abstract Lie and embedding theory of semigroups. This program has two parts. Firstly, we need a theory for the forming of quotient groups of semigroups, locally and globally, in the topological setting. This is the appropriate generalisation of the construction of the group of integers from the semigroup of natural numbers.

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