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THE GEOMETRY OF MONOIDS many faces. Since there is a natural bijection between the faces of C and the faces of C we may as well assume in the proof of (3) that C ∗ = 0. Let C := F0 ⊂ · · · ⊂ Fd = C be a maximal chain of faces of C. Since each Fi is an exact submonoid of C, the inclusions F0gp ⊆ F1gp ⊂ · · · ⊂ Fdgp of linear subspaces of C gp are all strict. Since C gp has dimension d, d ≤ d. We prove gp the opposite inequality by induction on the dimension d of C . If d = 0, C = 0 and the result is trivial.
For example, if ui : P → →Qi are maps of monoids for i = 1, 2 and Q is their amalgamated sum, then R[Q] ∼ = R[Q1 ]⊗R[P ] R[Q2 ]. Similarly, if S is a Q-set, we denote by R[S] the free R-module with basis S, endowed with the unique structure of R[Q]-module which is compatible with the action of Q on S. Then if T → S is a basis for S as a Q-set, the induced map T → R[S] is a basis for R[S] as a Q-module, and if S and S are Q-sets, there is a natural isomorphism R[S ⊗Q S ] ∼ = R[S] ⊗R[Q] R[S ]. If A is an R-algebra, a morphism from a monoid Q to the monoid (A, ·, 1) underlying A is sometimes called an A-valued character of Q.
3. The evaluation mapping ev: Q → H(H(Q)) factors through an isomorphism ev: Qsat → H(H(Q)). 36 CHAPTER I. THE GEOMETRY OF MONOIDS The key geometric tool is the following. Let P be a submonoid of an abelian group G, and let φ be a homomorphism G → Z which maps P to N. Suppose that t is an element of G and φ(t) < 0, and let Q be the submonoid of G generated by P and t. Then the homomorphism ψ: G → Ker(φ) : g → tφ(g) − gφ(t) induces multiplication by |φ(t)| on Ker(φ) and maps Q into P . The following result is a corollary of the theorem, but in fact it is one of the main ingredients in the proof.