By Alfred North Whitehead

Whitehead explains in wide phrases what arithmetic is set, what it does, and the way mathematicians do it.Generations of readers who've stayed with the thinker from the start to the top have came across themselves amply rewarded for taking this trip. As

*The big apple Times*saw a long time in the past, "Whitehead does not popularize or make palatable; he's easily lucid and cogent ... A finely balanced mix of wisdom and urbanity .... may still satisfaction you."

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**Example text**

Sip si = si1 . . sij . . sip . 31) Let x = si1 . . sip , b = x(i), and a = x(i+1). 30) means that b > a. Therefore, a is to the left of b in the complete notation of the identity, but is to the right of b in that of x. Hence, there exists j ∈ [p] such that a is to the left of b in si1 . . sij−1 but a is to the right of b in si1 . . sij . Hence, the complete notation of si1 . . sij . . sip is the same as that of si1 . . sip , except that a and b are interchanged. 31). 2. 22 1. The basics Exercises 1.

Deﬁnition and ﬁrst examples 29 that of Bn and several other Coxeter groups in Chapter 8. 2 is the diagram of B3 . 2. Bruhat order of B3 . 3 Inclusion order of Bruhat cells. Here we sketch the algebraic–geometric origin of Bruhat order in a central case — that of cell decompositions of ﬂag manifolds. Let G = GLn (C), and let B be the subgroup of upper-triangular matrices. It turns out that the quotient G/B has the structure of a smooth projective algebraic variety, called the ﬂag variety. 1) where W = Sn .

It is isomorphic to the lattice of Ferrers diagrams that ﬁt into a k × (n − k) box, ordered by inclusion. 7. 5 Bruhat order on quotients Quotients W J (and, more generally, descent classes) have interesting poset structure under Bruhat order. 8 for examples. The latter depicts W J for (W, S) = E6 and J ⊂ S such that (WJ , J) = D5 . Much of the structure found in Bruhat order on all of W is inherited when restricting to the subposet W J . This can to some extent be understood as transfer of structure via the projection maps deﬁned as follows.

### An introduction to mathematics by Alfred North Whitehead

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