Machine Theory

Categories and Computer Science by R. F. C. Walters PDF

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A. It turns out that the definition of product given above allows a great variety of constructions used in mathematics connected with functions of several variables.

We then say that the product is strictly associative. Example 15. In Sets the usual cartesian product of sets is not strictly associative. The isomorphism a is given by a (X1 X X2) X X3 - X1 X (X2 X X3) ((XI, X2), x3) F > (x1, (x2, x3 )) Notice, however, that there is an appropriate triple product (or, more generally, an n-ary product) f X1 X X2 X X3 = 1(x1,x2,x3);x1 E X1,x2 E X2,x3 E X3}. 42 2. PRODUCTS AND SUMS To avoid a proliferation of brackets we shall usually use the n-ary product rather than repeated binary products.

Then rk, 11 rk = inf kEK kEK E rk = sup rk. kEK kEK A larger example: Example 22. IR = {(x, 0) : x E IR}U{(x,1) : x E IR}U. U{(x, m-1) : x E IR} (m > 1); arrows: all functions between these sets. The category Flow has sums which are strictly associative. IR. 46 2. IR -f Z? IR (x, k) (x, k) I (x, l +M) < (x, l) It is easy to see that the property of a sum holds. IR --+ Z and g : n. I R --+ Z then if 0 < i < m - 1; f (x, i) g(x,i-m) ifm

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Categories and Computer Science by R. F. C. Walters


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