By Michael Barr, Charles Wells

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**Sample text**

Also, f is a total function and f (n) = nf (n − 1), so that φ(f ) is defined and by definition of φ, φ(f )(n) = nf (n − 1) = f (n), so φ(f ) = f . If also φ(g) = g, then g(0) = 1 and g(n) = φ(g)(n) = ng(n − 1) so by induction g is the factorial function. The following proposition, applied to the poset P of partial functions, warrants the general recursive construction of functions. 10 Proposition Let (S, ≤) be a strict ω-CPO and f : S − → S a continuous function. Then f has a least fixed point, that is an element p ∈ S with the property that f (p) = p and for any q ∈ S, if f (q) = q then p ≤ q.

A small discrete graph is essentially a set; small discrete graphs and sets are usefully regarded as the same thing for most purposes. 6 Definition A graph is finite if the number of nodes and arrows is finite. 7 Example It is often convenient to picture a relation on a set as a graph. 2) ❄ 4 Of course, graphs that arise this way never have more than one arrow with the same source and target. Such graphs are called simple graphs. 5), and so corresponds to a graph in the sense just described. The resulting picture has an arrow from each element x of the domain to f (x) so it is not the graph of the function in the sense used in calculus.

For example, in the category f ; idB = f , but in the language f and f ; idB are different source programs. This is in contrast to the treatment of languages using context free grammars: a context free grammar generates the actual language. 5 Example As a concrete example, we will suppose we have a simple such language with three data types, NAT (natural numbers), BOOLEAN (true or false) and CHAR (characters). We give a description of its operations in categorical style. (i) NAT should have a constant 0 : 1 − → NAT and an operation succ : NAT − → NAT.