Hello Category Community, N (P, <=) ---------------------> (P/E, <=) \ | \ | \ | \ | \ | \ | g \ | f \ | \ | \ | \ | \ | \ | \ v (Q, <=) - This is a problem from Arbib's book on category theory - P is a pre-order - (P/E) and Q are posets - N and f are order-preserving functions from a pre-order to a poset. - g is a order-preserving function between posets P/E and G, i.e. a poset homomorphism, that is UNIQUE. - Question 1: we are building category where the objects are the collection of order-preserving functions from P to posets and morphisms are order-preserving functions between posets that make a diagram like the following commute, i.e. h.g = f Is the above statement true? h (P, <=) ---------------------> (R, <=) \ | \ | \ | \ | \ | \ | g \ | f \ | \ | \ | \ | \ | \ | \ v (Q, <=) - Question 2: if indeed this forms a category, then (P/E, N) is an initial object in this category. True?? - Question 3: the whole notion introduced by this Arbib problem is that of "free" poset, i.e. something very akin to a free algebra, free group, etc. I.e. the same kind of categoric construct. Yes???? Thank you, Bill Halchin ________________________________________________________________________ Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com