"free" poset question

26 Jul 2000

      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

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