What would an example of a category where some pair of objects lacks a product? Do posets form such an category? I also read the statement that "discrete categories have no products". What does this mean exactly?
I've no formal background in category theory and I'm reading "Basic Category Theory for Computer Scientist".
Posets can fail to have products. The product of two elements in a poset is their greatest lower bound, and it's possible for elements of a poset to have no greatest lower bound. (On the other hand, there are some posets (e.g. lattices) that do have a greatest lower bound for every pair.)
"Discrete categories have no products" is probably better written "discrete categories do not have products." A category "has products" if every pair (equivalently every nonempty finite set) of objects has a product. The fact that discrete categories do not have products is straightforward from the definition. A product of a pair of different objects needs to have an arrow going into each object. But in discrete categories, any object only has one arrow going into itself.
(Actually, this is not quite right, as Derek points out. The discrete category with one object does have products, since the object's arbitrary product with itself is itself and there are no other different objects for the product to fail to exist with.
edit 20 And the fun doesn’t end there on this seemingly simple question. The empty category vacuously has all products except a nullary product so whether it “has products” or not depends on if we count nullary products as needing to exist. If so, I’d have to rephrase a couple things above. Having all binary products would not imply having products, and in order for a poset to have products it would have to also have to have a greatest element as well as be a lattice .)