> I've always felt that order theory, like category theory, lacks reasonably strong theorems. Perhaps that's because the corresponding objects simply don't have

> I've always felt that order theory, like category theory, lacks reasonably strong theorems. Perhaps that's because the corresponding objects simply don't have enough structure to be interesting of themselves. However, I've known several people who were raving on about how great especially categories are. I think that, from the point of view of computer science, these ideas are not primarily useful as theories, but as languages: ways to think and talk about things. For example, any decent programmer has some kind of concept in his head that corresponds to the mathematical idea of a function, another for the special case of a function with a finite domain, and yet another for the special-special case of a finite sequence. They know implementations for these, time-space trade-offs & whatnot. That doesn't mean they need to study any subject with a name like "Theory of Functions". It does mean that familiarity with mathematical concepts like "function" and "domain" is going to be useful in the programming field. Similarly, it would be good to be able to toss around concepts like "lattice" and "Cartesian product" and "monoid". But I'm not sure how much of the theory a programmer really needs to know. By the way, I don't agree with your assessment of order theory and its lack of theorems. Check out -- for example -- Dilworth's Theorem and the generalizations by Greene and Kleitman, for some terribly strong statements about partial orders. OTOH I have no idea what connection these might have to computing. (And as far as I know, you are quite correct about category theory.) > Summaries like this would gain tremendously if they were augmented with some useful examples of where taking the partial order / lattice perspective of the world actually helps you in understanding what is going on. Definitely.