Hi guys, so I just finished reading this thread, moving on to the other ones and getting started on participating in this amazing project. BTW: I am quite happy with this plan of trying to finish the course by the time school starts in the northern hemisphere. After reading through the thread I feel coming up with examples is a good form of introduction so here I go (heh):

I liked the chess ruminations so I'll name an example using Go; the set of all possible kifu
- that is possible ways to play a game of go from start to any legal position (because the game can end in any position due to one player resigning) is a preorder. While there is a rule to prevent the board from going back to the state it was in on the previous move there is no such rule to prevent cycles taking more than one move so I think the kifu's are not partially ordered (but they could be if we count number of moves recorded on the kifu?).

If you don't include the history then we definitely have a preorder since for any positions \\(x, y, z\\) such that \\(x \le y\\) and \\(y \le z\\) there is a kifu in the kifu set that describes the path from \\(x \le z\\) but we have lost the information containing the order of moves (did I just make a functor? ;)

The reasons \\(x \le x\\) is that passing is a legal move (however three passes end the game so actually we'll either need to include number of passes in our definition of the gamestate in the second example or assume that no passing has happened when presented with a board state and actually that rule means that \\(x \le x\\) isn't true! or is it? The morphism needs a player to make a move so an identity morphism would be an empty move, but passing isn't cause it adds information to the gamestate).

I wanted to make a couple of more general examples but I'm afraid my ability is not up to the task of stating / defining the scenarios. Still I'll make an attempt since I've heard the best way to get good explanations on the internet is to be slightly wrong about something but claiming it's truth vehemently.

Given a 'game' that has rules
1. for going from one game state to another
2. for possible initial states

We define a 'story' as what has happened in a game up to a point. Stories are a preorder. Actually this is just putting fun labels on automata / state machines.

Then finally I wanted to say that an 'explanation' given a 'language' form a partial order with respect to kolmogorov complexity in that language. Now this may actually be false in certain languages since different explanations could have the same level of complexity (i.e. preorder not poset) but I'd like to know if you guys have any way of figuring out which attributes of a language would determine that.