Partial order

A partial order is like a game where we have different objects, and we want to put them in a particular order, but we can't decide which one is better or worse. Instead, we can only figure out if one thing is better than another, or if they are the same.

Let's imagine we have three different toys: a teddy bear, a toy car, and a puzzle. We are going to put them in order, but we can only compare them two at a time.

First, we can decide that the teddy bear is better than the toy car because it's softer and more cuddly. So we put the teddy bear above the toy car.

Next, we can compare the toy car and the puzzle. We can't decide which is better, but we know they are different. So we don't put them on top of each other, but we don't put one above the other, either.

Finally, we compare the teddy bear and the puzzle. Again, we can't decide which is better, but we know the teddy bear is better than the toy car, and the toy car is NOT better than the puzzle. So we can't put the puzzle on top of the teddy bear, but we also can't put the teddy bear on top of the puzzle.

In the end, we have a partial order of our three toys, where the teddy bear is on top, followed by the toy car and the puzzle side by side, because we can't decide which one is better.

That's basically how a partial order works – we can put things in order, but sometimes we can't decide if one thing is better or worse than another, so we just make a partial order with some things above others and some things side by side.