An ordered set can be regarded as a category. What are its subcategories? Which of them are full?
Here's what I understand.
The elements of the ordered set correspond to the objects of the category. There are at most one arrow between any two objects, and there is an arrow $a\to b$ iff $a\le b$ in the ordered set. A subcategory consists of a subset of object and a subset of arrows subject to some conditions. So at least a subcategory is a subset of the ordered set. It also comes with a collection of arrows. For any two objects $a,b$ in the subcategory, if there is no arrow from $a$ to $b$ in the original category, then there cannot be arrow in the subcategory. If there is an arrow $a\to b$ in the original category, then this arrow may or may not belong to the subcategory. In particular, it cannot be the case that there is an arrow $a\to b$ in the subcategory but there is no arrow $a\to b$ in the original category. If $a\to b$ and $b\to c$ are in the subcategory, then $a\to c$ must be in the subcategory. But I don't see how to extract a reasonable classification of subcategories (including full subcategories) from all of the above.
There is a question that includes my question: What are the subcategories of ordered sets / groups?
But I don't understand the answer (see my comment to the accepted answer and see the above for an explanation/proof of the claim of the comment).
Best Answer
You're just about there. Your subcategory consists of a bunch of objects with at most one arrow between any two of them. That makes it a candidate for being an ordered set. Now check whether the arrows in your subcategory must fulfill the conditions to actually be an ordered set.....