Having trouble in understanding what subset of a poset means.

Question

I am having a problem to visualise a subset of a poset. Let $S = \{a,b,c\}$. Then $\mathcal{P}(S) = \{\{\},\{a\}, \{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\} \}$. Now if I impose $\subseteq$ on $\mathcal{P}(S)$, then the partial order relation is $ R = \{(\{\},\{a\}),(\{\},\{b\}),(\{a\},\{a,b\})(\{c\},\{a,b,c\}),(\{a,b\},\{a,b,c\}), \cdots\}$. Now, the partial order set or poset is $(S, R)$. I am not understanding what should be the subset of a poset like this. Can any kind-hearted one help me understanding this?

Answer 1

A "subset of a poset" is just what a subset always is: Take however many or few elements of the poset you like, put them into a set and give them a name. That's a subset of the poset!

Note that this is an entirely different thing than "the poset of subsets of $S$" which is what you're describing. This is shorthand for "the poset whose elements are all the subsets of $S$, ordered by set inclusion".

Apart from the name, your example of the poset of subsets of $\{a,b,c\}$ looks almost right -- the actual poset is $(\mathcal P(S),R)$, not $(S,R)$. It is not clear to me from the question that there's any understanding you actually lack.