Non decreasing tuple for strict order

Question

Let's say that we have a (non necessarily linear) strict order $(P, \prec )$ and a tuple of elements $a_1, ..., a_n \in P$. Can we always find a permutation $\sigma : \lbrace 1, ... , n\rbrace \to \lbrace 1, ... , n\rbrace $ such that $a_{\sigma(n)} \not \succ a_{\sigma(m)}$ for all $n < m$ ?

Answer 1

Yes. This is called topological sorting (where the directed graph to be topologically sorted is the given partial order).