If the set of partitions of $[a,b]$ $S$ is taken as the directed set, along with the order $\subseteq$, then the directed set has the property that "for every pair $(c,d)$ in the directed set there is a lower bound $e$ in the directed set (such that $e\subseteq c$ and $e\subseteq d$)". In this case, if we write $$U(P,f) = \sum_{i=1} ^n M_i (x_i - x_{i-1})$$ where $M_i=sup_{[x_i, x_{i-1}]}(f)$ and $P=(x_0 =a,...,x_i,..., x_n =b)$ we note that $(U_P)_{P\in S}$ is decreasing.
On the other hand, if we take the same directed set but with the little change that the order is now $\supseteq$. Then, the directed set has the property that "for every pair $(c,d)$ in the directed set there is an upper bound $e$ in the directed set (such that $c\supseteq e$ and $d\supseteq e$). And now the same $(U_P)_{P\in S}$ is increasing.
My question is: are upward directed set equivalent to downward directed set ? After all, we just have to switch the order relation to it's dual in order to get something equivalent ?
I am asking because the introduction of the Wikipedia article on directed sets let's think that these notions are not equivalent. "Other authors call a set directed if and only if it is directed both upward and downward." This sentence let's me think they are not equivalent.
One possible attempt to answer my question is that maybe we don't always have a dual to an order relation ?
Thanks in advance