Consider the relations of order $\circ_1, \circ_2$ and $\leq = \circ_1 \cap \circ_2$ on set $M = \{a,b,c,d\}$ given by: $$a \circ_1 b \circ_1 d \qquad \qquad (1)$$ $$a \circ_1 b \circ_1 c \qquad \qquad (2)$$ $$b \circ_2 d \circ_2 c \circ_2 a \qquad (3)$$ The task is to find all minimal and maximal elements in poset $(M, \leq)$. My approach is to simply select intersections from given inequalities:
From $(1)$ and $(3)$ we get:
$$b\circ_1 d, \qquad b\circ_2 d \qquad \Rightarrow \qquad b \leq d$$ From $(2)$ and $(3)$ we get: $$b \circ_1 c, \qquad b\circ_2 d \circ_2 c \Rightarrow b \circ_2 c \qquad \Rightarrow \qquad b \leq c$$
So minimal element is $\{b\}$ and maximal elements are $\{c,d\}$.
How true are these arguments and should we add element $\{a\}$ to minimal or/and maximal elements set, since it is not comparable to others?