1.
For each of the posets below, determine if the poset is an interval order. If a poset is an interval order, use the algorithm to find a representation.
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Solution.
For the first poset, we have
\begin{align*}
D_6 \amp = D(a)=D(b) = \set{i,g,h,j,c,e} \\
D_5 \amp = D(d) = \set{i,c,h,j,e}\\
D_4 \amp = D(f)=D(g) = \set{h,c,j,e}\\
D_3 \amp = D(i) = \set{h,j,e}\\
D_2 \amp = D(h) = D(j) = \set{e}\\
D_1 \amp = D(c) = D(e) = \set{}
\end{align*}
and
\begin{align*}
U_6 \amp = U(a) = U(b) = U(d) = U(f) = \set{}\\
U_5 \amp = U(g) = \set{a,b}\\
U_4 \amp = U(i) = \set{a,d,b}\\
U_3 \amp = U(c) = \set{d,f,g,a,b}\\
U_2 \amp = U(h) = U(j) = \set{i,f,g,a,d,b}\\
U_1 \amp = U(e) = \set{h,i,j,f,g,a,d,b}
\end{align*}
From these, we can put together
\begin{align*}
I(a) \amp = [6,6] \amp I(f)\amp = [4,6]\\
I(b) \amp = [6,6] \amp I(g)\amp = [4,5]\\
I(c) \amp = [1,3] \amp I(h)\amp = [2,2]\\
I(d) \amp = [5,6] \amp I(i)\amp = [3,4]\\
I(e) \amp = [1,1] \amp I(j)\amp = [2,2]\text{.}
\end{align*}
The second poset is not an interval order because the points \(\set{d,h,g,c}\) form a \(\mathbf{2}+\mathbf{2}\text{.}\)
