On posets with isomorphic interval posets
Judita Lihová · Czechoslovak Mathematical Journal · 1999
Let $$\mathbb{A} = (A, \leqslant )$$ be a partially ordered set, Int $$\mathbb{A}$$ the system of all (nonempty) intervals of $$\mathbb{A},$$ partially ordered by the set-theoretical inclusion $$ \subseteq$$ . We are interested in partially ordered sets $$\mathbb{B} = (B, \leqslant )$$ with Int $$\mathbb{B}$$ isomorphic to Int $$\mathbb{A}$$ . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If $$\mathbb{A}$$ is a directed partially ordered set, the only $$\mathbb{B}$$ with Int $$\mathbb{B}$$ isomorphic to Int $$\mathbb{A}$$ are $$\mathbb{A}_1^\delta \times \mathbb{A}_2 $$ corresponding to direct decompositions $$\mathbb{A}_1 \times \mathbb{A}_2 $$ of $$\mathbb{A}$$ ( $$\mathbb{A}_1^\delta $$ denotes the dual of $$\mathbb{A}_1 $$ . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].