BACKGROUND
Assume a poset $\langle P, \le \rangle$. For two points $a,b \in P$ with $a \le b$, then $I = [a,b] = \{ x : a \le x \le b \}$ is the interval between $a$ and $b$.
When $P$ is a chain (e.g. ${\mathbb Z}, {\mathbb R}$), then the $I$ are just standard intervals. Two real intervals $I=[a,b],J=[c,d] \subseteq {\mathbb R}$ are ordered usually to mean that $I \le J$ iff $b \le c$. Call this the "strong order", which isn't actually a proper order (it needs to be "reflexivized" to require that $I \le I$). Two other true orders are also available, namely that $a \le c$ and $b \le d$ (the product order of the endpoints), or that $a \le c$ and $b \ge d$ (subset order). These last two are conjugate orders. All of these are defined in the context of Allen's alegbra, enumerating all the possible relations between $I,J$ given combinations of both equal and unequal endpoints.
Additionally, the intersection graphs of sets of real intervals are interval graphs, which are well studied.
MOTIVATION
We work with data objects represented as finite, bounded posets. Analyzing the intervals therein, and their orderings and intersections, is very useful in a range of applications in layout and display.
QUESTION
We are thus seeking extensions from real intervals to poset intervals for the concepts of interval order, Allen's algebra, and inteveral graphs. Our preliminary literature reviews haven't turned up anything, and we're preparing to start the development from first principles. Pointers appreciated, thanks!
