Traditional interval computations deal with the cases when we are interested in the values of some quantity $x$ and we do not know it precisely; instead, we know two numerical bounds $x^-\le x^+$ that contain $x$: $x^-\le x\le x^+$. The set of all numbers $x$ within this bounds is an {\it interval} $[x^-,x^+]$; inside the computer, a computer can be represented as a pair of numbers $(x^-,x^+)$ with $x^-\le x^+$.
In a more complicated situation, we want to know a set $S$. This set can be the shape of a geometric object (e.g., in graphics or in manufacturing), or the set of situations that satisfy a certain property (e.g., in expert systems, when we are interested, e.g., in the set of situations for which a patient has to have a surgery). Often, we do not have the complete information about what elements belong to the set $S$: about some elements, we are sure that they belong to $S$, about some, we are sure that they don't; about some other elements, we do not know. The resulting information about $S$ is that $S^-\subseteq S\subseteq S^+$, where $S^-$ denotes the set of all elements that are known to belong to $S$, and $S^+$ is the set of all elements that {\it may} belong to $S$ (i.e., that are either known to belong to $S$, or for which the answer to the question ``$s\in S$?'' is ``I don't know''). This information can be thus represented as a pair $(S^-,S^+)$, with $S^-\subseteq S^+$; this pair of {\it sets} can be viewed as a set analogue of an {\it interval}.
For uncertain knowledge, the queries that we can ask may contain logical connectives ``and'', ``or'', ``not'', etc. (of the type ``describe the situations in which the patient has the fever and the headache''). In other words, we have ``basic'' queries $Q_1$,..., $Q_m$, and we can form more complicated queries by using logical connectives. The resulting set of possible queries ${\cal Q}$ is a Boolean algebra.
Ideally, for each query $Q$, we would like to know the set of situations $t(Q)$ in which $Q$ is true. Since our knowledge is incomplete, in general, we will only produce a {\it pair} $(t^-(Q),t^+(Q))$ with the property that the unknown set $t(q)$ is in between $t^-(Q)$ and $t^+(Q)$. Therefore, our goal is, for every $Q$, to produce $t^-(Q)$ and $t^+(Q)$.
First, the authors notice that it is sufficient to produce $t^-(Q)$ (the set of all situations $s$ for which know that $Q$ is true), because $t^+(Q)$ can be described as a complement to $t^-(\neg Q)$ (i.e., as the set of all situations in which we do not know that $Q$ is false).
Second, for every two queries $Q$ and $Q'$, the set of all $s$ for which we know that both $Q$ and $Q'$ are true is exactly the intersection of the set of all $s$ for which $Q$ is true and the set of all $s$ for which $Q'$ is true: $t^-(Q\wedge Q')=t^-(Q)\cap t^-(Q')$. In other words, $t^-$ is a {\it $\wedge-$homomorphism} from the Boolean algebra of queries into a Boolean algebra of subsets. The authors define an {\it interval structure} as such a homomorphism, and illustrate this definition on two types of incomplete information: rough sets and expert-system style incomplete information.
A rough set, crudely speaking, describes the case when we have subdivided the set $A$ of all situations into finitely many mutually disjoint subsets $S_1,...,S_p$, and for each of these subsets $S_i$ and for each of the basic queries $Q_j$, we know whether $Q_j$ is true for all elements of $S_j$ or not. Then, for each $Q$, $t^-(Q)$ is the union of all subsets $S_j$ for which $Q$ is known to be true.
In expert-system style incomplete information, for each query $Q$, we describe the set of situations $s^-(Q)$ for which the expert is sure that $Q$ is true. The expert is not a perfect logician, so it can happen that the statement $Q(s)$ was not part of the expert's belief, but it can be deduced from the other beliefs of this expert. Therefore, the goal is to generate the set $t^-(Q)$ of all situations in which $Q(s)$ can be deduced from the expert's beliefs. The function $t^-$ is also an interval structure.
For each query $Q$, in addition to the set of situations in which $Q$ is true, we may also want to describe, e.g., the {\it probability} $p(Q)$ of $Q$. If we know the probability $p(s)$ of each situation $s$, then we can conclude that the desired probability $p(Q)$ belongs to the interval $[p(t^-(Q)),p(t^+(Q))]$. The authors show that intervals generated by Dempster-Shafer formalism can be thus interpreted.