Interval orders based on arbitrary ordered sets (Q1898345)

Interval orders are defined as ordered sets having interval representations on linearly ordered sets. In this paper the author investigates a more general situation in that the underlying sets may be members of any class of ordered sets. If \(C\) is a class of ordered sets then \(I(C)\) denotes the class of ordered sets having an interval representation in \(C\). The author proves, for example, that if \(C\) is a class of ordered sets which is closed under Dedekind-MacNeille completion and retracts, then \(P \in I(C)\) iff \(B(P,P,<) \in C\), where \(B(P,P,<)\) is the concept lattice of \(P\). If \(Q\) is an ordered set, denote by \(C_Q\) the class of ordered sets for which \(Q\) is a forbidden suborder. Finite ordered sets \(Q\) for which \(C_Q\) is closed under Dedekind-MacNeille completion are described. The author also shows connections between order dimension and interval dimension for general classes of ordered sets.