On the Interplay between Interval Dimension and Dimension

This paper investigates a transformation P ! Q between partial orders P; Q that transforms the interval dimension of P to the dimension of Q, i.e., idim(P ) = dim(Q). Such a construction has been shown before in the context of Ferrer's dimension by Cogis [2]. Our construction can be shown to be equivalent to his, but it has the advantage of (1) being purely order-theoretic, (2) providing a geometric interpretation of interval dimension similar to that of Ore [15] for dimension, and (3) revealing several somewhat surprising connections to other order-theoretic results. For instance, the transformation P ! Q can be seen as almost an inverse of the well-known split operation, it provides a theoretical background for the influence of edge subdivision on dimension (e.g., the results of Spinrad [17]) and interval dimension, and it turns out to be invariant with respect to changes of P that do not alter its comparability graph, thus providing also a simple new proof for the comparability in..