Type: Article
Publication Date: 1994-02-01
Citations: 18
DOI: https://doi.org/10.1137/s089548019121885x
This paper investigates a transformation $P \to Q$ between partial orders $P,Q$ that transforms the interval dimension of P to the dimension of Q, i.e., $\text{idim} ( P ) = \dim ( Q )$. Such a construction has been shown before in the context of Ferrer's dimension by Cogis [Discrete Math., 38 (1982), pp. 47–52]. The construction in this paper 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 [Amer. Math. Soc. Colloq. Publ., Vol. 38, 1962] for dimension, and (3) revealing several somewhat surprising connections to other order-theoretic results. For instance, the transformation $P \to Q$ 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 [Order, 5 (1989), pp. 143–147]) and interval dimension, and it turns out to be invariant with respect to changes of P that do not alter its comparability graph, thus also providing a simple new proof for the comparability invariance of interval dimension.