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 i n 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 equiv alent to his, but it has the advantage oft I) being purely order-theor etic, (2) providing a geometric interpretation of interval dimension s imilar to that of Ore [Amer. Math. Soc. Colloq. Publ., Vol. 38, 1962] for dimension, and (3) revealing several somewhat surprising connectio ns to other order-theoretic results. For instance, the transformation P->Q and can be seen as almost an inverse of the well-known split oper ation; 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 invari ant with respect to changes of P that do not alter its comparability g raph, thus also providing a simple new proof for the comparability inv ariance of interval dimension.