Weak dimension and chain-weak dimension of ordered sets
Bull. Korean Math. Soc. 2005 Vol. 42, No. 2, 315-326 Printed June 1, 2005
Jong Youl Kim and Jeh Gwon Lee Sogang University, Sogang University
Abstract : In this paper, we define the weak dimension and the chain-weak dimension of an ordered set by using weak orders and chain-weak orders, respectively, as realizers. First, we prove that if $P$ is not a weak order, then the weak dimension of $P$ is the same as the dimension of $P$. Next, we determine the chain-weak dimension of the product of $k$-element chains. Finally, we prove some properties of chain-weak dimension which hold for dimension.