A hierarchy of filters smaller than $CF_\kappa\lambda-->$

This research was partially supported by Grant-in-Aid for Scientific Research (No. 06640178 and No. 06640336), Ministry of Education, Science and Culture of Japan Mathematics Subject Classification: 03E05 --> Abstract. Following Carr's study on diagonal operations and normal filters on ${\cal P}_{\kappa}\lambda$ in [2], several weakenings of normality have been investigated. One of them is to consider normal filters without $\kappa$-completeness, for example, see DiPrisco-Uzcategui [3]. The other is weakening normality itself while keeping $\kappa$-completeness such as in Mignone [10] and Shioya [11]. We take the second one so that all filters are assumed to be $\kappa$-complete. In Sect. 1 a hierarchy of filters on ${\cal P}_{\kappa}\lambda$ is presented which corresponds to the length of diagonal intersections under which the filters are closed. It turns out that many ranks exist between $FSF_{\kappa\lambda}$ and $CF_{\kappa\lambda}$. We consider seminormal ideals in Sect. 2 and determine the minimal seminormal ideal extending Johnson's result in [6]. Its precise descripti on changes according to $cf(\lambda )$ although we can write it in a single form as well. We also prove that a nonnormal seminormal ideal $I\supset NS_{\kappa\lambda}$ exists if and only if $\lambda$ is regular.