Irrelevance, a notion which was first put forward by this author jointly with A. Sgarro, is a convenient tool to speed up computations in the arithmetic of interactive fuzzy numbers. In this paper we are trying to understand what happens if the fuzzy quantities one is considering are incomplete, or sub-normal, that is if one allows that a fuzzy quantity is "cut" at a height h which is less than 1. We motivate the reasons why we deem it important to extend fuzzy arithmetic to fuzzy quantities which may be incomplete, and we show that irrelevance keeps proving a convenient tool. Interactivity is described by suitable monotone joins, which generalize t-norms.