A natural extension of the Young partition lattice

Recently Andrews introduced the concept of signed partition: a {\it signed partition} is a finite sequence of integers $a_k , \dots , a_1,a_{-1} ,\dots , a_{-l} $ such that $a_k \ge \dots \ge a_1 > 0 > a_{-1} \ge \dots \ge a_{-l} $. So far the signed partitions have been studied from an arithmetical point of view. In this paper we first generalize the concept of signed partition and we next use such a generalization to introduce a partial order on the set of all the signed partitions. Furthermore, we show that this order has many remarkable properties and that it generalizes the classical order on the Young lattice.