Summary: The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff [Duke Math. J. 3, 443–454 (1937; Zbl 0017.19403); Lattice theory. Third (new) ed. American Mathematical Society (AMS), Providence, RI (1967; Zbl 0153.02501)]. This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic applications bring us distributive lattices as a ring family of sets which is closed with respect to the set union and intersection. When it comes to a ring family of sets, the underlying set is partitioned into subsets (or components) and we have a poset structure on the partition. This is a set-theoretical variant of the Birkhoff theorem revealing the correspondence between finite ring families and finite posets on partitions of the underlying sets, which was pursued by Masao Iri around 1978 [M. Iri, in: Combinatorial mathematics, 2nd int. Conf., New York 1978, Ann. New York Acad. Sci. 319, 306–319 (1979; Zbl 0478.05023); in: Mathematical programming, 11th int. Symp., Bonn 1982, 158–201 (1983; Zbl 0542.05024); in: Progress in combinatorial optimization, Conf. Waterloo/Ont. 1982, 197–219 (1984; Zbl 0544.05018)], especially concerned with what is called the principal partition of discrete systems such as graphs, matroids, and polymatroids. In the present paper we investigate a signed-set version of the Birkhoff-Iri decomposition in terms of signed ring family, which corresponds to Reiner’s result on signed posets, a signed counterpart of the Birkhoff theorem. We show that given a signed ring family, we have a signed partition of the underlying set together with a signed poset on the signed partition which represents the given signed ring family. This representation is unique up to certain reflections.