Abstract
This chapter presents the analysis of the sums of double systems of lattices and DS-congruences of lattices. It defines a double system of lattices. The equivalence classes obtained by means of a congruence of a lattice are sublattices. Thus, the question arises for which congruences it is possible to represent the lattice as the sum of a double system of all its sublattices resulting from congruence classes. The chapter presents the analysis of a family of such congruences of lattices (DS-congruences). It describes DS-congruences in terms of principal ideals and filters and presents the proof that the set DS(L) of all DS-congruences of a lattice L is a meet sub-semilattice of the lattice C(L) of all congruences of L.
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