In their 1936 seminal paper, Birkhoff and von Neumann made the proposal of a non-classical logic for quantum mechanics founded on the basic lattice-order properties of all closed subspaces of a Hilbert space. These lattice-order properties are captured in the orthomodular lattice structure, characterized by a weak form of distributivity called orthomodular law. This “weak distributivity”, which...

The closest vector problem for general lattices is NP-hard. However, we can efficiently find the closest lattice points for some special lattices, such as root lattices (An, Dn and some En). In this paper, we discuss the closest vector problem on more general lattices than root lattices.

Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D. Micciancio in [16] and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen [19] showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem S...

This paper shows that any compactly generated lattice is a subdirect product of subdirectly irreducible lattices which are complete and upper continuous. An example of a compactly generated lattice which cannot be subdirectly decomposed into subdirectly irreducible compactly generated lattices is given. In the case of an ideal lattice of a lattice L, the decomposition into subdirectly irreducib...