In this paper we formalize the notions of information elements and information lattices, first proposed by Shannon. Exploiting this formalization, we identify a comprehensive parallelism between information lattices and subgroup lattices. Qualitatively, we demonstrate isomorphisms between information lattices and subgroup lattices. Quantitatively, we establish a decisive approximation relation ...

The methods to classify extremal unimodular lattices with given automorphisms are extended the situation of modular lattices. A slightly more general notion than type from PhD thesis [11] is det-type. det-type an automorphism on L determines one all partial dual L. This easy observation allows exclude quite a few det-types left open in [11]. Passing suitable p-maximal lattices, ?-modular compos...

In this paper we review a particular connection between information theory and group theory. We formalize the notions of information elements and information lattices, first proposed by Shannon. Exploiting this formalization, we expose a comprehensive parallelism between information lattices and subgroup lattices. Qualitatively, isomorphisms between information lattices and subgroup lattices ar...

We address the existence of vortex solitons supported by azimuthally modulated lattices and reveal how the global lattice discrete symmetry has fundamental implications on the possible topological charges of solitons. We set a general "charge rule" using group-theory techniques, which holds for all lattices belonging to a given symmetry group. Focusing on the case of Bessel lattices allows us t...

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...

We provide a generalization of one-sided (crisp-fuzzy) concept lattices, based on Galois connections. Our approach allows analysis of object-attribute models with different structures for truth values of attributes. Moreover, we prove that this method of creating one-sided concept lattices is the most general one, i.e., with respect to the set of admissible formal contexts, it produces all Galo...

Three-way concept analysis is an important tool for information processing, and rule acquisition one of the research hotspots three-way analysis. However, compared with lattices, semi-concept lattices have operators weaker constraints, which can generate more concepts. In this article, problem discussed in general. The authors construct finer relation propose a method lattices. also discuss set...

We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in the case of a boolean algebra, the obtained topology is the discrete one. Thus, our construction provides a general tool for studying orthomodular lattices b...

let $l$ be a lattice in $zz^n$ of dimension $m$. we prove that there exist integer constants $d$ and $m$ which are basis-independent such that the total degree of any graver element of $l$ is not greater than $m(n-m+1)md$. the case $m=1$ occurs precisely when $l$ is saturated, and in this case the bound is a reformulation of a well-known bound given by several authors. as a corollary, we show t...

We extend the notion of the spectral function of shift-invariant spaces introduced by the authors in [BRz] to the case of general lattices. The main feature is that the spectral function is not dependent on the choice of the underlying lattice with respect to which a space is shift-invariant. We also show that in general the spectral function is not additive on the orthogonal infinite sums of S...