scholarly journals Entropy Inequalities for Lattices

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 784 ◽  
Author(s):  
Peter Harremoës
Keyword(s):  
Group Theory ◽  
Conditional Independence ◽  
Lattice Theory ◽  
Boolean Lattice ◽  
Functional Dependencies ◽  
Modular Lattices ◽  
Special Lattice ◽  
Entropy Inequalities ◽  
Lattice Of Subgroups

We study entropy inequalities for variables that are related by functional dependencies. Although the powerset on four variables is the smallest Boolean lattice with non-Shannon inequalities, there exist lattices with many more variables where the Shannon inequalities are sufficient. We search for conditions that exclude the existence of non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group. In order to formulate and prove the results, one has to bridge lattice theory, group theory, the theory of functional dependences and the theory of conditional independence. It is demonstrated that the Shannon inequalities are sufficient for planar modular lattices. The proof applies a gluing technique that uses that if the Shannon inequalities are sufficient for the pieces, then they are also sufficient for the whole lattice. It is conjectured that the Shannon inequalities are sufficient if and only if the lattice does not contain a special lattice as a sub-semilattice.

scholarly journals Entropy Inequalities for Lattices

2018 ◽  
Author(s):  
Peter Harremoës
Keyword(s):  
Boolean Lattice ◽  
Functional Dependencies ◽  
Modular Lattices ◽  
Power Set ◽  
Entropy Inequalities ◽  
Lattice Of Subgroups

We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that excludes the existence of non-Shannon inequalities. It is demonstrated that planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group.

scholarly journals Entropy Inequalities for Lattices

2018 ◽  
Author(s):  
Peter Harremoës
Keyword(s):  
Boolean Lattice ◽  
Functional Dependencies ◽  
Modular Lattices ◽  
Power Set ◽  
Entropy Inequalities ◽  
Lattice Of Subgroups

We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that excludes the existence of non-Shannon inequalities. It is demonstrated that planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group.

scholarly journals MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing

2014 ◽  
Author(s):  
Daniel J. Greenhoe
Keyword(s):  
Boolean Lattice ◽  
Symbolic Sequence ◽  
Sequence Processing ◽  
Linear Subspaces ◽  
Genomic Signal Processing ◽  
Set Inclusion ◽  
Symbolic Sequences ◽  
Special Lattice ◽  
Boolean Lattices ◽  
Selection Of

The linear subspaces of a multiresolution analysis (MRA) and the linear subspaces of the wavelet analysis induced by the MRA, together with the set inclusion relation, form a very special lattice of subspaces which herein is called a "primorial lattice". This paper introduces an operator R that extracts a set of 2^{N-1} element Boolean lattices from a 2^N element Boolean lattice. Used recursively, a sequence of Boolean lattices with decreasing order is generated---a structure that is similar to an MRA. A second operator, which is a special case of a "difference operator", is introduced that operates on consecutive Boolean lattices L_2^n and L_2^{n-1} to produce a sequence of orthocomplemented lattices. These two sequences, together with the subset ordering relation, form a primorial lattice P. A logic or probability constructed on a Boolean lattice L_2^N likewise induces a primorial lattice P. Such a logic or probability can then be rendered at N different "resolutions" by selecting any one of the N Boolean lattices in P and at N different "frequencies" by selecting any of the N different orthocomplemented lattices in P. Furthermore, P can be used for symbolic sequence analysis by projecting sequences of symbols onto the sublattices in P using one of three lattice projectors introduced. P can be used for symbolic sequence processing by judicious rejection and selection of projected sequences. Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in "genomic signal processing").

An extension of Boolean lattice theory

1963 ◽  
Vol 151 (4) ◽  
pp. 332-345 ◽  
Author(s):  
N. V. Subrahmanyam
Keyword(s):  
Lattice Theory ◽  
Boolean Lattice

On the Structure of the Schild Group in Relativity Theory

2017 ◽  
Vol 60 (4) ◽  
pp. 774-790
Author(s):  
Gerd Jensen ◽  
Christian Pommerenke
Keyword(s):  
Group Theory ◽  
Relativity Theory ◽  
Matrix Group ◽  
Lorentz Transformations ◽  
Full Group ◽  
Generating Sets ◽  
Lattice Of Subgroups

AbstractAlfred Schild has established conditions that Lorentz transformationsmap world-vectors (ct, x, y, z) with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations.This paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature, we associate Lorentz transformations with matrices in SL(z, ℂ). We consider the lattice of subgroups of the group originated in Schild’s paper and obtain generating sets for the full group and its subgroups.

scholarly journals The descending chain condition in join-continuous modular lattices

1969 ◽  
Vol 10 (1-2) ◽  
pp. 1-4 ◽  
Author(s):  
L. G. Kovács
Keyword(s):  
Distributive Lattice ◽  
Lattice Theory ◽  
Chain Condition ◽  
Modular Lattices ◽  
Algebraic Systems ◽  
The Third ◽  
Irreducible Elements ◽  
Descending Chain Condition ◽  
A Chain

If L is a distributive lattice in which every element is the join of finitely many join-irreducible elements, and if the set of join-irreducible elements of L satisfies the descending chain condition, then L satisfies the descending chain condition: this follows easily from the results of Chapter VIII, Section 2, in the Third (New) Edition of Garrett Birkhoff's ‘Lattice Theory’ (Amer. Math. Soc., Providence, 1967). Certain investigations (M. S. Brooks, R. A. Bryce, unpublished) on the lattice of all subvarieties of some variety of algebraic systems require a similar result without the assumption of distributivity. Such a lattice is always join-continuous: that is, it is complete and (∧X) ∨ y = ∧ {x ∨ y: x ∈ X} whenever X is a chain in the lattice (for, the dual of such a lattice is complete and ‘algebraic’, in Birkhoff's terminology). The purpose of this note is to present the result:

scholarly journals Matroid And Its Relations

2021 ◽  
Author(s):  
Henry Garrett
Keyword(s):  
Graph Theory ◽  
Group Theory ◽  
Algebraic Structure ◽  
Lattice Theory ◽  
Structure Graph

In this article, the connections amid matroid and other notions have been studied. The structure of matroid could be a reflection of some other structure in lattice theory, group theory, other algebraic structure, graph theory, combinatorics and enumeration theory.

scholarly journals MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing

2014 ◽  
Author(s):  
Daniel J. Greenhoe
Keyword(s):  
Boolean Lattice ◽  
Symbolic Sequence ◽  
Sequence Processing ◽  
Linear Subspaces ◽  
Genomic Signal Processing ◽  
Set Inclusion ◽  
Symbolic Sequences ◽  
Special Lattice ◽  
Boolean Lattices ◽  
Selection Of

The linear subspaces of a multiresolution analysis (MRA) and the linear subspaces of the wavelet analysis induced by the MRA, together with the set inclusion relation, form a very special lattice of subspaces which herein is called a "primorial lattice". This paper introduces an operator R that extracts a set of 2^{N-1} element Boolean lattices from a 2^N element Boolean lattice. Used recursively, a sequence of Boolean lattices with decreasing order is generated---a structure that is similar to an MRA. A second operator, which is a special case of a "difference operator", is introduced that operates on consecutive Boolean lattices L_2^n and L_2^{n-1} to produce a sequence of orthocomplemented lattices. These two sequences, together with the subset ordering relation, form a primorial lattice P. A logic or probability constructed on a Boolean lattice L_2^N likewise induces a primorial lattice P. Such a logic or probability can then be rendered at N different "resolutions" by selecting any one of the N Boolean lattices in P and at N different "frequencies" by selecting any of the N different orthocomplemented lattices in P. Furthermore, P can be used for symbolic sequence analysis by projecting sequences of symbols onto the sublattices in P using one of three lattice projectors introduced. P can be used for symbolic sequence processing by judicious rejection and selection of projected sequences. Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in "genomic signal processing").

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