An extension of Boolean lattice theory

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

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.

Partitioning the Boolean lattice into a minimal number of chains of relatively uniform size

2003 ◽  
Vol 24 (2) ◽  
pp. 219-228 ◽  
Author(s):  
Tim Hsu ◽  
Mark J. Logan ◽  
Shahriar Shahriari ◽  
Christopher Towse
Keyword(s):  
Boolean Lattice ◽  
Uniform Size

Short-range order and concentration fluctuations in binary molten alloys

1987 ◽  
Vol 65 (3) ◽  
pp. 309-325 ◽  
Author(s):  
R. N. Singh
Keyword(s):  
Lattice Theory ◽  
Order Parameter ◽  
Short Range ◽  
Short Range Order ◽  
Range Order ◽  
Concentration Fluctuations ◽  
Mixing Properties ◽  
Chemical Theory ◽  
Local Ordering ◽  
Interchange Energy

The quasi-chemical theory and the quasi-lattice theory are discussed with a view to obtaining information about concentration fluctuations, SCC(0), and the short-range order parameter, α1, for regular and compound-forming molten alloys. The influence of the coordination number z and the interchange energy ω on the mixing properties of the alloy is critically examined. SCC(0) and α1 have been found to be very useful in extracting microscopic information, like local ordering and segregation in molten systems. The problem of glass formation in compound-forming binary molten alloys is also briefly discussed.

Criticism of the O-lattice theory

1973 ◽  
Vol 7 (6) ◽  
pp. 557-563 ◽  
Author(s):  
K. Sadananda ◽  
M.J. Marcinkowski
Keyword(s):  
Lattice Theory
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