Lattice Theory

2021 ◽  
pp. 81-109
Author(s):  
Gerhard X. Ritter ◽  
Gonzalo Urcid
Keyword(s):  
Lattice Theory

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

scholarly journals Stone algebras form an equational class: (Remarks on Lattice Theory III)

1969 ◽  
Vol 9 (3-4) ◽  
pp. 308-309 ◽  
Author(s):  
G. Grätzer
Keyword(s):  
Lattice Theory ◽  
Equational Class ◽  
Distributive Lattices ◽  
Image Position

To prove the statement given in the title take a set Σ1 of identities characterizing distributive lattices 〈L; ∨, ∧, 0, 1〉 with 0 and 1, and let Then is Σ redundant set of identities characterizing Stone algebras = 〈L; ∨, ∧, *, 0, 1〉. To show that we only have to verify that for a ∈ L, a* is the pseudo-complement of a. Indeed, a ∧ a* 0; now, if a ∧ x = 0, then a* ∨ x* 0* = 1, and a** ∧ = 1* = 0; since a** is the complement of a*, the last identity implies x** ≦ a*, thus x ≦ x** ≦ a*, which was to be proved.

scholarly journals Identities Generalizing the Theorems of Pappus and Desargues

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1382
Author(s):  
Roger D. Maddux
Keyword(s):  
Projective Plane ◽  
Vector Space ◽  
Invariant Theory ◽  
Lattice Theory ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

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