Lattice Theory| Lattice Theory |
In General > s.a. posets.
$ Def: A poset in which
every pair of elements (x, y) has a least upper bound
x ∨ y (l.u.b., or join) and a greatest lower
bound x ∧ y (g.l.b., or meet).
* Examples: The lattice of
propositions in (quantum) logic; The lattice of partitions of a positive
integer; The lattice of subsets of a set X, or of causally closed
subsets of spacetime M, or of all closed vector subspaces of a
Hilbert space \(\cal H\) (these are all orthocomplemented).
> Online resources:
see Wikipedia page.
Special Types and Related Concepts
* Complete lattice: One in
which the meet and the join exist for arbitrary families of elements.
* Unit and zero elements: The
greatest and least elements in A (if they exist), i.e., elements
0 and 1 such that for all x in A, 0 ≤ x ≤ 1.
* Complement of an element:
An operation x \(\mapsto\) x' defined on a bounded lattice
with greatest element 1 and least element 0, such that x ∨ x'
= 1 and x ∧ x' = 0.
* Orthomodular lattice: One such
that for all x, y in A, we have y ⊆
x implies x ∧ (y ∨ x') = y;
This condition is weaker than distributivity.
* Result: An orthomodular
lattice L is determined by its lattice of subalgebras Sub(L),
as well as by its poset of Boolean subalgebras BSub(L).
@ Orthomodular: Harding IJTP(04) [concrete];
Greechie & Legan IJTP(06) [three classes];
Brunet IJTP(07)qp [intrinsic topology];
Harding & Navara a1009 [subalgebras].
References
@ General: Szasz 63; Birkhoff 67;
Grätzer 03,
09,
11.
@ Examples: Latapy & Phan DM(09) [partitions of a positive integer].
@ Special approaches:
Grätzer 05 [finite lattices, proof-by-picture].
@ Number of paths: Mohanty 79;
Coker DM(03).
Lattices in the Sense of Spatial Patterns
> Types: see crystals;
ising model [Archimedean and Laves lattices]; non-commutative geometry.
> Physical theories on lattices:
see lattice field theory; Measurements;
optics [optical lattices]; spin models.
main page
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 17 jan 2016