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.

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send feedback and suggestions to bombelli at olemiss.edu – modified 17 jan 2016