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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