Lattice Theory

In General > s.a. posets.
\$ Def: A poset in which every pair of elements (x, y) has a least upper bound xy (l.u.b., or join) and a greatest lower bound xy (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).

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 xx' = 1 and xx' = 0.
* Orthomodular lattice: One such that for all x, y in A, we have yx implies x ∧ (yx') = 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.