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 (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 elements O, I in A (if they exist) such that for all x in A, O x I.
* Complement of an element: An operation x x' such that...
* Orthomodular lattice: One such that for all x, y in A, we have y x implies x (y x') = y; Weaker than distributivity.
@ Orthomodular: Harding IJTP(04) [concrete]; Greechie & Legan IJTP(06) [three classes]; Brunet IJTP(07)qp [intrinsic topology].

References
@ General: Szasz 63; Birkhoff 67; Grätzer 03.
@ 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 > see crystals; lattice field theory; Measurements; non-commutative geometry; spin models.

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