06: Order, lattices, ordered algebraic structures
Ordered sets, or lattices, give a uniform structure to, for example, the set of subfields of a field. Various special types of lattices have particularly nice structure and have applications in group theory and algebraic topology, for example.
A large portion of this field involves simple combinatorial structures on arbitrary sets; see 05: Combinatorics and 03E: Set Theory.
Linear orderings especially on infinite sets is the study of Ordinals in Set Theory; these are traditionally considered in 03: Mathematical Logic, especially 03G: Algebraic Logic. See 03G05: Boolean algebras and 03G10: Lattices and related structures.
Ordered sets may be viewed as topological spaces; see 54: General Topology, especially 54F05: Ordered topological spaces, for more detail.
There is significant overlap with 08: General algebraic structures, and orderings (e.g. subgroup lattices) are a natural part of many particular algebraic structures; see 20: Group Theory, 13: Commutative Rings, 16: Associative Rings. For ordered (algebraic) categories in general see section 18B35 of 18: Homological Algebra.
Boolean algebra is used in circuit design and pattern matching; see 94: Information and Circuits and 68: Computer Science.
Lattices in the sense of section 06 are essentially unrelated to the lattices of number theory.
Other fields with some overlap seen in the diagram are areas 81: Quantum Theory, 46: Functional Analysis, 90: Operations Research, 28: Measure Theory and Integration, 52: Convex Geometry, and 51: Geometry
Prior to 1973, articles in this area were assigned to another 2-digit discipline (with -06 suffix). Also appropriate were headings 02.42 (Boolean algebras, lattices, topologies) 1959-1972.
Browse all (old) classifications for this area at the AMS.
For a survey see Birkhoff, Garrett: "What is a lattice?" Amer. Math. Monthly 50, (1943). 484--487. MR5,31b
Good texts in lattice theory and partially ordered sets include those by a couple of researchers particularly closely associated with this area:
This section also includes Boolean algebras and rings. We mention a few sources of information:
It must be pointed out that readers of Russian have considerably more latitude in selecting their reading material!