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06: Order, lattices, ordered algebraic structures


Introduction

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.

History

Applications and related fields

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. [Schematic of subareas and related areas]

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

Subfields

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.


Textbooks, reference works, and tutorials

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!

Software and tables

Other web sites with this focus

Selected Topics at this site


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Last modified 2000/01/14 by Dave Rusin. Mail: