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[Texts]## 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

- 06A: Ordered sets
- 06B: Lattices, see also 03G10
- 06C: Modular lattices, complemented lattices
- 06D: Distributive lattices
- 06E: Boolean algebras (Boolean rings), see also 03G05
- 06F: Ordered structures

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:

- Birkhoff, Garrett: "Lattice theory", American Mathematical Society, Providence, R.I., 1979. ISBN 0-8218-1025-1
- Birkhoff also gave a nice survey of the field later in "Lattices and their applications" (Darmstadt conference, 1991), pp. 7--25, Res. Exp. Math., 23, Heldermann, Lemgo, 1995.
- Grätzer, George, "General lattice theory", Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. 381 pp. ISBN 0-12-295750-4, and Birkhäuser Verlag, Basel-Stuttgart, ISBN 3-7643-0813-3
- Grätzer's earlier book is also good: "Lattice theory. First concepts and distributive lattices", W. H. Freeman and Co., San Francisco, Calif., 1971. 212 pp.
- Davey, B. A.; Priestley, H. A.: "Introduction to lattices and order", Cambridge University Press, Cambridge, 1990 ISBN 0-521-36584-8; 0-521-36766-2
- Erné, Marcel: "Einfuhrung in die Ordnungstheorie", Bibliographisches Institut, Mannheim, 1982. ISBN 3-411-01638-8

This section also includes Boolean algebras and rings. We mention a few sources of information:

- There is an (expensive!) three-volume "Handbook of Boolean algebras" edited by J. Donald Monk and Robert Bonnet, North-Holland Publishing Co., Amsterdam-New York, 1989. 1367 pp., ISBN 0-444-70261-X, 0-444-87152-7, 0-444-87153-5. See especially the survey vol. 1 by Sabine Koppelberg.
- Whitesitt, J. Eldon: "Boolean algebra and its applications" Dover Publications, Inc., New York, 1995 ISBN 0-486-68483-0
- Faure, R.; Heurgon, E.: "Structures ordonnées et algèbres de Boole", Gauthier-Villars Éditeur, Paris 1971

It must be pointed out that readers of Russian have considerably more latitude in selecting their reading material!

- Here are the AMS and Goettingen resource pages for area 06.

- Generating linear extensions of posets.
- Number of partial orders on a finite set
- Any countable linear ordering can be embedded in the rationals.
- Characterizing eta, the order type of the rationals (as the unique countable densely packed linearly ordered set without min or max)
- Real line is unique uncountable complete linearly ordered set with a countable dense subset and no endpoints
- Sperner's theorem on maximal collections of incomparable subsets
- Better-quasi-ordering transfinite sequences
- Schreier Refinement Theorem for modular lattices (and thus to groups etc.)
- An ordered group with least upper property is the integers or reals.
- Constructing the (Boolean) ring from a Boolean algebra
- lattice for positive Boolean functions
- What is the Möbius inversion function?
- There cannot be an ordering of the complex numbers consistent with the expected rules of arithmetic.

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