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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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.

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

• 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.

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:

• 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!

Software and tables

Other web sites with this focus

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

Selected Topics at this site

• 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.