Struktur matematika: Perbedaan antara revisi

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Revisi per 7 Agustus 2012 17.37

Di dalam matematika, struktur pada sebuah himpunan, atau lebih umumnya tipe, terdiri dari objek-objek matematika tambahan yang dalam beberapa cara melekat (atau berhubungan) dengan himpunan, membuatnya lebih mudah untuk memvisualkan atau bekerja dengannya, atau memberkati koleksi dengan makna atau keberartian/signifikansi.

Sebuah daftar sebagian dari struktur-struktur yang mungkin adalah ukuran, struktur aljabar (grup, lapangan, dst.), Topologi, struktur metrik (geometri), urutan, relasi ekivalen, struktur diferensial, dan kategori.

Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly. For example, an order induces a topology. As another example, if a set both has a topology and is a group, and the two structures are related in a certain way, the set becomes a topological group.

Mappings between sets which preserve structures (so that structures in the domain are mapped to equivalent structures in the codomain) are of special interest in many fields of mathematics. Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.

N. Bourbaki suggested an explication of the concept "mathematical structure" in their book "Theory of Sets" (Chapter 4. Structures) and then defined on that base, in particular, a very general concept of isomorphism.

Contoh: bilangan real

The set of real numbers has several standard structures:

an order: each number is either less or more than every other number.
algebraic structure: there are operations of multiplication and addition that make it into a field.
a measure: intervals along the real line have a specific length, which can be extended to the Lebesgue measure on many of its subsets.
a metric: there is a notion of distance between points.
a geometry: it is equipped with a metric and is flat.
a topology: there is a notion of open sets.

There are interfaces among these:

Its order and, independently, its metric structure induce its topology.
Its order and algebraic structure make it into an ordered field.
Its algebraic structure and topology make it into a Lie group, a type of topological group.

Lihat pula

Referensi

Structure, PlanetMath.org. (provides a model theoretic definition.)
D.S. Malik and M. K. Sen (2004) Discrete mathematical structures: theory and applications, ISBN 978-0-619-21558-3 .
M. Senechal (1993) "Mathematical Structures", Science 260:1170–3.
Bernard Kolman, Robert C. Ross, and Sharon Cutler (2004) Discrete mathematical Structures, ISBN 978-0-13-083143-9 .
Stephen John Hegedes and Luis Moreno-Armella (2011)"The emergence of mathematical structures", Educational Studies in Mathematics 77(2):369–88.
Journal: Mathematical structures in computer science, Cambridge University Press ISSN 0960-1295.

@@ Baris 3: / Baris 3: @@
 Di dalam [[matematika]], '''struktur''' pada sebuah [[himpunan (matematika)|himpunan]], atau lebih umumnya [[teori tipe intuisionistik|tipe]], terdiri dari [[objek matematika|objek-objek matematika]] tambahan yang dalam beberapa cara melekat (atau berhubungan) dengan himpunan, membuatnya lebih mudah untuk memvisualkan atau bekerja dengannya, atau memberkati koleksi dengan makna atau keberartian/signifikansi.
-A partial list of possible structures are [[Measure theory|measures]], [[algebraic structure]]s ([[group (mathematics)|group]]s, [[field (mathematics)|field]]s, etc.), [[Topology|topologies]], [[Metric space|metric structures]] ([[Geometry|geometries]]), [[Order theory|orders]], [[equivalence relation]]s, [[differential structure]]s, and [[Category (category theory)|categories]].
+Sebuah daftar sebagian dari struktur-struktur yang mungkin adalah [[ukuran (matematika)|ukuran]], [[struktur aljabar]] ([[grup (matematika)|grup]], [[lapangan (matematika)|lapangan]], dst.), [[Topologi]], [[ruang metrik|struktur metrik]] ([[geometri]]), [[teori urutan|urutan]], [[relasi ekivalen]], [[struktur diferensial]], dan [[kategori (matematika)|kategori]].
 Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly. For example, an order induces a topology. As another example, if a set both has a topology and is a group, and the two structures are related in a certain way, the set becomes a [[topological group]].