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单词 Structure
释义

structure


Let τ be a signaturePlanetmathPlanetmathPlanetmathPlanetmath.A τ-structureMathworldPlanetmath 𝒜 comprises of a set A, called the (or underlying set or ) of 𝒜, and an interpretationMathworldPlanetmath of the symbols of τ as follows:

  • for each constant symbol cτ, anelement cAA;

  • for each n-ary function symbol fτ,a function (or operationMathworldPlanetmath) fA:AnA;

  • for each n-ary relation symbol Rτ,a n-ary relationMathworldPlanetmathPlanetmath RA on A.

Some authors require that A be non-empty.

If 𝒜 is a structure, then the cardinality (or power) of 𝒜, |𝒜|, is the cardinality of its A.

Examples of structures abound in mathematics. Here are some of them:

  1. 1.

    A set is a structure, with no constants, no functions, and no relations on it.

  2. 2.

    A partially ordered setMathworldPlanetmath is a structure, with one binary relation call partial orderMathworldPlanetmath defined on the underlying set.

  3. 3.

    A group is a structure, with one binary operationMathworldPlanetmath called multiplicationPlanetmathPlanetmath, one unary operation called inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and one constant called the multiplicative identityPlanetmathPlanetmath.

  4. 4.

    A vector spaceMathworldPlanetmath is a structure, with one binary operation called addition, unary operations called scalar multiplications, one for each element of the underlying set, and one constant 0, the additive identity.

  5. 5.

    A partially ordered group is a structure like a group, but with the addition of a partial order on the underlying set.

If τ contains only relation symbols, then a τ-structure is called a relational structure. If τ contains only function symbols, then a τ-structure is called an algebraic structure. In the examples above, 2 is a relation structure, while 3,4 are algebraic structures.

Titlestructure
Canonical nameStructure
Date of creation2013-05-20 18:26:21
Last modified on2013-05-20 18:26:21
OwnerCWoo (3771)
Last modified byunlord (1)
Numerical id23
AuthorCWoo (1)
Entry typeDefinition
Classificationmsc 03C07
Related topicSubstructure
Related topicAlgebraicStructure
Related topicModel
Related topicRelationalSystem
Definesstructure
Definesinterpretation
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