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, an element 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.

Title structure
Canonical name Structure
Date of creation 2013-05-20 18:26:21
Last modified on 2013-05-20 18:26:21
Owner CWoo (3771)
Last modified by unlord (1)
Numerical id 23
Author CWoo (1)
Entry type Definition
Classification msc 03C07
Related topic Substructure
Related topic AlgebraicStructure
Related topic Model
Related topic RelationalSystem
Defines structure
Defines interpretation