grounded relation

A grounded relation over a sequence of sets is a mathematical object consisting of two components. The first component is a subset of the cartesian product taken over the given sequence of sets, which sets are called the domains of the relationMathworldPlanetmath. The second component is just the cartesian product itself.

For example, if L is a grounded relation over a finite sequence of sets, X1,,Xk, then L has the form L=(F(L),G(L)), where F(L)G(L)=X1××Xk.

1 Remarks

  • In various languagePlanetmathPlanetmath that is used, F(L) may be called the figure or the graph of L, while G(L) may be called the ground of L.

  • The default assumptionPlanetmathPlanetmath in almost all applications is that the domains of the grounded relation are nonempty sets, hence departures from this assumption need to be noted explicitly.

  • In many applications all relations are considered relative to explicitly specified grounds. In these settings it is conventional to refer to grounded relations somewhat more simply as “relations”.

  • One often hears or reads the usage ``LX1××Xk" when the speaker or writer really means ``F(L)X1××Xk". Be charitable in your interpretationsMathworldPlanetmathPlanetmath.

  • The cardinality of G(L) is referred to as the adicity or the arity of the relation. For example, in the finite case, L may be described as k-adic or k-ary.

  • The set domj(L):=Xj is referred to as the jth domain of the relation.

  • In the special case where k=2, the set X1 is called “the domain” and the set X2 is called “the codomain” of the relation.

Title grounded relation
Canonical name GroundedRelation
Date of creation 2013-03-22 17:48:38
Last modified on 2013-03-22 17:48:38
Owner Jon Awbrey (15246)
Last modified by Jon Awbrey (15246)
Numerical id 11
Author Jon Awbrey (15246)
Entry type Definition
Classification msc 08A70
Classification msc 08A02
Classification msc 03G15
Classification msc 03E20
Classification msc 03C05
Classification msc 03B10
Related topic Relation