intersection structure

Intersection structures

An intersection structure is a set C such that

  1. 1.

    C is a subset of the powerset P(A) of some set A, and

  2. 2.

    intersectionMathworldPlanetmath of a non-empty family of elements of C is again in C.

If order C by set inclusion, then C becomes a poset.

There are numerous examples of intersection structures. In algebra, the set of all subgroups of a group, the set of all ideals of a ring, and the set of all subspacesMathworldPlanetmathPlanetmath of a vector space. In topologyMathworldPlanetmath, the set of all closed setsPlanetmathPlanetmath of a topological space is an intersection structure. Finally, in functional analysis, the set of all convex subsets of a topological vector spaceMathworldPlanetmath is also an intersection structure.

The set of all partial orderings on a set is also an intersection structure. A final example can be found in domain theory: let C be the set of all partial functionsMathworldPlanetmath from a non-empty set X to a non-empty set Y. Since each partial function is a subset of X×Y, C is a subset of P(X×Y). Let :={fiiI} be an arbitrary collectionMathworldPlanetmath of partial functions in C and f=. f is clearly a relationMathworldPlanetmath between X and Y. Suppose x is in the domain of f. Let E={yYxfy}. Then xfiy for each fi where x is in the domain of fi. Since fi is a partial function, y=fi(x), so that y is uniquely determined. This means that E is a singleton, hence f is a partial function, so that C, meaning that C is an intersection structure.

The main differencePlanetmathPlanetmath between the last two examples and the previous examples is that in the last two examples, C is rarely a complete latticeMathworldPlanetmath. For example, let be a partial ordering on a set P. Then its dual is also a partial ordering on P. But the join of and does not exist. Here is another example: let X={1} and Y={2,3}. Then C={,(1,2),(1,3)}. (1,2) and (1,3) are the maximal elementsMathworldPlanetmath of C, but the join of these two elements does not exist.

Topped intersection strucutres

If, in condition 2 above, we remove the requirement that be non-empty, then we have an intersection structure called a topped intersection structure.

The reason for calling them topped is because the top element of such an intersection structure always exists; it is the intersection of the empty family. In additionPlanetmathPlanetmath, a topped intersection structure is always a complete lattice. For a proof of this fact, see this link (

As a result, for example, to show that the subgroups of a group form a complete lattice, it is enough to observe that arbitrary intersection of subgroups is again a subgroup.


  • A topped intersection structure is also called a closure system. The reason for calling this is that every topped intersection structure CP(X) induces a closure operatorPlanetmathPlanetmathPlanetmath cl on P(X), making X a closure space. cl:P(X)P(X) given by


    is well-defined.

  • Conversely, it is not hard to see that every closure space (X,cl) gives rise to a closure system C:={cl(A)AP(X)}.

  • An intersection structure C is said to be algebraic if for every directed setMathworldPlanetmath BC, we have that BC. All of the examples above, except the set of closed sets in a topological space, are algebraic intersection structures. A topped intersection structure that is algebraic is called an algebraic closure system if,

  • Every algebraic closure system is an algebraic lattice.


  • 1 B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge (2003)
  • 2 G. Grätzer: Universal AlgebraMathworldPlanetmathPlanetmath, 2nd Edition, Springer, New York (1978).
Title intersection structure
Canonical name IntersectionStructure
Date of creation 2013-03-22 17:06:28
Last modified on 2013-03-22 17:06:28
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 03G10
Classification msc 06B23
Synonym closure system
Related topic CriteriaForAPosetToBeACompleteLattice
Defines topped intersection structure
Defines algebraic intersection structure
Defines algebraic closure system