intersection structure
Intersection structures
An intersection structure is a set $C$ such that

1.
$C$ is a subset of the powerset $P(A)$ of some set $A$, and

2.
intersection^{} of a nonempty family $\mathcal{F}$ 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 subspaces^{} of a vector space. In topology^{}, the set of all closed sets^{} of a topological space is an intersection structure. Finally, in functional analysis, the set of all convex subsets of a topological vector space^{} 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 functions^{} from a nonempty set $X$ to a nonempty set $Y$. Since each partial function is a subset of $X\times Y$, $C$ is a subset of $P(X\times Y)$. Let $\mathcal{F}:=\{{f}_{i}\mid i\in I\}$ be an arbitrary collection^{} of partial functions in $C$ and $f=\bigcap \mathcal{F}$. $f$ is clearly a relation^{} between $X$ and $Y$. Suppose $x$ is in the domain of $f$. Let $E=\{y\in Y\mid xfy\}$. Then $x{f}_{i}y$ for each ${f}_{i}$ where $x$ is in the domain of ${f}_{i}$. Since ${f}_{i}$ is a partial function, $y={f}_{i}(x)$, so that $y$ is uniquely determined. This means that $E$ is a singleton, hence $f$ is a partial function, so that $\bigcap \mathcal{F}\in C$, meaning that $C$ is an intersection structure.
The main difference^{} between the last two examples and the previous examples is that in the last two examples, $C$ is rarely a complete lattice^{}. For example, let $\le $ be a partial ordering on a set $P$. Then its dual ${\le}^{\partial}$ is also a partial ordering on $P$. But the join of $\le $ and ${\le}^{\partial}$ does not exist. Here is another example: let $X=\{1\}$ and $Y=\{2,3\}$. Then $C=\{\mathrm{\varnothing},(1,2),(1,3)\}$. $(1,2)$ and $(1,3)$ are the maximal elements^{} 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 $\mathcal{F}$ be nonempty, 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 addition^{}, a topped intersection structure is always a complete lattice. For a proof of this fact, see this link (http://planetmath.org/CriteriaForAPosetToBeACompleteLattice).
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.
Remarks.

•
A topped intersection structure is also called a closure system. The reason for calling this is that every topped intersection structure $C\subseteq P(X)$ induces a closure operator^{} $\mathrm{cl}$ on $P(X)$, making $X$ a closure space. $\mathrm{cl}:P(X)\to P(X)$ given by
$$\mathrm{cl}(A)=\bigcap \{B\in C\mid A\subseteq B\}$$ is welldefined.

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

•
An intersection structure $C$ is said to be algebraic if for every directed set^{} $B\subseteq C$, we have that $\bigcup B\in C$. 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.
References
 1 B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge (2003)
 2 G. Grätzer: Universal Algebra^{}, 2nd Edition, Springer, New York (1978).
Title  intersection structure 
Canonical name  IntersectionStructure 
Date of creation  20130322 17:06:28 
Last modified on  20130322 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 