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.

intersection of a non-empty 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 non-empty set $X$ to a non-empty 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 $xf_{i}y$ 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 $\leq$ be a partial ordering on a set $P$. Then its dual $\leq^{\partial}$ is also a partial ordering on $P$. But the join of $\leq$ and $\leq^{\partial}$ does not exist. Here is another example: let $X=\{1\}$ and $Y=\{2,3\}$. Then $C=\{\varnothing,(1,2),(1,3)\}$. $(1,2)$ 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 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 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 $\operatorname{cl}$ on $P(X)$, making $X$ a closure space. $\operatorname{cl}:P(X)\to P(X)$ given by

 $\operatorname{cl}(A)=\bigcap\{B\in C\mid A\subseteq B\}$

is well-defined.

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

• 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: , 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