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 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 $\mathcal{F}:=\lbrace f_i\mid i\in I\rbrace$ be an arbitrary collection of partial functions in $C$ and $f=\bigcap \mathcal{F}$ . $f$ is clearly a relation between $X$ and $Y$ . The domain $D$ of $f$ is the intersection of the domains of each of the $f_i$ . Suppose $x\in D$ . Let $E=\lbrace y\in Y\mid xfy\rbrace$ . Then $xf_i y$ for each $f_i$ . 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 $X=\lbrace 1\rbrace$ and $Y=\lbrace 2,3\rbrace$ . Then $C=\lbrace \varnothing, (1,2),(1,3)\rbrace$ . $(1,2)$ 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.

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

Bibliography

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).