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[parent] intersection structure (Definition)

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



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See Also: criteria for a poset to be a complete lattice

Other names:  closure system
Also defines:  topped intersection structure, algebraic intersection structure, algebraic closure system

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Cross-references: algebraic lattice, directed set, algebraic, well-defined, closure space, closure operator, induces, proof, top, maximal elements, join, complete lattice, singleton, relation, collection, partial functions, theory, domain, partial orderings, topological vector space, convex subsets, functional analysis, topological space, closed sets, topology, vector space, subspaces, ring, ideals, group, subgroups, algebra, poset, set inclusion, order, intersection, powerset, subset
There are 4 references to this entry.

This is version 5 of intersection structure, born on 2007-05-19, modified 2007-05-21.
Object id is 9406, canonical name is IntersectionStructure.
Accessed 1967 times total.

Classification:
AMS MSC: 06B23 (Order, lattices, ordered algebraic structures :: Lattices :: Complete lattices, completions)
 03G10 (Mathematical logic and foundations :: Algebraic logic :: Lattices and related structures)
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