An intersection structure is a set such that
If order by set inclusion, then 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 be the set of all partial functions from a non-empty set to a non-empty set . Since each partial function is a subset of , is a subset of . Let be an arbitrary collection of partial functions in and . is clearly a relation between and . Suppose is in the domain of . Let . Then for each where is in the domain of . Since is a partial function, , so that is uniquely determined. This means that is a singleton, hence is a partial function, so that , meaning that is an intersection structure.
The main difference between the last two examples and the previous examples is that in the last two examples, is rarely a complete lattice. For example, let be a partial ordering on a set . Then its dual is also a partial ordering on . But the join of and does not exist. Here is another example: let and . Then . and are the maximal elements of , 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 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.
Conversely, it is not hard to see that every closure space gives rise to a closure system .
An intersection structure is said to be algebraic if for every directed set , we have that . 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.
|Date of creation||2013-03-22 17:06:28|
|Last modified on||2013-03-22 17:06:28|
|Last modified by||CWoo (3771)|
|Defines||topped intersection structure|
|Defines||algebraic intersection structure|
|Defines||algebraic closure system|