closure of a relation with respect to a
property∗
CWoo†
2013-03-21 23:57:25
Introduction
Fix a set A. A property P of n-ary relations on a set A may be thought of as
some subset of the set of all n-ary relations on A. Since an n-ary relation is just
a subset of An , P ⊆ P (An ), the powerset of An . An n-ary relation is said to
have property P if R ∈ P.
For example, the transitive property is a property of binary relations on A;
it consists of all transitive binary relations on A. Reflexive and symmetric properties are sets of reflexive and symmetric binary relations on A correspondingly.
Let R be an n-ary relation on A. By the closure of an n-ary relation R with
respect to property P, or the P-closure of R for short, we mean the smallest
relation S ∈ P such that R ⊆ S. In other words, if T ∈ P and R ⊆ T , then
S ⊆ T . We write ClP (R) for the P-closure of R.
Given an n-ary relation R on A, and a property P on n-ary relations on A,
does ClP (R) always exist? The answer is no. For example, let P be the antisymmetric property of binary relations on A, and R = A2 . For another example,
take P to be the irreflexive property, and R = ∆, the diagonal relation on A.
However, if An ∈ P and P is closed under arbitrary intersections, then P is
a complete lattice according to this fact, and, as a result, ClP (R) exists for any
R ⊆ An .
Reflexive, Symmetric, and Transitive Closures
From now on, we concentrate on binary relations on a set A. In particular, we
fix a binary relation R on A, and let X the reflexive property, S the symmetric
property, and T be the transitive property on the binary relations on A.
Proposition 1. Arbitrary intersections are closed in X , S, and T . Furthermore, if R is any binary relation on A, then
∗ hClosureOfARelationWithRespectToAPropertyi
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• R= := ClX (R) = R ∪ ∆, where ∆ is the diagonal relation on A,
• R↔ := ClS (R) = R ∪ R−1 , where R−1 is the converse of R, and
• R+ := ClT (R) is given by
[
Rn = R ∪ (R ◦ R) ∪ · · · ∪ (R ◦ · · · ◦ R) ∪ · · · ,
|
{z
}
n∈N
n-fold power
where ◦ is the relational composition operator.
• R∗ := R=+ = R+= .
R= , R↔ , R+ , and R∗ are called the reflexive closure, the symmetric closure,
the transitive closure, and the reflexive transitive closure of R respectively. The
last item in the proposition permits us to call R∗ the transitive reflexive closure
of R as well (there is no difference to the order of taking closures). This is true
because ∆ is transitive.
Remark. In general, however, the order of taking closures of a relation is
important. For example, let A = {a, b}, and R = {(a, b)}. Then R↔+ = A2 6=
{(a, b), (b, a)} = R+↔ .
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