<P> A transitive relation T satisfies aTb ∧ bTc ⇒ aTc . An arbitrary homogeneous relation R may not be transitive but it is always contained in some transitive relation: R ⊆ T . The operation of finding the smallest such T corresponds to a closure operator called transitive closure . </P> <P> Among heterogeneous relations there are properties of difunctionality and contact which lead to difunctional closure and contact closure . The presence of these closure operators in binary relations leads to topology since open - set axioms may be replaced by Kuratowski closure axioms . Thus each property P, symmetry, transitivity, difunctionality, or contact corresponds to a relational topology . </P> <P> In the theory of rewriting systems, one often uses more wordy notions such as the reflexive transitive closure R--the smallest preorder containing R, or the reflexive transitive symmetric closure R--the smallest equivalence relation containing R, and therefore also known as the equivalence closure . When considering a particular term algebra, an equivalence relation that is compatible with all operations of the algebra is called a congruence relation . The congruence closure of R is defined as the smallest congruence relation containing R . </P> <P> For arbitrary P and R, the P closure of R need not exist . In the above examples, these exist because reflexivity, transitivity and symmetry are closed under arbitrary intersections . In such cases, the P closure can be directly defined as the intersection of all sets with property P containing R . </P>

In whole numbers closure property is not satisfied with respect to operations