irreflexive

A binary relation $ℛ$ on a set $A$ is said to be irreflexive (or antireflexive) if $\forall a\in A$, $\mathrm{\neg }aℛa$. In other words, “no element is $ℛ$-related to itself.”

For example, the relation (“less than”) is an irreflexive relation on the set of natural numbers.

Note that “irreflexive” is not simply the negation of “reflexive (http://planetmath.org/Reflexive).” Although it is impossible for a relation (on a nonempty set) to be both reflexive (http://planetmath.org/Reflexive)and irreflexive, there exist relations that are neither. For example, the relation $\{(a,a)\}$ on the two element set $\{a,b\}$ is neither reflexive nor irreflexive.

Here is an example of a non-reflexive, non-irreflexive relation “in nature.” A subgroup in a group is said to be self-normalizing if it is equal to its own normalizer. For a group $G$, define a relation $ℛ$ on the set of all subgroups of $G$ by declaring $HℛK$ if and only if $H$ is the normalizer of $K$. Notice that every nontrivial group has a subgroup that is not self-normalizing; namely, the trivial subgroup $\{e\}$ consisting of only the identity. Thus, in any nontrivial group $G$, there is a subgroup $H$ of $G$ such that $\mathrm{\neg }HℛH$. So the relation $ℛ$ is non-reflexive. Moreover, since the normalizer of a group $G$ in $G$ is $G$ itself, we have $GℛG$. So $ℛ$ is non-irreflexive.