Consider a relation $\sim$ which is irreflexive, that is:
$$ \forall A \:A\nsim A $$
Further suppose the reflexive closure of $\sim$ is an equivalence relation. The reflexive closure of $\sim$, denoted $\simeq$, is defined like this:
$$ \forall A \forall B \:A\simeq B \iff A \sim B \vee A = B $$
(in other words, it's formed by unioning the $\sim$ relation with the equality relation)
Is there a standard name for relations like this?
I thought of this while reading a rather interesting discussion of database schemas in the context of same-sex and plural marriages. The relation I describe above corresponds to "is-married-to" in one of those schemas. Since people cannot marry themselves, it is irreflexive, but it's otherwise structured as an equivalence relation.