What's the term for antisymmetry where equal elements are not in the relation?

Question

The most common definition of antisymmetry of a relation $R$ on a set $S$ is

$$ \forall a, b \in S, R(a, b) \land R(b, a) \to a = b. $$

However, this doesn't cover a relation such as $<$, for example, since there is no way to have $a<b$ and $b>a$ simultaneously (according to traditional definitions). A better way to describe the antisymmetry of $<$, then, is to write

$$ \forall a, b \in S, R(a,b) \to \lnot R(b, a) $$

Of course, a relation which satisfies the second definition will satisfy the first (because the hypothesis of the first would then be a contradiction) but not the other way around, so this is a distinct and more specific notion. Is there a name for this kind of antisymmetry?

Answer 1

The definition of antisymmetry does cover relations like $<$. If you examine that definition carefully, you’ll see that in order for a relation $R$ to violate it, there must be elements $a,b\in S$ such that $R(a,b)$, $R(b,a)$, and $a\ne b$. If you can’t even find elements $a,b\in S$ such that $R(a,b)$ and $R(b,a)$, then you certainly can’t find elements that violate antisymmetry of $R$. Thus, all totally irreflexive relations like $<$ are automatically antisymmetric. (The technical expression is that they are vacuously antisymmetric.)

Your second property is asymmetry: a relation with that property is said to be asymmetric.

Answer 2

The $\le$ formulation of a partial order says that the relation is reflexive, transitive and antisymmetric. The $<$ formulation that it is most familiar to me says that it is transitive and irreflexive, where irreflexive means that $x < x$ never holds. Irreflexivity and asymmetry (as in Brian M. Scott's answer) are equivalent for transitive relations, but I have encountered "irreflexive" much more frequently than "asymmetric".