Group-theory,discrete mathematics, trichotomous property Is the intersection of trichotomous relations trichotomous? Generally, trichotomy is the property of an order relation < on a set X that for any$ x$ and $y$, exactly one of the following holds: $x<y, x=y, $ or $ x>y$.

Not necessarily. Let $<$ be a strict linear order on a set with at least two elements. Then $>$ is also a strict linear order and hence trichtomous, but the intersection of these is is the empty relation, which is not trichotomous.