When meet is strictly monotone? What is the name?

Question

In a meet semilattice $(S, \wedge)$, the meet operation is monotonic. I.e. if $x\geq x'$ and $y\geq y'$, then $x\wedge y\geq x'\wedge y'$.

What if we assume that this is true for strict inequalities? Is there a name for such semilattices?

If $x> x'$ and $y> y'$, then $x\wedge y > x'\wedge y'$.

What is the name of such semilattices?

Answer 1

I don't think there is any such name because that is not an interesting property.
Indeed, assume $x$ and $y$ are not related (that is, if the semilattice is not a chain);
it follows that $x \wedge y < x$ and $x \wedge y < y$.
If the operation would have such a property, then we would have $$x \wedge y > (x\wedge y) \wedge (x \wedge y) = x\wedge y,$$ a contradiction.

So that property is only hold for semilattices which are chains, but in that case, it is trivial.