Consider a binary relation $\mathsf{R}$ such that $x\mathsf{R}y$ is the case only if there is some $z$ such that both $x\mathsf{R}z$ and $y\mathsf{R}z$ are the case. Is there a well-known name for the property enjoyed by this relation? And are there natural examples in mathematics (or in ARS) of relations characterized by this property?
In normal modal logic, the above property characterizes frames axiomatized by the schema $\lozenge\square\alpha\to\lozenge\alpha$ (or, dually, by $\square\alpha\to\square\lozenge\alpha$). It amounts to a stronger version of the Church-Rosser property, according to which $x\mathsf{R}y$ and $y\mathsf{R}x$ are simultaneously the case only if there is a $z$ such that both $x\mathsf{R}z$ and $y\mathsf{R}z$ are the case. I have however been unable to determine so far whether there is an established name for the above mentioned property in the literature.
(This is cross-posted with TCS.SE, where the question received a couple of comments but no answers were posted in the first few days.)
Had a look around and Hughes "introduction to model logic" and Boolos" The logic of provability" call it CONVERGENT, athough Hughes only calls <>[]p -> []<>p convergent Boolos is more flexibel: see Boolos page 88