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.(EDIT: note that this may be formalized as $\forall x\forall y(x\mathsf{R}y\to\exists z(x\mathsf{R}z\land y\mathsf{R}z)))$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.(I am now cross-posting this with Math.SE, to see if anyone there has an aswer to offer.)
Is there a name for this property of a binary relation?
graph theory;terminology;term rewriting systems;modal logic