# Is there a name for this property of a binary relation? $\forall x\forall y(x\mathsf{R}y\to\exists z(x\mathsf{R}z\land y\mathsf{R}z)))$

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.)

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If you had "if and only if" in the definition instead of just "only if" then you could call such a relation a symmetric idempotent (with respect to relational composition). But as it stands I can't immediately think of a suitable name. – Dave Wilding Jan 21 '14 at 18:52
can you add all relations ( is it transitive and or serial) maybe the combination of relations has a name, as afar as i can see refexive includes yor relation and it is a bit a combination of serial and connective , in G' notation it is an G(0,1,1,1) but this one doesn't has a name – Willemien Jan 27 '14 at 14:41
@Willemien Indeed, from the comments I received so far it seems that the axiom G(0,1,1,1) ---or, equivalently, G(1,1,0,1)--- has not been baptized, or even studied for its own sake. It does seem to correspond, anyway, to a qualified version of the notion of joinability, in ARS, where one demands $x\mathsf{R}y$ as a necessary condition for $x\downarrow y$. – J Marcos Jan 28 '14 at 14:17
Abstract rewriting is a different logic branch, (I think it belongs to combinatory logic or Lamda calculus, branches I am not familiar with. I am not sure if it is possible to put Abstract rewriting in a one-to-one relation with modal logc, I can only advice you to be very careful here, take nothing for obvious. (the arrow with star looks very transitive but then is []p -> [][]p an axiom? The same for []p -> p (reflexivity) is it vacously true or false? and most important: Is |-p => |-[]p (rule of nesecitation, needed for normal modal logics) valid? all questions I don't know the answer. – Willemien Jan 28 '14 at 15:06

## 1 Answer

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

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In the study of abstract rewriting systems, such diamond property is a particular case of the notion of confluence. You get convergence if you add termination to confluence. – J Marcos Jan 28 '14 at 14:01