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Suppose that $X$ is a set and $\sim$ is a binary relation on $X$ that satisfies for all $x,y \in X$; if $x \sim y$ then $x \sim x$ and $y \sim y$. Is there a name for this type of relation?

I am thinking of using the name "partly reflexive". I prefer this to "partially reflexive" because the set $X$ will usually be a partially ordered set. In case it matters this property will be used to define a generalization of the notion of extreme subset. In the context of extreme subsets the property says: if $x$ is an extreme subset of $y$ then $x$ is an extreme subset of $x$ and $y$ is an extreme subset of $y$.

If there is no common name for this property and you can think of a better name I would appreciate that. Also, if there is a reason why the name "partly reflexive" should be avoided I would appreciate that information as well.

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The problem with "partially reflexive" is that it doesn't say when it's reflexive. Any relation that has some elements that are reflexive could be described the same way. I don't know a catchier way than "related to itself if it's related to anything". "Reflexive or disjoint", maybe? – wnoise Apr 20 '11 at 2:09
Your property is not very interesting, because the elements not guaranteed to be reflexive are those that don't relate in any way to anything else. – Yuval Filmus Apr 20 '11 at 6:44
Reflexive orders are usually less interesting in the confines of set theory, it is somewhat of a "dummy" property that can be added or removed as one likes. Furthermore, the problem with your property being applied to partial orders is that partial order may require reflexivity - which renders your property useless, or they usually turn out as irreflexive which renders this property useless. And one last observation is that if $\sim$ is a full relation (i.e. every $x\in X$ is in $\sim$-relation with someone else) then this implies $\sim$ is just reflexive. – Asaf Karagila Apr 20 '11 at 8:34
Related but maybe not exactly what you described – quanta Apr 20 '11 at 11:08
Why is everyone saying that this property is uninteresting when the OP already stated why he's interested in it, and I mentioned in my answer that it's relevant in modal logic? – joriki Apr 20 '11 at 11:13
up vote 12 down vote accepted

The term for this seems to be "quasi-reflexive". You'll find some examples when you Google that term, among them an entry in Encyclopedia Britannica. This appears to be relevant in modal logic, where a possible world is accessible from itself if it is accessible from some possible world.

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I added the definition in Wikipedia: – joriki Apr 20 '11 at 8:23
By the way, "quasi-reflexive and serial" is equivalent to "reflexive". – joriki Apr 20 '11 at 8:30
Thanks for the answer and comments. – Jay Apr 21 '11 at 1:42

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