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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?2011-04-20
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    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.2011-04-20
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    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.2011-04-20
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    Related but maybe not exactly what you described http://en.wikipedia.org/wiki/Partial_equivalence_relation2011-04-20
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    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?2011-04-20
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    @Asaf: The term "full relation" is ambiguous, since it's also used for a relation relating all elements to all elements (e.g. http://www.thomasalspaugh.org/pub/fnd/relation.html, http://en.wikipedia.org/wiki/Epimorphism). A less ambiguous term for this is "serial relation".2011-04-20
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    @joriki: I agree about that. The meaning I was aiming for in my previous comment was that $\forall x\in X\exists y\in X (x\sim y)$.2011-04-20
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    @quanta: That's not "maybe not exactly what you described", it's something quite different :-) (though related in that one can usefully describe it by talking about the elements that are related to something and the ones that aren't).2011-04-20

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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: http://en.wikipedia.org/wiki/Quasi-reflexive_relation2011-04-20
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    By the way, "quasi-reflexive and serial" is equivalent to "reflexive".2011-04-20
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    Thanks for the answer and comments.2011-04-21