# Terminology: What to call a set where equivalence is defined over all members?

Math amateur here, and new to the site. I have to write a technical specification, and am rather unsure about how to express things mathematically properly.

So I have a set of “entities” of some sort. Let's call it A. It shouldn't matter what the entities are for this question.

I also have a notion of equivalence between the entities in A.

But, for any two entities, it may or may not be possible to tell whether they are equivalent. So, the equivalence relationship is not defined for all pairs of entities.

Now, I need to talk a lot about subsets of A where the equivalence relationship is defined for all members of the subset. (For any a and b in the subset, “a equivalent b” is defined.)

What should I call these subsets? Is there a good term for describing exactly that? So I'm looking for a good term for XXX in this sentence: “If a subset B of A is a XXX (that is, “a equivalent b” is defined for any a and b in B), then…”

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You can define a new relation $\approx$ where $a\approx b$ means "$a$ can be compared to $b$ using the old equivalence relation." It is defined on all of $A$, and if you're lucky, it's an equivalence relation. That being said, I think you just have to make up a name, like clique. –  Arthur Sep 12 '12 at 21:34
@cygri You may call $B$ a subsetoid. –  goblin Mar 23 '13 at 20:15
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## 1 Answer

The notion of a partial equivalence relation $R$ on $A$ ( http://en.wikipedia.org/wiki/Partial_equivalence_relation ) might be what you are looking for. It is a symmetric, transitive relation on $A$. It is like an equivalence relation expect that it is not required to be reflexive, i.e., there may be elements $x \in A$ such that $\neg(x \mathbin{R} x)$. These are the elements for which "equivalence is undefined" in your terminology.

This assumes that whenever $x$ is such that "the equivalence of $x$ and $y$ is undefined" for some $y$, then "the equivalence of $x$ and $y$ is undefined" for all $y$. If this is not the case then you may need something more complicated.

EDIT: I just realized that I didn't really answer the question. I don't know if there is a name for the set $\{x \in A : x \mathbin{R} x\}$. The name "clique" might not be good, because it suggests that all elements in the subset are related to one another.

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As I understood cygri, he was looking for a term not for the set $\{x \in A : x \mathbin{R} x\}$, but for the name of a set $B \subseteq A$ satisfying $(x \mathbin{R} y)$ or $\neg \: (x\mathbin{R} y)$ for all $x, y \in B$, I still feel clique is an applicable word for a set like that. There might be better. If you draw a graph where the elements of $A$ are vertices and there is a line between two points if they can be compared using the relation, he wants a name for a complete subgraph. If the graph represented friendship between people instead, a clique is a very good word. –  Arthur Sep 13 '12 at 9:42
Arthur is right. In my case the equivalence relationship is reflexive. “Clique” seems like an okay term. I was hoping that there's a more specific term, but that may just not be the case. Thanks anyway for your help Trevor! –  cygri Sep 13 '12 at 11:38
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