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The tags and seem to cover similar ground (judging from the questions which have them).

However, the former can also be misunderstood as talking about the general congruence relation (i.e. a relation $R$ such that $aRb$ and $cRd$ imply $(a\circ c)R(b\circ d)$ with respect to some binary operation $\circ$).

As Willie Wong points out in the comments, there's also the "congruence" of triangles from Euclidean geometry.

Therefore, in the interest of descriptive tagging, I propose to:

  1. Retag those questions in which are about modular equations with . A synonym for mentioning "congruence" would be needed -- avoiding the current pitfall;
  2. Retag the ones about congruence relations with (if they exist -- I didn't check);
  3. Retag the ones about triangles with and/or a new tag, e.g. ;
  4. Perhaps blacklist .
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    $\begingroup$ Some of the questions are about en.wikipedia.org/wiki/Congruence_%28geometry%29 ; they need to be separately retagged. Also, technically the $\pmod p$ congruences are a special type of congruence-relations. $\endgroup$
    – Willie Wong
    Oct 10, 2013 at 11:03
  • $\begingroup$ @WillieWong Ooh, hadn't thought of that (I'm a very algebraic person I guess :)). I know $\pmod p$ is a special case of congruence-relations, but in the current schema it's too easy to confuse the general and the particular. $\endgroup$
    – Lord_Farin
    Oct 10, 2013 at 11:07
  • $\begingroup$ Do polynomial congruences fall under (modular-arithmetic)? At the moment, there are 3 questions tagged congruences+polynomials. $\endgroup$
    – Martin Sleziak
    Dec 1, 2013 at 13:28
  • $\begingroup$ (By polynomial congruences I meant something like $x^2 \equiv -1 \pmod{x^2-1}$, i.e. congruences in the rings of polynomials.) $\endgroup$
    – Martin Sleziak
    Dec 2, 2013 at 9:27

4 Answers 4

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There seems to be considerable support for taking on this tag (+8/-0), which is "high", for questions about tagging).

This answer intends to poll the specific approach I suggested.

Concretely, I suggest to:

$^\dagger$: Martin Sleziak suggested in the Tagging chatroom.

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  • $\begingroup$ Note that 'congruences-and-remainders' has 26 characters, which is 1 too much. I have suggested 'congruences-remainders' here. $\endgroup$
    – Bart Michels
    Mar 22, 2015 at 12:01
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It is too soon to say what is the consensus on this. (Only two people voted on the question.) But I thought that it would be good to be prepared for dividing the tags into several tags by making a list of questions that need to be retagged. It seems that this list is very short. (Of course, I could have missed some questions.) I am making this post CW, feel free to edit it, if needed.

Congruences in geometry

(ordered by id)

Congruence relations

We have some questions about congruence relations in the sense of universal algebra. (Probably many hits from the search for congruence relation universal algebra site:math.stackexchange.com are such questions.) But I did not find any such questions among the questions currently tagged . Here are questions which I found among questions tagged :

(ordered by id)

Congruences of polynomials

(ordered by id)

Matrix congruence:

(ordered by id)

Not sure

Perhaps it is worth mentioning that in the meantime the tag congruences has also a tag-excerpt (it was created after this discussion on meta was started):

For question about the properties or calculation of congruences and congruences equation, and related theorems of congruences like chinese remainder theorem, Fermat's little theorem and Euler's totient theorem.

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  • $\begingroup$ Those special cases have been retagged in the meanwhile. $\endgroup$
    – Bart Michels
    Mar 22, 2015 at 17:38
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I am not sure whether I like having synonym of (as suggested by Lord_Farin.) So here is an alternative suggestion

  • Create a and keep it as a separate tag or make it a synonym of .
  • Make a synonym of . (This is the most frequent use of this tag so far.)
  • Create and keep it as a separate tag (or, if it is not suitable as a standalone tag, make it a synonym of .
  • If needed, we can also create . (Which would be for questions about congruences in rings of polynomials, for example How to find $u(x)$ such that $u(x)\cdot (x+1) \equiv 1 \pmod{x^2+x+1}$ holds in $\mathbb Z_2[x]$. Not for questions about modular arithmetic where one side of the congruence is a polynomial. For example How to show that $x\equiv y \pmod n$ $\Rightarrow$ $f(x) \equiv f(y) \pmod n$, where $f$ is a polynomial with integer coeffcients?)
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In practice, congruences tag is already an equivalent of modular-arithmetic.

I found only $6$ questions with the words "congruence relation" that are not about modular arithmetic or equivalence relations, subjects that already have several tags.

is:q "congruence relation" -[modular-arithmetic] -[number-theory] -[elementary-number-theory]
https://math.stackexchange.com/search?q=is%3Aq+%22congruence+relation%22+-%5Bmodular-arithmetic%5D+-%5Bnumber-theory%5D+-%5Belementary-number-theory%5D

The added expressive power of a new tag beyond , , or , which do appear on the existing congruence relation questions, is small.

For geometric congruences, there are only 3 (of 78) geometry questions in , identifiable by clicking on all the non-number theoretic secondary tags like

geometry
triangle
homework
proof-strategy

https://math.stackexchange.com/questions/tagged/congruences

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    $\begingroup$ The very point of this thread is to make the observation that it is "practically" an equivalent official by taking it all the way. That there are few questions that need to be retagged for this is only good. I plainly disagree on the assessment of congruence-relations. $\endgroup$
    – Lord_Farin
    Oct 31, 2013 at 7:49
  • $\begingroup$ Yes, but there is a choice of whether 'all the way' means removing one duplicative tag (blacklist/synonym/merge of congruences with modular-arithmetic) or replacing it by three duplicative tags. Congruences in geometry are arguably the same thing as euclidean-geometry (they are the morphisms in that category). I would guess that most mathematicians have never encountered the axioms for a congruence relation (as a named concept distinct from equivalence relation), and many would not consider it distinct after having seen the definition! $\endgroup$
    – zyx
    Oct 31, 2013 at 17:24
  • $\begingroup$ It is not relevant (at least to me) how many synonyms a tag has. The only thing I'd like to avoid with this scheme is wrong tagging. Letting a tag sit around that has multiple, unrelated interpretations is simply bad. Do you have an alternative solution for that? $\endgroup$
    – Lord_Farin
    Oct 31, 2013 at 17:46
  • $\begingroup$ Your proposal but with synonyms for all three interpretations instead of one plus two new tags. Such as (congruence[geometry]) --> (euclidean-geometry) and (congruence-relations) --> (equivalence-relations). $\endgroup$
    – zyx
    Oct 31, 2013 at 17:55
  • $\begingroup$ Some reading and thinking have won me over. I've edited my proposal according to your suggestions. $\endgroup$
    – Lord_Farin
    Nov 1, 2013 at 9:19

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