# When a name is used for two different notions!

What would you do if you see a notion which you used to work with has two names and the same name you use for this notion is used for another notion too. For example, after the work of J.B. Bost, A. Connes, (Hecke algebras, type III factors and phase transition with spontaneous symmetry breaking in number theory. Selecta Math. (New Series) Vol.1, no.3, 411-456, (1995)), in operator algebras community people call a subgroup $H$ of a group $G$ almost normal if every double coset of $H$, like $HgH$ is the union of finitely left cosets. It is equivalent to say that $H$ is almost normal if it is commensurable with its conjugates. Therefore in group theory people call such a subgroup conjugate commensurable. In stead, in group theory people call a subgroup $H$ of a group $G$ almost normal if its normalizer is of finite index in $G$. Now if I use the word conjugate commensurable for the above notion it is confusing in operator algebras community and if I use almost normal it can be confused with group theorists' notion of almost normal subgroups. What is the best way to deal with these sort of problems.

To start with the obvious, in a given work you should conform to the usage that the majority of your readers expect. A second principle is that you should work out some reasonable system for your own work and follow that consistently. In cases where you expect readers from both camps, you will need to frequently draw your readers' attention to the meaning of your notation.

This is not a problem you can solve, you just have to find a solution you can live with. And whatever you find, your referees will disagree with you (and possibly with each other).

• @Chris: Actually that is what I had to do in my last work. I explained the situation in the footnote.
– user23860
May 27, 2012 at 15:07
• Why not giving it some space in your introduction about notation etc.? May 27, 2012 at 19:38
• I don't think that a scholarly article is the wrong place to make a brief case for intelligently standardizing mathematical terminology or notation. It's actually a bit odd that mathematics has never evolved a standards bureau, say, as the organic chemists have. May 27, 2012 at 19:47
• @David Feldman: I think having a standards bureau for notation is a fabulous idea. Perhaps one might argue that Bourbaki started one decades ago, but it didn't ever get updated and now need serious updating? Could the nLab fulfil this role I wonder? May 27, 2012 at 20:27
• In practice, when a standards association attempts to deal with two competing systems of the notation, the usual outcome is a third competing system. See xkcd.com/927 May 27, 2012 at 22:03

About your example, "almost something" is often a bad choice of terminology, and as ambiguous as possible. When a terminology is ambiguous and some other better terminology is available (in your example, "commensurated subgroup" is used, see for instance this paper by Conner and Mihalik and the references therein).

PS: It seems to me that these types of questions are preferably ticked as Community Wiki.

• @Yves: I think I have to stick to the terminology used by the majority of operator algebraists. \\ It's getting more interesting and involved yet. Now, we have one more name for the same notion. The funny thing is a pair $(G,H)$ is called a Hecke pair if $H$ is an almost normal (or conjugate commensurable or commensurated ) subgroup of $G$. So, somehow we have four different terminology for this notion. \\ By the way the paper you cited is interesting, thanks.
– user23860
May 28, 2012 at 2:52