# User talk:Dfeuer/Definition:Strictly Positive Cone

A more regular name would be "strictly positive cone" though if I tried I could probably grammatically justify the current version. --Lord_Farin (talk) 18:33, 1 February 2013 (UTC)

I made the move, but that makes it even more awkward to name the total version, no? It's forced to be Total Strictly Positive Cone. --Dfeuer (talk) 18:46, 1 February 2013 (UTC)

That's unfortunate. I am currently crawling through the sewers of the interhole to try and locate some literature about this. --Lord_Farin (talk) 18:51, 1 February 2013 (UTC)
Terminology seems to indicate that "total strictly positive cone" is usually called "positive cone". Key name appears to be Dale Rolfsen. See e.g. this pdf for an overview. Theory appears extensive; connections with braid groups and topology crop up most. --Lord_Farin (talk) 19:02, 1 February 2013 (UTC)
Wikipedia has positive cones for other structures, such as vector spaces (from which the terminology likely arose), which are not total. --19:07, 1 February 2013 (UTC)
One option if we're desperate might be to use a name like "pone", which probably doesn't have any mathematical meaning at all. --Dfeuer (talk) 19:14, 1 February 2013 (UTC)
Whatever it's called in whatever sources you are sourcing, that's what it is to be called. Period. Please do not make up names for things because you don't like the name that is given them by your sources, and most importantly of all, do not dismiss such sources with emotively insulting words like "stupid" and "crazy" because that will get you blocked. --prime mover (talk) 23:12, 1 February 2013 (UTC)
If you were paying attention to what's going on here, you would understand why I am looking for a name for this concept. As you weren't, please go do something useful rather than saying nasty things about me. An awkward but not terrible answer to my question seems to have appeared, as you can see below. --Dfeuer (talk) 23:35, 1 February 2013 (UTC)
For the record: The point Dfeuer makes is valid in the present situation. More so as this concerns user space. --Lord_Farin (talk) 23:37, 1 February 2013 (UTC)
Pointing out the fact that you have a predilection to call established naming conventions "stupid" and "crazy" (I can go and find instances if you like) is not "saying nasty things" about you. My point stands. Using your own names for things, unless you are crafting whole new areas of mathematics, is not a good idea. If the names already exist, whether you like them or not, it's a very good idea to use them. --prime mover (talk) 06:26, 2 February 2013 (UTC)
Which is why considerable effort has gone into verifying that this is not a new area of mathematics, and subsequently trying to locate the most consistent nomenclature found in the literature. The suggestion "pone" may not have been the best we've ever seen, but I took the rather less provocative way of simply ignoring it. After all, the premises for the suggestion weren't satisfied yet. --Lord_Farin (talk) 10:58, 2 February 2013 (UTC)
Okay, I'll let you get on with it ... as long as the above guidelines are kept in mind. --prime mover (talk) 13:15, 2 February 2013 (UTC)
To me, such goes without saying. --Lord_Farin (talk) 19:28, 2 February 2013 (UTC)

## Connecting a bit to group theory

If $N$ is a normal subgroup of $G$, then by definition $a N a^{-1}= N$.

If $ab \in N$, then $a^{-1} (ab) a \in N$, so $ba \in N$.

Thus any normal subgroup is what I call a "cone".

Let $C$ be a cone.

Suppose $x \in C$ and $g \in G$. Then $gg^{-1} x g g^{-1} \in C$ so $(g^{-1} x g) g^{-1}g = g^{-1}xg \in C$.

This is very normal subgroup-like, but it's not actually always a subgroup, because it doesn't necessarily contain the identity (reflexivity) or inverses (symmetry). Is there a "normal subset"? --Dfeuer (talk) 20:23, 1 February 2013 (UTC)

The internet does show there is such a thing as a normal subset. Thus what I call a cone in a group is perhaps most descriptively called a normal submagma of the group. That definition of course works only for groups, and is horribly awkward, but at least it shouldn't clash with anything. --Dfeuer (talk) 20:45, 1 February 2013 (UTC)

In view of the proposed restriction to groups, this seems adequate. It would be good (IMO) to subsequently connect to established language (i.e. taking the literature definition for Pos. Cone) because it's of course not a maximal submagma. --Lord_Farin (talk) 22:00, 1 February 2013 (UTC)