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convex subgroup (Definition)

We begin this article with something more general. Let $ P$ be a poset. A subset $ A\subseteq P$ is said to be convex if for any $ a,b\in A$ with $ a\le b$, the poset interval $ [a,b]\subseteq A$ also. In other words, $ c\in A$ for any $ c\in P$ such that $ a\le c$ and $ c\le b$. Examples of convex subsets are intervals themselves, antichains, whose intervals are singletons, and the empty set.

One encounters convex sets most often in the study of partially ordered groups. A convex subgroup $ H$ of a po-group $ G$ is a subgroup of $ G$ that is a convex subset of the poset $ G$ at the same time. Since $ e\in H$, we have that $ [e,a]\subseteq H$ for any $ e\le a\in H$. Conversely, if a subgroup $ H$ satisfies the property that $ [e,a]\subseteq H$ whenever $ a\in H$, then $ H$ is a convex subgroup: if $ a,b\in H$, then $ a^{-1}b\in H$, so that $ [e,a^{-1}b]\subseteq H$, which implies that $ [a,b]=a[e,a^{-1}b]\subseteq H$ as well.

For example, let $ G=\mathbb{R}^2$ be the po-group under the usual Cartesian ordering. $ G$ and 0 are both convex, but these are trivial examples. Let us see what other convex subgroups $ H$ there are. Suppose $ P=(a,b)\in H$ with $ (a,b)\ne (0,0)=O$. We divide this into several cases:

  1. $ ab>0$. If $ a>0$, then $ b>0$ ($ P$ in the first quadrant), so that $ O\le P$, which means $ [O,P]\subseteq H$. If $ a<0$, then $ b<0$ ($ P$ in the third quandrant), so that $ O\le -P$. In either case, $ H$ contains a rectangle ($ [O,P]$ or $ [O,-P]$) that generates $ G$, so $ H=G$.
  2. One of $ a$ or $ b$ is 0. Suppose $ a=0$ for now. Then either $ 0<b$ so that $ [O,P]\subseteq H$ or $ b<0$ so that $ [O,-P]\subseteq H$. In either case, $ H$ contains a line segment on the $ y$-axis. But this line segment generates the $ y$-axis. So $ y$-axis $ \subseteq H$. If $ H$ is a subgroup of the $ y$-axis, then $ H$=$ y$-axis.

    Otherwise, another point $ Q=(c,d)\in H$ not on the $ y$-axis. We have the following subcases:

    1. If $ cd>0$, then $ H=G$ as in the previous case.
    2. If $ cd< 0$, say $ d<0$ (or $ 0<c$), then for some positive integer $ n$, $ 0<d+nb$, so that $ O\le Q+nP$, and $ H=G$ as well. On the other hand, if $ c<0$ (or $ 0<d$), then $ -Q$ returns us to the previous argument and $ H=G$ again.
    3. If $ d=0$ (so $ c\ne 0$), then either $ O\le P+Q$ (when $ 0<c$) or $ O\le P-Q$ (when $ c<0$), so that $ H=G$ once more.

    A similar set of arguments shows that if $ H$ contains a segment of the $ x$-axis, then either $ H$ is the $ x$-axis or $ H=G$. In conclusion, in the case when $ ab=0$, $ H$ is either one of the two axes, or the entire group.

  3. $ ab<0$. It is enough to assume that $ 0<a$ and $ b<0$ (that $ P$ lies in the fourth quadrant), for if $ P$ lies in the second quadrant, $ -P$ lies in the fourth.

    Since $ O,P\in H$, $ H$ could be a subgroup of the line group $ L$ containing $ O$ and $ P$. No two points on $ L$ are comparable, for if $ (r,s)<(t,u)$ on $ L$, then the slope of $ L$ is positive

    $\displaystyle 0<\frac{u-s}{t-r},$
    a contradiction. So $ L$, and hence $ H$, is an antichaine. This means that $ H$ is convex.

    Suppose now $ H$ contains a point $ Q=(c,d)$ not on $ L$. We again break this down into subcases:

    1. $ Q$ is in the first or third quandrant. Then $ H=G$ as in the very first case above.
    2. $ Q$ is on either of the axes. Then $ H=G$ also, as in case 2(b) above.
    3. $ Q$ is in the second or fourth quadrant. It is enough to assume that $ Q$ is in the same quadrant as $ P$ (fourth). So we have $ 0<c$ and $ d<0$. Since $ L$ passes through $ P$ and not $ Q$, we have that
      $\displaystyle \frac{a}{c}\ne \frac{b}{d}.$
      Let $ 0<r=a/c$ and $ 0<s=b/d$ and assume $ r<s$. Then there is a rational number $ m/n$ (with $ 0<m,n$) such that
      $\displaystyle r<\frac{m}{n}<s.$
      This means that $ na< mc$ and $ nb<md$, or $ nP<mQ$. But $ nP,mQ\in H$, so is $ R=mQ-nP\in H$, which is in the first quadrant. This implies that $ H=G$ too.
    In summary, if $ H$ contains a point in the second or fourth quadrant, then either $ H$ is a subgroup of a line with slope $ <0$, or $ H=G$.
The three main cases above exhaust all convex subgroups of $ \mathbb{R}^2$ under the Cartesian ordering.

If the Euclidean plane is equipped with the lexicographic ordering, then the story is quite different, but simpler. If $ H$ is non-trivial, say $ P=(a,b)\in H$, $ P\ne O$. If $ 0<a$, then $ (c,d)\le (a,b)$ for any $ c< a$ regardless of $ d$. Choose $ Q=(c,d)$ to be in the first quadrant. Then $ [O,Q]\subseteq H$, so that $ H=G$. If $ a<0$, then $ -P$ takes us back to the previous argument. If $ a=0$, then either $ [O,P]$ (when $ 0<b$), or $ [O,-P]$ (when $ b<0$) is a positive interval on the $ y$-axis. This implies that $ H$ is at least the $ y$-axis. If $ H$ contains no other points, then $ H=y$-axis. In summary, the po-group $ \mathbb{R}^2$ with lexicographic order has the $ y$-axis as the only non-trivial proper convex subgroup.

Bibliography

1
G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).



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Cross-references: lexicographic order, lexicographic ordering, Euclidean plane, rational number, passes through, contradiction, slope, comparable, line, group, entire, conclusion, segment, similar, argument, integer, positive, point, line segment, generates, rectangle, contains, quadrant, divide, Cartesian ordering, implies, property, subgroup, partially ordered groups, empty set, singletons, antichains, intervals, words, poset interval, convex, subset, poset
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This is version 1 of convex subgroup, born on 2007-05-10.
Object id is 9360, canonical name is ConvexSubgroup.
Accessed 837 times total.

Classification:
AMS MSC: 06A99 (Order, lattices, ordered algebraic structures :: Ordered sets :: Miscellaneous)
 06F15 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered groups)
 06F20 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered abelian groups, Riesz groups, ordered linear spaces)
 20F60 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Ordered groups)
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