convex subgroup

We begin this article with something more general. Let P be a poset. A subset AP is said to be convex if for any a,bA with ab, the poset interval [a,b]A also. In other words, cA for any cP such that ac and cb. Examples of convex subsets are intervalsMathworldPlanetmath themselves, antichainsMathworldPlanetmath, whose intervals are singletons, and the empty setMathworldPlanetmath.

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

For example, let G=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)H with (a,b)(0,0)=O. We divide this into several cases:

  1. 1.

    ab>0. If a>0, then b>0 (P in the first quadrantMathworldPlanetmath), so that OP, which means [O,P]H. If a<0, then b<0 (P in the third quandrant), so that O-P. In either case, H contains a rectangleMathworldPlanetmathPlanetmath ([O,P] or [O,-P]) that generates G, so H=G.

  2. 2.

    One of a or b is 0. Suppose a=0 for now. Then either 0<b so that [O,P]H or b<0 so that [O,-P]H. In either case, H contains a line segmentMathworldPlanetmath on the y-axis. But this line segment generates the y-axis. So y-axis H. If H is a subgroup of the y-axis, then H=y-axis.

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

    1. (a)

      If cd>0, then H=G as in the previous case.

    2. (b)

      If cd<0, say d<0 (or 0<c), then for some positive integer n, 0<d+nb, so that OQ+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. (c)

      If d=0 (so c0), then either OP+Q (when 0<c) or OP-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 conclusionMathworldPlanetmath, in the case when ab=0, H is either one of the two axes, or the entire group.

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


    a contradictionMathworldPlanetmathPlanetmath. 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. (a)

      Q is in the first or third quandrant. Then H=G as in the very first case above.

    2. (b)

      Q is on either of the axes. Then H=G also, as in case 2(b) above.

    3. (c)

      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


      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


      This means that na<mc and nb<md, or nP<mQ. But nP,mQH, so is R=mQ-nPH, 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 2 under the Cartesian ordering.

If the Euclidean planeMathworldPlanetmath is equipped with the lexicographic ordering, then the story is quite different, but simpler. If H is non-trivial, say P=(a,b)H, PO. If 0<a, then (c,d)(a,b) for any c<a regardless of d. Choose Q=(c,d) to be in the first quadrant. Then [O,Q]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 2 with lexicographic orderMathworldPlanetmath has the y-axis as the only non-trivial proper convex subgroup.


Title convex subgroup
Canonical name ConvexSubgroup
Date of creation 2013-03-22 17:04:04
Last modified on 2013-03-22 17:04:04
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Definition
Classification msc 06A99
Classification msc 06F15
Classification msc 06F20
Classification msc 20F60
Defines convex subset