We begin this article with something more general. Let be a poset. A subset is said to be convex if for any with , the poset interval also. In other words, for any such that and . 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 of a po-group is a subgroup of that is a convex subset of the poset at the same time. Since , we have that for any . Conversely, if a subgroup satisfies the property that whenever , then is a convex subgroup: if , then , so that , which implies that as well.
For example, let be the po-group under the usual Cartesian ordering. and are both convex, but these are trivial examples. Let us see what other convex subgroups there are. Suppose with . We divide this into several cases:
One of or is . Suppose for now. Then either so that or so that . In either case, contains a line segment on the -axis. But this line segment generates the -axis. So -axis . If is a subgroup of the -axis, then =-axis.
Otherwise, another point not on the -axis. We have the following subcases:
If , then as in the previous case.
If , say (or ), then for some positive integer , , so that , and as well. On the other hand, if (or ), then returns us to the previous argument and again.
If (so ), then either (when ) or (when ), so that once more.
A similar set of arguments shows that if contains a segment of the -axis, then either is the -axis or . In conclusion, in the case when , is either one of the two axes, or the entire group.
. It is enough to assume that and (that lies in the fourth quadrant), for if lies in the second quadrant, lies in the fourth.
a contradiction. So , and hence , is an antichaine. This means that is convex.
Suppose now contains a point not on . We again break this down into subcases:
is in the first or third quandrant. Then as in the very first case above.
is on either of the axes. Then also, as in case 2(b) above.
is in the second or fourth quadrant. It is enough to assume that is in the same quadrant as (fourth). So we have and . Since passes through and not , we have that
Let and and assume . Then there is a rational number (with ) such that
This means that and , or . But , so is , which is in the first quadrant. This implies that too.
In summary, if contains a point in the second or fourth quadrant, then either is a subgroup of a line with slope , or .
The three main cases above exhaust all convex subgroups of under the Cartesian ordering.
If the Euclidean plane is equipped with the lexicographic ordering, then the story is quite different, but simpler. If is non-trivial, say , . If , then for any regardless of . Choose to be in the first quadrant. Then , so that . If , then takes us back to the previous argument. If , then either (when ), or (when ) is a positive interval on the -axis. This implies that is at least the -axis. If contains no other points, then -axis. In summary, the po-group with lexicographic order has the -axis as the only non-trivial proper convex subgroup.
- 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
|Date of creation||2013-03-22 17:04:04|
|Last modified on||2013-03-22 17:04:04|
|Last modified by||CWoo (3771)|