Directed arcs and suprema

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

An arc of a pretree with a chosen direction runs towards something: towards its last point if it has one, and otherwise towards a place that may or may not be a point of the pretree. This lesson makes this precise in two ways. First, every directed arc $A$ defines a flow, $\operatorname{flow}(A)$, which is the flow towards the last point if there is one (Proposition 2.4). Two directed arcs that share a final part define the same flow, and conversely two directed arcs that meet and define the same flow share a final part (Lemmas 2.3 and 2.6). Second, a point $b$ is a supremum of a directed arc if the arc ends at $b$ or runs up to $b$ without reaching it. The main result (Theorem 4.3) says that a pretree is complete, in the sense that every arc is an interval, exactly when every directed arc has a supremum.

The lesson follows Lemmas 3.5 to 3.13 of [Bowditch]. It uses the lessons Betweenness and pretrees, for arcs, directions and intervals, and Flows on a pretree.

1. Directed arcs and final segments

Let $T$ be a pretree. A directed arc is an arc $A$, that is, a nonempty full linear subset, together with a direction $<$: a total order on $A$ such that for $x,y,z\in A$ the relation $xyz$ holds exactly when $x<y<z$ or $z<y<x$. Since $A$ is full, for $x<y$ in $A$, $[x,y]=\{w\in A:\ x\le w\le y\}.$

A final segment of $(A,<)$ is a nonempty subset $B\subseteq A$ such that $x\in B$, $y\in A$ and $x\le y$ imply $y\in B$. With the restricted order it is a directed arc. For $x\in A$ the set $\{y\in A:x\le y\}$ is a final segment. A point $a\in A$ is the maximum of $A$ if $x\le a$ for all $x\in A$; it then lies in every final segment. $A$ is endless if it has no maximum. A final segment of an endless arc is endless.

Two directed arcs $(A,<)$ and $(A',<')$ are cofinal if there is a set $C$ that is a final segment of both and on which the two orders agree.

Lemma 1.1. Let $(A,<)$ be a directed arc.

If $B\subseteq A$ is an arc that contains a final segment of $A$, then $B$ is a final segment of $A$.
If $(A',<')$ is a directed arc cofinal with $A$, then $A\cap A'$ is a final segment of both, and the two orders agree on it.

Proof. (1) Let $C\subseteq B$ be a final segment of $A$ and $z\in C$. Let $x\in B$ and $y\in A$ with $x\le y$. If $y\ge z$, then $y\in C\subseteq B$. If $y\le z$, then $y\in[x,z]\subseteq B$, because $B$ is full.

(2) Let $C$ be a common final segment on which the orders agree. The set $A\cap A'$ is full and linear and contains $C$, so by (1) it is a final segment of $A$ and of $A'$. The restrictions of $<$ and $<'$ to $A\cap A'$ are directions of this linear set. If $C$ has two points, the two directions agree on them and are equal (Proposition 5.2 of the first lesson). If $C=\{c\}$, then $c$ is the maximum for both; a linear set with more than one point has two directions, each the reverse of the other, and only one of them has $c$ as its maximum. $\blacksquare$

2. The flow of a directed arc

Lemma 2.1. Let $(A,<)$ be an endless directed arc and $x,y\in T$. The following are equivalent.

For every $z\in A$ there is $w\in A$ with $z<w$ and $xyw$.
There is $z\in A$ such that $xyw$ for all $w\in A$ with $w>z$.

Proof. (2) implies (1): let $z_0$ be as in (2); given $z$, choose $w\in A$ larger than $z$ and $z_0$.

(1) implies (2): suppose (1) holds and (2) fails. Choose $z_0\in A$, with $z_0=y$ if $y\in A$. Since (2) fails there is $a>z_0$ with $xya$ false; by (1) there is $b>a$ with $xyb$; since (2) fails there is $c>b$ with $xyc$ false. The points $a,b,c$ differ from $y$. By (T3), from $xyb$ and the point $a$: $xya$ or $bya$; so $bya$. In the same way $byc$. So $y$ lies in $(a,b)$ and in $(b,c)$. But $a<b<c$ gives $abc$, and then $[a,b]\cap[b,c]=\{b\}$ (Lemma 3.2 of the first lesson). $\blacksquare$

Definition 2.2. Let $(A,<)$ be a directed arc. If $A$ has a maximum $a$, then $\operatorname{flow}(A)$ is the flow $\bar a$ towards $a$. If $A$ is endless, $\operatorname{flow}(A)$ is the binary relation, written $xy.A$, that holds when the two equivalent conditions of Lemma 2.1 hold.

Lemma 2.3. Cofinal directed arcs have the same flow.

Proof. Let $C$ be a common final segment of $A$ and $B$ on which the orders agree. If $A$ has a maximum $a$, then $a\in C$, and $a$ is the maximum of $B$: an element $b\in B$ above $a$ would lie in $C\subseteq A$ and be above $a$ there. So both flows are $\bar a$. The same holds if $B$ has a maximum. Let $A$ and $B$ be endless; it suffices to compare $A$ with $C$. If $xy.A$ holds with $z$ as in condition 2, replace $z$ by a larger element of $C$; condition 2 then holds for $C$. If $xy.C$ holds with $z\in C$ as in condition 2, the elements of $A$ above $z$ lie in $C$, so condition 2 holds for $A$. $\blacksquare$

Proposition 2.4. Let $(A,<)$ be a directed arc.

If $x,y\in A$, then $xy.A$ if and only if $x<y$.
If $x,y\notin A$, then $xy.A$ if and only if $xyz$ for some $z\in A$, and if and only if $xyz$ for all $z\in A$.
$\operatorname{flow}(A)$ is a flow on $T$.
If $x\in A$ and $y\in T$, then $xy.A$ if and only if $y\in A$ and $x<y$.

Proof. (1) If $A$ has the maximum $a$: $y\in(x,a]$ means $x<y\le a$, by the description of intervals in $A$. Let $A$ be endless. If $x<y$, then $xyw$ for all $w>y$, which is condition 2. If $xy.A$, condition 1 with $z=y$ gives $w>y$ with $xyw$; then $y$ lies between $x$ and $w$ in the order and $y<w$, so $x<y$.

(2) Let $x,y\notin A$ and $z,w\in A$. Then $zyw$ fails, because $A$ is full and $y\notin A$. So if $xyz$, then (T3) with the point $w\ne y$ gives $xyw$. Hence $xyz$ holds for some $z\in A$ if and only if it holds for all. If $A$ has the maximum $a$, then $xy.\bar a$ means $xya$, since $y\ne a$. If $A$ is endless, condition 2 of Lemma 2.1 holds if and only if $xyw$ for all $w\in A$.

(3) For $A$ with a maximum this is Lemma 2.1 of the lesson on flows. Let $A$ be endless and $x,y,z\in T$. The set of $w\in A$ that are larger than those of $x,y,z$ that lie in $A$ is a final segment $A'$ of $A$ containing none of the three points, and $\operatorname{flow}(A')=\operatorname{flow}(A)$ by Lemma 2.3. Fix $a\in A'$. By (2), for $u,v\in\{x,y,z\}$: $uv.A'$ if and only if $uva$, that is, if and only if $uv.\bar a$. Each of the axioms (F1), (F2), (F3) for the points $x,y,z$ involves only such pairs and the betweenness of $T$, and $\bar a$ is a flow. So the axioms hold for $\operatorname{flow}(A)$.

(4) If $A$ has the maximum $a$: $y\in(x,a]$ if and only if $y\in A$ and $x<y$. Let $A$ be endless. If $xy.A$, condition 1 gives $w\in A$ with $xyw$, so $y\in[x,w]\subseteq A$, and $x<y$ by (1). The converse is (1). $\blacksquare$

So the flow of a directed arc flows along the arc in its direction, and from outside it flows towards the arc.

Lemma 2.5. Let $p=\operatorname{flow}(A)$ for a directed arc $(A,<)$, and consider the pretree $T\sqcup\{p\}$ of Proposition 3.1 of the lesson on flows. Then $A\cup\{p\}$ is linear, and for $w\in A$ and $y\in T$: $w\,y\,p$ holds if and only if $y\in A$ and $w<y$.

Proof. The second statement is Proposition 2.4(4), since $w\,y\,p$ means $wy.A$. For $u<v$ in $A$ it gives $u\,v\,p$; with the linearity of $A$ this shows that $A\cup\{p\}$ is linear. $\blacksquare$

Lemma 2.6. Let $(A,<)$ and $(B,<')$ be directed arcs with a common point and with $\operatorname{flow}(A)=\operatorname{flow}(B)$. Then they are cofinal.

Proof. Let $x\in A\cap B$, and let $C$ be the set of all $y\in T$ with $xy.A$. By Proposition 2.4(4), $C=\{y\in A:x<y\}$, and also $C=\{y\in B:x<'y\}$. If $C$ is empty, $x$ is the maximum of both, and $\{x\}$ is a common final segment. Otherwise $C$ is a final segment of both. The orders agree on $C$: for $y,y'\in C$, $y<y'$ holds if and only if $x\,y\,y'$, and the same is true for $<'$. $\blacksquare$

3. Full subsets

A subset $S\subseteq T$ is a pretree, and an arc of $T$ that lies in $S$ is an arc of $S$ when $S$ is full. For a directed arc $B$ of $S$, $\operatorname{flow}_S(B)$ denotes its flow as a relation on $S$, and $\operatorname{flow}_T$ the flow on $T$.

Lemma 3.1. Let $S\subseteq T$ be full and $(A,<)$ a directed arc of $T$ that meets $S$. Then $B=A\cap S$, with the restricted order, is a directed arc. If $B$ is endless, then $\operatorname{flow}_T(A)\,|\,S=\operatorname{flow}_S(B).$

Proof. $B$ is nonempty, linear, and full as an intersection of full sets.

Let $B$ be endless, $p=\operatorname{flow}_T(A)$, and $x,y\in S$. In $T\sqcup\{p\}$, $xy.A$ means $x\,y\,p$. Call $w\in B$ late if $w>y$ in case $y\in B$; if $y\notin B$, every $w\in B$ is late. Since $B$ is endless, above every element of $B$ there is a late one. For late $w$, the relation $w\,y\,p$ fails: by Lemma 2.5 it would require $y\in A$ and $w<y$; but $y\in A$ gives $y\in A\cap S=B$, and then $w>y$.

If $x\,y\,p$, then for every late $w$, (T3) with the point $w\ne y$ gives $xyw$ or $p\,y\,w$; so $xyw$. The late elements contain all elements of $B$ above some element, so condition 2 of Lemma 2.1 holds for $B$: $xy.B$.

If $xy.B$, choose a late $w$ with $xyw$, by condition 1. By (T3) in $T\sqcup\{p\}$ with the point $p$: $x\,y\,p$ or $w\,y\,p$. The second fails. So $xy.A$. $\blacksquare$

4. Suprema and completeness

Intervals as directed arcs. For $x\ne b$ the interval $[x,b)$ is an arc, and we direct it by the direction of $[x,b]$ in which $x$ is the least point. For $y\in[x,b)$ the set $[y,b)$ is the final segment $\{w\in[x,b):w\ge y\}$, with the same direction (Corollary 5.3 of the first lesson).

Lemma 4.1. Let $(A,<)$ be a directed arc, $x\in A$ and $b\in T$, $b\ne x$. If $[x,b)$ is cofinal with $A$, then $[x,b)$ is a final segment of $A$, with the same order.

Proof. Let $C$ be a common final segment on which the orders agree, and $y\in C$. Then $[y,b)$ is a final segment of $[x,b)$ contained in $C$, so it is a final segment of $C$ and hence of $A$. Now $[x,b)\subseteq[x,y]\cup[y,b)$ by Lemma 3.1 of the first lesson, and $[x,y]\subseteq A$ because $A$ is full. So $[x,b)$ is an arc inside $A$ containing a final segment of $A$; by Lemma 1.1(1) it is a final segment of $A$, and the orders agree on it by the argument of Lemma 1.1(2). $\blacksquare$

Lemma 4.2. Let $(A,<)$ be an endless directed arc, $x,y\in A$ and $b\in T$. Then $[x,b)$ is a final segment of $A$ with the order of $A$ if and only if $[y,b)$ is.

Proof. Let $x<y$. If $[x,b)$ is such a final segment, it contains $y$, and $[y,b)$ is a final segment of $[x,b)$, hence of $A$.

Conversely let $[y,b)$ be such a final segment. Since $A$ is endless there is $z\in A$ with $z>y$; then $z\in(y,b)$, so $yzb$, and $byz$ fails by (T2). From $x<y<z$ we have $xyz$, and (T3) with the point $b\ne y$ gives $xyb$ or $zyb$; so $xyb$. Hence $y\in[x,b)$, and $[y,b)$ is a final segment of $[x,b)$ and of $A$, with the same orders: $[x,b)$ is cofinal with $A$. By Lemma 4.1 it is a final segment of $A$. $\blacksquare$

Definition. A point $b\in T$ is a supremum of a directed arc $(A,<)$ if $b$ is the maximum of $A$, or if $A$ is endless and for some $x\in A$ the interval $[x,b)$ is a final segment of $A$ with the order of $A$. By Lemma 4.2 this then holds for every $x\in A$.

So a supremum of an endless arc is a point $b\notin A$ such that $A$ runs up to $b$: from each of its points $x$ on, $A$ is the interval $[x,b)$.

Theorem 4.3. A pretree is complete if and only if every directed arc has a supremum.

Proof. Let $T$ be complete and $(A,<)$ a directed arc. If $A$ has a maximum, that is a supremum. Let $A$ be endless. Then $A$ has more than one point, and it is an interval with ends $p\ne q$. So $A\subseteq[p,q]$, and $<$ is the restriction of one of the two directions of $[p,q]$; name the ends so that it is the one with $p<q$. Since $A$ has no maximum, $q\notin A$. So $A$ is $[p,q)$ or $(p,q)$, and for $x\in A$ the set $\{w\in A:w\ge x\}$ is $\{w\in[p,q]:x\le w<q\}=[x,q)$. Hence $q$ is a supremum.

Let every directed arc have a supremum and let $A$ be an arc. If $A$ is a single point $x$, it is the interval $[x,x]$. Otherwise choose a direction $<$ (Proposition 5.2 of the first lesson), let $b$ be a supremum of $(A,<)$ and $a$ a supremum of $A$ with the reverse direction.

If $A$ has a least point and a greatest point, these are $a$ and $b$, and $A=[a,b]$: every other point lies between them, and $A$ is full.

If $A$ has a least point $a$ and no greatest point, then $[a,b)$ is a final segment of $A$ containing the least point, so $A=[a,b)$. In the same way, if $A$ has a greatest point $b$ and no least point, $A=(a,b]$.

Let $A$ have neither. For $x\in A$ we have $\{w\in A:w\ge x\}=[x,b)$ and $\{w\in A:w\le x\}=(a,x]$. Choose $u<x<w$ in $A$. Then $aux$, $uxw$ and $xwb$. By Lemma 2.3(1) of the first lesson, $aux$ and $uxw$ give $axw$; then $axw$ and $xwb$ give $axb$. So $x\in(a,b)$, and by Lemma 3.2 of the first lesson $(a,b)=(a,x]\cup[x,b)=A$. $\blacksquare$

5. Exercises

5.1. In the pretree $\mathbb Q$, find a directed arc without a supremum, and describe its flow.

5.2. Let $T$ be the complete dense pretree of Proposition 3.2 of Dense, complete and median pretrees. Find a supremum of the arc formed by the points of leg 2, directed towards the centre, and show that the arc formed by the points of legs 2 and 3, in either direction, has its end $y$ or $z$ as supremum.

5.3. Give two directed arcs that have the same underlying set and are not cofinal, and compare their flows.

5.4. Show that in Lemma 2.6 the assumption that the arcs have a common point cannot be dropped.

5.5. Let $(A,<)$ be an endless directed arc with a supremum $b$. Show that $\operatorname{flow}(A)$ and $\bar b$ agree on all pairs $(x,y)$ with $y\ne b$, and that $xb.A$ fails for $x\in A$. So $\operatorname{flow}(A)\ne\bar b$.

6. Solutions

5.1. Let $A$ be the set of rational numbers $x$ with $x^2<2$, with the usual order. It is endless. A supremum would be a rational $b$ such that for $x\in A$ the set of $w\in A$ with $w\ge x$ is $[x,b)$; this set consists of the rationals in $[x,\sqrt2)$, so $b$ would be $\sqrt2$. The flow: $xy.A$ holds exactly when $y$ lies strictly between $x$ and $\sqrt2$ on the real line. Indeed, by Lemma 2.1, $xy.A$ means that $y$ lies strictly between $x$ and $w$ for all $w\in A$ close enough to $\sqrt2$; since $y$ is rational, this is the stated condition.

5.2. The points of leg 2 form the interval $[y,q)$ with $y=(2,1)$ and $q=(1,\frac12)$ (Proposition 3.2 there); directed towards the centre it is $[y,q)$ with $y$ least, and $q$ is a supremum by definition. The points of legs 2 and 3 form the interval $[y,z]$, which has a greatest point in either direction.

5.3. The real line with the usual order and with the reverse order. A common final segment on which the orders agree would be a set unbounded above and below on which $<$ and $>$ agree. The flows are $x<y$ and $y<x$.

5.4. Let $T$ be the real line without the point $0$, a pretree as a subset of the line. Let $A$ be the set of negative numbers with the usual order and $B$ the set of positive numbers with the reverse order. Both are endless directed arcs of $T$, and they are disjoint, so they are not cofinal. As in the solution of 5.1, $xy.A$ holds exactly when $y$ lies strictly between $x$ and $0$, and the same is true for $B$. So the two flows are equal.

5.5. Recall that for $w\in A$ the set of elements of $A$ above $w$ is $[w,b)$; in particular $b\notin A$, and $(w,b)\subseteq A$. Let $y\ne b$.

$x\in A$. By Proposition 2.4(4), $xy.A$ means $y\in A$ and $x<y$, that is, $y\in(x,b)$; for $y\ne b$ this is $xy.\bar b$.

$x\notin A$, $y\notin A$. By Proposition 2.4(2), $xy.A$ means $xyw$ for all $w\in A$, and $xy.\bar b$ means $xyb$. If $xyb$ and $w\in A$, then (T3) gives $xyw$ or $byw$; the second would put $y$ into $(w,b)\subseteq A$. If $xyw$, then (T3) with the point $b$ gives $xyb$ or $wyb$, and again the second is impossible.

$x\notin A$, $y\in A$. For $w\in A$ with $w>y$ we have $ywb$, so $wyb$ fails by (T2). If $xyb$, then (T3) with the point $w$ gives $xyw$ or $byw$; so $xyw$ for all $w>y$, which is condition 2 of Lemma 2.1. If $xy.A$, choose $w>y$ with $xyw$; (T3) with the point $b$ gives $xyb$ or $wyb$; so $xyb$.

Finally, for $x\in A$ the relation $xb.A$ fails by Proposition 2.4(4), because $b\notin A$, while $xb.\bar b$ holds.

References

[Bowditch] B. H. Bowditch, Treelike structures arising from continua and convergence groups, Memoirs of the American Mathematical Society 662 (1999), Section 3. Author's version: http://homepages.warwick.ac.uk/~masgak/papers/bhb-treelike.pdf