# Convex hull lattices point generated

Carles Cardó

###### Abstract.

The simplest way to generate a lattice of convex sets is to consider an
initial set of points and draw segments, triangles, and any convex hull from
it, then intersect them to obtain new points, and so forth. The result is an
infinite lattice for most sets, while only a few initial sets of points
perform a finite lattice. By giving an adequate notion of the configuration of
points, we identify which sets in the plane define a finite convex hull
lattice: four regular families and one sporadic configuration. We explore
configurations in the space and higher dimensions.

MSC(2010): 52A10, 06B99, 52B99.
Keywords: Convex hull lattices point generated, Convex hull lattice, Discrete
geometry, Convex geometry, Planar configuration.
$*$ Under review at _Journal of Discrete & Computational Geometry_.

##  1\. Introduction

The class of lattices of convex sets in $\mathbb{R}^{d}$ is enormous. However,
a straightforward way to get one of such lattices is to consider an initial
finite set of points, draw segments, triangles and any convex hull, and then
intersect them to obtain new points. For most of the initial sets, the
generated lattice is infinite. For example, take the vertices of a regular
pentagon. By drawing all possible segments, we obtain a pentagon inside the
initial pentagon again, and by repeating the operation, we get a third
pentagon and so forth. The lattice is finite for only a few configurations of
points. We aim to identify such sets of points, first in the plane and later
to gain insight into higher dimensions. As far as we know, the literature does
not refer to that simple problem close to recreative mathematics, [3, 10, 17].
Although the answer is simple in the case of the plane, we will need to study
a combinatorial version of the convex lattices and perform an adequate point
configuration notion.

Let us define the main object of interest for this article. Recall that the
_convex hull_ of a set $X\subseteq\mathbb{R}^{d}$, denoted by
$\operatorname{ch}(X)$ is the union of all the possible segments with extremes
in $X$. A set $X$ is said to be a _convex_ if $\operatorname{ch}(X)=X$. The
symbol $\operatorname{ch}$ is a _closure operator_ [8, 4], and hence the set
of convex sets in $\mathbb{R}^{d}$, denoted by $\mathscr{K}^{d}$ is a lattice
with operations $X\vee Y=\operatorname{ch}(X\cup Y)$ and $X\wedge Y=X\cap Y$.
We will call to abbreviate _ch-lattices_ the sublattices of $\mathscr{K}^{d}$.

######  Definition 1.1.

Given a set $X\subseteq\mathbb{R}^{d}$, we denote by $\mathscr{K}^{d}(X)$ the
least sublattice of $\mathscr{K}^{d}$ containing the set $\\{\\{x\\}\mid x\in
X\\}$. We call those lattices _point generated_ , and we say that they are
_finitely point generated_ when $X$ is finite.

Since the intersection and the convex hull of a pair of bounded polytopes are
also bounded, by the principle of structural induction for algebras, elements
of $\mathscr{K}^{d}(X)$ are bounded polytopes.

Fix the notation and elementary concepts. When necessary, we will abbreviate
point singletons as $a=\\{a\\}$ and the segments as
$ab=\\{a\\}\vee\\{b\\}=a\vee b$. Given three non-collinear points $a,b,c$ in
the plane, $\triangle abc$ denotes the triangle with vertices a,b,c, that is,
$\triangle abc=a\vee b\vee c$. Given a lattice $\mathcal{L}=(L,\vee,\wedge)$
with the order $\leq$, we said that $y\in L$ _covers_ $x\in L$ if $x<y$ and
for any other $z\in L$ or $x=z$ or $z=y$. If the lattice has a bottom element
$0$, we say that $a\in L$ is an _atom_ if $a$ covers $0$. We denote the set of
atoms by $\operatorname{Atom}(\mathcal{L})$. A lattice with a bottom element
$0$ is _atomic_ if, for each non-zero element $x$, there exists an atom $a$
such that $a\leq x$. A lattice is atomistic if it is atomic and every element
is a join of some finite set of atoms. For other elementary concepts on
lattice and order theory, we will follow [8] or [4].

We write $\overline{abc}$ to mean that the points $a$,$b$, and $c$ are
collinear. A subset $S\subseteq X$ is said _maximal collinear in $X$_ if
$|S|\geq 3$, all the points in $S$ are collinear and for any $x\in X\setminus
S$, the points in $S\cup\\{x\\}$ are not collinear. If we want to note that
$b$ is between $a$ and $c$, we will write $a{-}b{-}c$, provided $a,b,c$ are
different and collinear. A set of points in the plane is said to be _in
general position_ if it does not contain three collinear points. A set of
points $X$ is said to be in _convex position_ if for any $x\in X$,
$x\not\in\operatorname{ch}(X\setminus\\{x\\})$. If $X$ is in convex position,
it is in general position. $H^{+}(ab,c)$ denotes the unique open half-plane
delimited by the line that passes by $a$ and $b$ and contains $c$. We define
the set $\Theta(a,b,c)=H^{+}(ab,c)\cap H^{+}(ac,b)\cap H^{-}(bc,a)$. If
$a,b,c$ are not collinear and $d\in\Theta(a,b,c)$, then $ad\wedge
bc\not=\emptyset$. $\operatorname{int}(X)$ denotes the interior of a set
$X\subseteq\mathbb{R}^{d}$ and $\partial(X)$ denotes the boundary in the usual
Euclidean topological sense. $\operatorname{ext}(X)$ denotes the set of
extreme points of $X$. For other elementary terms on convex geometry, we will
follow [5] or [14] for topology.

##  2\. Configurations and morphisms of configurations

Any set of a fixed number of collinear points defines the same isomorphic ch-
lattice point generated, but this is just a particularity of the one-
dimensional space. The goal is to define an equivalence of sets of points that
reflects the algebraic properties of its ch-lattice. The literature on
discrete geometry offers multiple non-equivalent meanings of point
configuration, which are inadequate here. A natural attempt is to consider
that two configurations $X,Y$ are equivalent if a mapping
$f:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$ preserving the convexity such
that $f(X)=Y$ exists. These mappings, so-called _rational affine_ , have been
well studied [1]. However, these mappings are too rigid for our proposes. The
solution is to consider the relative convex hull [2].

######  Definition 2.1.

By a _configuration of dimension $d$_, we mean simply a set of points in
$\mathbb{R}^{d}$, for some $d\geq 0$, such that its affine hull has dimension
$d$. Given a subset $A\subseteq X$, The _convex hull of $A$ relative to a
configuration $X$_ is the set

$\operatorname{Rch}_{X}(A)=\operatorname{ch}(A)\cap X.$

When the context gives the configuration $X$, we will abbreviate
$\operatorname{Rch}(A)=\operatorname{Rch}_{X}(A)$. We say that a subset
$A\subseteq X$ is relatively convex (in the context $X$) if
$\operatorname{Rch}(A)=A$. Given two configurations $X,Y$, a mapping
$f:X\longrightarrow Y$ is said a _morphism of configurations_ if for any
subset $A\subseteq X$

$f(\operatorname{Rch}_{X}(A))=\operatorname{Rch}_{Y}(f(A)).$

An _isomorphism_ $f$ is a bijective morphism. We say that $X$ and $Y$ are
_equivalent_ , denoted by $X\equiv Y$, if there is an isomorphism between
them.

The composition of morphisms is a morphism of configurations, and the inverse
of an isomorphism is also an isomorphism. The terms monomorphism, epimorphism,
and automorphism have the usual algebraic sense. The set of automorphisms is
denoted by $\operatorname{Aut}(X)$. We call an _abstract configuration_ an
equivalence class of $\equiv$, but, as it is usual in algebra, by a mild abuse
of terminology, we will omit the adjective “abstract”. Similarly, when a
monomorphism exists $X\longrightarrow Y$, we will say that $X$ is a
_subconfiguration_ of $Y$.

######  Example 2.2.

Consider a triangle $\triangle abc$ with a point $d$ in the interior,
$T=\\{a,b,c,d\\}$, and consider a square with vertices $S$. $T$ and $S$ are
not equivalent, because $d\in\operatorname{ch}(\\{a,b,c\\})$, but no triangle
in $S$ contains a point.

Now consider three collinear points $C=\\{x,y,z\\}$, $x{-}y{-}z$. There is an
epimorphism of configurations $f:T\longrightarrow C$, given by $f(a)=x$,
$f(b)=f(c)=z$, $f(d)=y$.

Contrary to our intuition, the dimension of a configuration, defined as the
dimension of its affine hull, is not invariant under the equivalence of
configurations. It turns out that a tetrahedron is equivalent to a convex
quadrilateral. Moreover, any set of $n$ points in convex position in
$\mathbb{R}^{d}$ is equivalent to the regular $n$-agon in $\mathbb{R}^{2}$.
Therefore, we need some invariants to discriminate non-equivalent
configurations. Equivalent configurations have the same size. If $f$ is a
morphism of configurations, then it preserves betweenness. If $a{-}b{-}c$ in
$X$, then $b\in\operatorname{ch}(\\{a,c\\})$, and $f(a),f(b),f(c)$ are
collinear. Therefore, $f(b)\in\operatorname{ch}(\\{f(a),f(c)\\}$ and
$f(a){-}f(b){-}f(c)$. Two useful invariants are the following. First, define
the set of _extreme points relative to $X$_ as

$\operatorname{Rext}X=\\{x\in X\mid X\setminus\\{x\\}\mbox{ is relatively
convex}\,\\}.$

It is easily seen that if $X\equiv Y$,
$\operatorname{Rext}X\equiv\operatorname{Rext}Y$. The following invariant is,
in fact, a family of invariants. Let $Z$ be a class of equivalence of
configurations. We count subconfigurations $Z$ in $X$:

$\\#_{Z}(X)=\mbox{ the number of subconfigurations equivalent to $Z$ in $X$}.$

The first obvious form of classifying finite configurations is by the number
of points. Then, we can group them by their sets of relative extreme points.
When two configurations have the same set of relative extreme points, we can
discern by $\\#_{z}(\cdot)$ for some adequate $Z$.

If $f:X\longrightarrow Y$ is an isomorphism of configurations and $x\in X$,
then $X\setminus\\{x\\}\equiv Y\setminus\\{f(x)\\}$. So, configurations of
order $n+1$ can be obtained from configurations of order $n$ by adding a
point. According to the location of the point, we obtain different
configurations. When the number of points grows, listing all configurations is
difficult. Fortunately, five points will suffice for the next sections to
prove the main theorems.

There is only one configuration with sizes one and two, $L_{1}$ and $L_{2}$.
See Figure 4.1 for the notations. There are two configurations of size three,
$L_{3}$ and $T_{2}$. They are not equivalent because
$\operatorname{Rext}L_{3}$ is a segment but $\operatorname{Rext}T_{2}$ is a
triangle. We obtain four-point configurations by adding a point in
configurations of size three: $L_{4},T_{3},I_{0,2}$, and $I_{1,1}$. Except by
$T_{3}$ and $I_{0,2}$, we know that all the pairs of configurations are not
equivalent because of their relative extreme points. However,
$\\#_{T_{2}}(T_{3})=3$ and $\\#_{T_{2}}(I_{0,2})=4$. In the same manner, we
obtain the twelve five-point configurations. Using the commented invariants,
the reader can check that these are all the configurations.

The invariant $\\#_{Z}(\cdot)$ satisfies the equalities:

$\\#_{L_{3}}(X)+\\#_{T_{2}}(X)={|X|\choose 3},$

provided $|X|\geq 3$, and:

$\\#_{L_{4}}(X)+\\#_{T_{3}}(X)+\\#_{I_{0,2}}(X)+\\#_{I_{1,1}}(X)={|X|\choose
4},$

if $|x|\geq 4$. More in general, when $|X|\geq k$,

$\sum_{Z}\\#_{Z}(X)={|X|\choose k},$

where the sum runs over all the configurations $Z$ of size $k$.

\begin{overpic}[scale={0.15}]{Fig1} \put(2.0,64.0){$L_{1}$}
\put(9.0,64.0){$L_{2}$} \put(20.0,64.0){$L_{3}$} \put(37.0,64.0){$L_{4}$}
\put(75.0,64.0){$L_{5}$} \par\put(17.0,53.0){$T_{2}$} \put(33.0,53.0){$T_{3}$}
\put(66.0,53.0){$T_{4}$} \put(78.0,53.0){$V_{5}$} \put(92.0,53.0){$D_{0,2}$}
\par\par\put(35.5,38.5){$I_{0,2}$} \put(63.0,38.0){$R$}
\put(78.0,38.0){$I_{0,3}$} \put(93.0,38.0){$G$} \par\put(36.0,24.0){$I_{1,1}$}
\par\par\put(58.5,24.0){$R^{\prime}$} \put(69.0,24.0){$D_{1,1}$}
\put(80.0,24.0){$I_{1,2}$} \put(92.0,24.0){$G^{\prime}$}
\par\put(70.5,8.0){$P_{5}$} \par\end{overpic} Figure 1. Planar configurations
for 1,2,3,4, and 5 points and its designations. See Definition 4.1 for
designations $L_{n}$, $T_{n}$, $D_{p,q}$, and $I_{p,q}$. Shadows indicate the
convex hulls. Lines either indicate collinear points or a line where a point
cannot lay to preserve the configuration.

##  3\. Complete and finitely completable configurations

######  Definition 3.1.

Given a configuration $X$ of dimension $d$, its _completion_ is the set of
singletons points of $\mathscr{K}^{d}(X)$, and it will be denoted by
$\overline{X}=\\{x\in\mathbb{R}^{d}\mid\\{x\\}\in\mathscr{K}^{d}(X)\\}$. The
set $X$ is said _complete_ if $\overline{X}=X$. $X$ is said _finitely
completable_ if $\overline{X}$ is finite.

Trivially, if $X$ is finite and complete, it is finitely completable. Notice
that $\overline{X}$ is always complete and that
$\mathscr{K}^{d}(X)=\mathscr{K}^{d}(\overline{X})$. Notice also that if
$X\subseteq Y$, then $\mathscr{K}^{d}(X)\subseteq\mathscr{K}^{d}(Y)$. Then, if
$X\subseteq Y$, and $Y$ is finitely completable, $X$ is.

######  Example 3.2.

If we add the centre to a convex quadrilateral configuration $Q$, we get a
non-equivalent configuration $\overline{Q}\equiv D_{1,1}$. We have that
$\mathscr{K}^{2}(Q)=\mathscr{K}^{2}(\overline{Q})$. Thus, $Q$ is finitely
completable.

######  Definition 3.3.

Given a configuration $X$, $\operatorname{Rch}_{X}(\cdot)$ is a closure
operator, whereby it makes sense to define the lattice
$\mathscr{R}^{d}(X)=\\{\operatorname{Rch}_{X}(Y)\mid Y\subseteq X\\}$ with
operations $A\vee_{R}B=\operatorname{Rch}_{X}(A\cup B)$ and
$A\wedge_{R}B=A\cap B$.

We will prove that if $X$ is complete, $\mathscr{R}^{d}(X)$ is the
combinatoric version of $\mathscr{K}^{d}(X)$, or more precisely, they are
isomorphic. We need first a lemma.

######  Lemma 3.4.

Let $X$ be complete. For any $A\in\mathscr{K}^{d}(X)$,
$\operatorname{ext}(A)\subseteq X$.

###### Proof.

To establish the result, we need to know the extreme points of $A\wedge B$ and
$A\vee B$ in terms of the extreme points of $A$ and $B$. For the join
operation, we have a direct relation: $\operatorname{ext}(A\vee
B)\subseteq\operatorname{ext}(A)\cup\operatorname{ext}(B)$. To calculate the
intersection of two polytopes, there are several algorithms from the beginning
of the computational geometry; see [16, 13, 6]. First, we can triangulate the
polytopes into simplices and calculate the intersection of each pair of
simplices. Let $S_{A}$ be a simplex of the triangulation of $A$ and $S_{B}$ of
$B$. Then, to calculate $S_{A}\wedge S_{B}$ it suffices calculate $\partial
S_{A}\cap\partial S_{B}$, [16, p. 271]. If $\partial S_{A}\cap\partial
S_{B}=\emptyset$, either one of the simplices is inside the other, or the
intersection is empty. In both cases $\operatorname{ext}(S_{A}\wedge
S_{B})\subseteq\operatorname{ext}(S_{A})\cup\operatorname{ext}(S_{B})$. If the
intersection is not empty, then $\operatorname{ext}(S_{A}\wedge
S_{B})\subseteq\operatorname{ext}(\partial S_{A}\wedge\partial S_{B})$. The
boundary of a convex polytope $P$ is conformed by its maximal facets, denoted
by $\mathrm{maxF}(P)$. Then, we must only intersect the maximal facets of
$S_{A}$ with the maximal facets of $S_{B}$. However, even we can improve it.
To find the extreme points, we only need to intersect the maximal facets of
$S_{A}$ with edges of $S_{B}$ and reversely. That is,

$\displaystyle\operatorname{ext}(\partial S_{A}\wedge\partial S_{B})$
$\displaystyle\subseteq\left(\bigcup_{\begin{subarray}{c}F\in\mathrm{maxF}(S_{A})\\\
x,y\,\in\,\operatorname{ext}(S_{B})\end{subarray}}xy\wedge
F\right)\cup\left(\bigcup_{\begin{subarray}{c}F\in\mathrm{maxF}(S_{B})\\\
x,y\,\in\,\operatorname{ext}(S_{A})\end{subarray}}xy\wedge F\right).$

Look at the first pair of brackets. Since $\mathscr{K}^{d}(X)$ is generated by
joins and meets of points in $X$, if we suppose that
$\operatorname{ext}(S_{A}),\operatorname{ext}(S_{B})\subseteq X$, then for
each $F\in\mathrm{maxF}(S_{A})$, $F\in\mathscr{K}^{d}(X)$, and for each
$x,y\in\operatorname{ext}(S_{B})$, $xy\in\mathscr{K}^{d}(X)$. Since $xy\wedge
F$ is a point (and therefore, an atom) and $X$ is complete, $xy\wedge F\in X$.
The same holds for the second pair of brackets. Therefore, for each pair of
simplices, $\operatorname{ext}(S_{A}\wedge S_{B})\subseteq X$. We can sum up
all in a single statement:

$\operatorname{ext}(A),\operatorname{ext}(B)\subseteq
X\implies\operatorname{ext}(A\vee B),\,\operatorname{ext}(A\wedge B)\subseteq
X.$

By the structural induction principle, the result follows. ∎

######  Proposition 3.5.

If $X$ is complete, $\mathscr{K}^{d}(X)\cong\mathscr{R}^{d}(X)$.

###### Proof.

Let $\Phi:\mathscr{K}^{d}(X)\longrightarrow\mathscr{R}^{d}(X)$, $\Phi(A)=A\cap
X$. Let us check that $\Phi$ is a bijection and its inverse is
$\operatorname{ch}(\cdot)$. Clearly, if $A\in\mathscr{R}^{d}(X)$, then $A$ is
relatively convex, $\operatorname{Rch}(A)=A$, and then
$\Phi(\operatorname{ch}(A))=\operatorname{ch}(A)\cap
X=\operatorname{Rch}(A)=A$. Let $B\in\mathscr{K}^{d}(X)$. By Lemma 3.4 above,
$\operatorname{ext}(B)\subseteq X$. Clearly $\operatorname{ext}(B)\subseteq
B$, and then $\operatorname{ext}(B)\subseteq B\cap X$. Since
$\operatorname{ch}$ is monotone,
$\operatorname{ch}(\operatorname{ext}B)\subseteq\operatorname{ch}(B\cap X)$,
that is $B\subseteq\operatorname{ch}(B\cap X)$. Next, consider the other
direction of the last inclusion. $B\cap X\subseteq B$. Therefore,
$\operatorname{ch}(B\cap X)\subseteq\operatorname{ch}(B)$. Since $B$ is a
convex set, $\operatorname{ch}(B)=B$, and then $\operatorname{ch}(B\cap
X)\subseteq B$. Hence, $\operatorname{ch}(\Phi(B))=\operatorname{ch}(B\cap
X)=B$. Thus, $\Phi\circ\operatorname{ch}$ is the identity on
$\mathscr{R}^{d}(X)$ and $\operatorname{ch}\circ\,\Phi$ is the identity on
$\mathscr{K}^{d}(X)$.

Next, see that $\Phi$ is a morphism of lattices. For the meet operation, it is
trivial:

$\displaystyle\Phi(A\wedge B)$ $\displaystyle=\Phi(A\cap B)=A\cap B\cap X$
$\displaystyle=A\cap X\cap B\cap X=\Phi(A)\cap\Phi(B)$
$\displaystyle=\Phi(A)\wedge_{R}\Phi(B).$

For the join operation, $\Phi(A\vee B)=\operatorname{ch}(A\cup B)\cap X$.
Notice that we cannot write $\operatorname{ch}(A\cup B)\cap
X=\operatorname{Rch}(A\cup B)$, because $A$ and $B$ are not subsets of $X$,
but of $\mathbb{R}^{d}$. However, using $\operatorname{ch}(B\cap X)=B$, proved
in the above paragraph,

$\displaystyle\Phi(A\vee B)$ $\displaystyle=\operatorname{ch}(A\cup B)\cap X$
$\displaystyle=\operatorname{ch}((A\cup B)\cap X)\cap X$
$\displaystyle=\operatorname{ch}((A\cap X)\cup(B\cap X))\cap X$
$\displaystyle=\operatorname{Rch}((A\cap X)\cup(B\cap X))$
$\displaystyle=\operatorname{Rch}(A\cap X)\vee_{R}\operatorname{Rch}(B\cap X)$
$\displaystyle=(\operatorname{ch}(A\cap X)\cap
X)\vee_{R}(\operatorname{ch}(B\cap X)\cap X)$ $\displaystyle=(A\cap
X)\vee_{R}(B\cap X)=\Phi(A)\vee_{R}\Phi(B).$

∎

######  Proposition 3.6.

We have the following properties.

  1. (i)

$\mathscr{K}^{d}(X)$ is atomistic.

  2. (ii)

$\overline{X}=\bigcup\operatorname{Atom}\mathscr{K}^{d}(X)$.

  3. (iii)

The facial lattice of any polytope in $\mathscr{K}^{d}(X)$ is a sublattice.

  4. (iv)

$\mathscr{K}^{d}(X)$ is finite iff $X$ is finitely completable.

###### Proof.

$\mathscr{K}^{d}(X)=\mathscr{K}^{d}(\overline{X})\cong\mathscr{R}^{d}(\overline{X})$.
Since $\mathscr{R}^{d}(\overline{X})$ is by definition atomistic, by
Proposition 3.5, $\mathscr{K}^{d}(X)$ is, and then
$\bigcup\operatorname{Atom}\mathscr{K}^{d}(X)=\bigcup\operatorname{Atom}\mathscr{R}^{d}(\overline{X})=\overline{X}$.
That proves (i) and (ii). (iii) follows from (i) and (ii). For (iv), the
direction $(\Rightarrow)$ follows from the fact that if $\mathscr{K}^{d}(X)$
is finite, then $\mathscr{R}^{d}(\overline{X})$ is finite and then
$\overline{X}$ is finite. $(\Leftarrow)$ Since $\mathscr{K}^{d}(X)$ is point
atomistic if $\overline{X}$ is finite, then any element in the lattice can be
expressed as the convex hull of a finite number of point atoms, therefore
$\mathscr{K}^{d}(X)=\mathscr{K}^{d}(\overline{X})$ must be finite. ∎

######  Definition 3.7.

Given a morphism of configurations $f:X\longrightarrow Y$, there is at most a
morphism of lattices $F:\mathscr{K}^{d}(X)\longrightarrow\mathscr{K}^{d}(Y)$,
such that $F(\\{x\\})=\\{f(x)\\}$ for any $x\in X$. That is an elementary fact
of universal algebra [4]. When such $F$ exists, we will call it the _lattice
extension of $f$_, or equivalently, we will say that $f$ is the _configuration
restriction of $F$_.

######  Theorem 3.8.

Let $X$ and $Y$ be complete configurations. We have that $X\equiv Y$ iff
$\mathscr{K}^{d}(X)\cong\mathscr{K}^{d}(Y)$.

###### Proof.

$(\Rightarrow)$ If $f:X\longrightarrow Y$ is an isomorphism of configurations,
the set extension defines an isomorphism of the relative lattices
$f:\mathscr{R}^{d}(X)\longrightarrow\mathscr{R}^{d}(Y)$. Let
$A,B\in\mathscr{R}^{d}(X)$. For the meet operation, since $f$ is injective,
$f(A\wedge B)=f(A\cap B)=f(A)\cap f(B)=f(A)\wedge f(B)$. For the join
operation $f(A\vee B)=f(\operatorname{Rch}(A\cup
B))=\operatorname{Rch}(f(A\cup B))=\operatorname{Rch}(f(A)\cup f(B))=f(A)\vee
f(B)$. If we assume that $X,Y$ are complete, then $\operatorname{ch}\circ
f\circ\Phi$ is an isomorphism from $\mathscr{K}^{d}(X)$ to
$\mathscr{K}^{d}(Y)$, where $\Phi(A)=A\cap X$, see proof of Theorem 3.5.

$(\Leftarrow)$ We prove that if
$F:\mathscr{K}^{d}(X)\longrightarrow\mathscr{K}^{d}(Y)$ is an isomorphism of
lattices, then the natural restriction $f:X\longrightarrow Y$ is an
isomorphism of configurations. An epimorphism of lattices is cover-preserving
[7]. Then,
$F(\operatorname{Atom}(\mathscr{K}^{d}(X)))\subseteq\operatorname{Atom}(\mathscr{K}^{d}(Y))$.
Since $F$ is an isomorphism,
$F^{-1}(\operatorname{Atom}(\mathscr{K}^{d}(Y)))\subseteq\operatorname{Atom}(\mathscr{K}^{d}(X))$.
Therefore,

$F(\operatorname{Atom}(\mathscr{K}^{d}(X)))=\operatorname{Atom}(\mathscr{K}^{d}(Y)).$

By Proposition 3.6(ii)
$\overline{X}=\bigcup\operatorname{Atom}\mathscr{K}^{d}(X)$ and
$\overline{Y}=\bigcup\operatorname{Atom}\mathscr{K}^{d}(Y)$, which means that
the natural restriction, defined by the relation $\\{f(x)\\}=F(\\{x\\})$, is a
bijective mapping $f:\overline{X}\longrightarrow\overline{Y}$. For any
$z\in\overline{X}$, we have that $z\in\operatorname{ch}(Z)$ iff
$\\{z\\}\subseteq\bigvee_{z^{\prime}\in Z}\\{z^{\prime}\\}$, with $Z$ finite.
Then,

$F(\\{z\\})\subseteq F\left(\bigvee_{z^{\prime}\in
Z}\\{z^{\prime}\\}\right)=\bigvee_{z^{\prime}\in
Z}F(\\{z^{\prime}\\})=\bigvee_{z^{\prime}\in
Z}\\{f(z^{\prime})\\}=\bigvee_{z^{\prime\prime}\in
f(Z)}\\{z^{\prime\prime}\\},$

which is equivalent to $f(z)\in\operatorname{ch}(f(Z))$. Therefore,
$f(\operatorname{Rch}_{X}(Z))\subseteq\operatorname{Rch}_{Y}(f(Z))$. Using the
inverse $F^{-1}$ we obtain the symmetric result
$f^{-1}(\operatorname{Rch}_{Y}(f(Z))\subseteq\operatorname{Rch}_{X}(Z)$, and
from there, the equality. ∎

##  4\. Finite ch-lattices point generated in the Euclidean plane

\begin{overpic}[scale={0.130}]{Fig2} \put(7.0,53.0){$L_{n}$}
\par\put(7.0,37.0){$T_{n}$} \put(22.0,42.0){$s$} \par\put(10.0,12.0){$S_{6}$}
\par\put(55.0,50.0){$D_{p,q}$} \put(76.0,53.5){$s$} \put(76.0,40.0){$c$}
\put(76.0,30.5){$s^{\prime}$}
\put(54.0,41.0){$\underbrace{\qquad\qquad\quad}_{p}$}
\put(81.0,41.0){$\underbrace{\qquad\qquad\quad}_{q}$}
\par\put(55.0,20.0){$I_{p,q}$} \put(76.0,24.0){$s$}
\put(76.0,0.5){$s^{\prime}$} \par\end{overpic} Figure 2. Configurations from
Definition 4.1.

######  Definition 4.1.

We define the following four families of planar configurations and an extra
configuration.

  1. (i)

The _Linear configuration_ , $L_{n}$, consisting in a set of $n>0$ collinear
points.

  2. (ii)

The _Triangular configuration_ , $T_{n}$, consisting in a set of $n>1$
collinear points $a_{1}{-}\cdots{-}a_{n}$ and an extra non-collinear point
$s$, which we will call _star point_. We will call the _axis_ the segment
$a_{1}\vee a_{n}$.

  3. (iii)

The _Diamond configuration_ , $D_{p,q}$, consisting in a set of $p+q+1$,
collinear points $a_{1}{-}\cdots{-}a_{p}{-}c{-}b_{1}{-}\cdots{-}b_{q}$, with
$(p,q)\not\in\\{(0,0),$ $(1,0),(0,1)\\}$, and two _star points_ , $s$ and
$s^{\prime}$, non-collinear with the above points such that $ss^{\prime}\wedge
r=c$, where $r=a_{1}b_{q}$ is the _axis_ of the configuration. The point $c$
is called the _centre_.

  4. (iv)

The _Subdiamond configuration_ , $I_{p,q}=D_{p,q}\setminus\\{c\\}$, where $c$
is the center of $D_{p,q}$.

  5. (v)

The _Sporadic configuration_ , $S_{6}$, consisting in six points $a,b,c$,
$a^{\prime},b^{\prime},c^{\prime}$ such that $a{-}a^{\prime}{-}c^{\prime}$,
$b{-}b^{\prime}{-}a^{\prime}$ and $c{-}c^{\prime}{-}b^{\prime}$, and no more
collinear subsets.

Figure 2 shows the configurations, the cardinalities of which are:

$|L_{n}|=n,\,\,\,\,|T_{n}|=n+1,\,\,\,\,|D_{p,q}|=p+q+3,\,\,\,\,|I_{p,q}|=p+q+2,\,\,\,\,|S_{6}|=6.$

By simply inspecting Figure 2, we see that all configurations are not
equivalent. Notice also that when $p=0$ or $q=0$, $I_{p,q}$ is complete. When
$p,q>0$, it is incomplete. These configurations suffice to characterise all
the finite planar ch-lattices. To streamline our analysis, we will exploit the
symmetries of the configurations to reduce the number of cases in the proofs
of the next propositions. In particular,
$\operatorname{Aut}(D_{1,1})\cong\operatorname{Aut}(I_{1,1})\cong\mathcal{D}_{4}$,
where $\mathcal{D}_{4}$ is the dihedral group of order $4$,
$\operatorname{Aut}(D_{p,q})\cong\operatorname{Aut}(I_{p,q})\cong\mathbb{Z}/(2)$,
for $p,q\not=1$, and $\operatorname{Aut}(S_{6})\cong\mathbb{Z}/(3)$.

######  Example 4.2.

Let us characterise some of the ch-lattices of the above configurations. Let
$\Sigma^{*}$ be the free monoid of words on the letters $\Sigma$. A word $x$
is said to be a _subword_ of $y$ (or that $x$ is a _factor word_ of $y$) if
there are words $z,t$ such that $y=zxt$. Let $W_{n}$ be the set of subwords of
the word $a_{1}\cdots a_{n}$, where $a_{1},\ldots,a_{n}$ are all different
letters of $\Sigma$. The order “be a subword” on $W_{n}$ defines a lattice
denoted by $\mathcal{W}_{n}$. Then, it is easy to see that
$\mathscr{K}^{2}(L_{n})\cong\mathcal{W}_{n}$. In addition, let
$\mathcal{C}_{2}$ be the chain lattice of length two. Then,
$\mathscr{K}^{2}(T_{n})\cong\mathcal{C}_{2}\times\mathcal{W}_{n}$.

Figure 3 shows the Hasse diagrams for the lattices $\mathscr{K}^{2}(L_{5})$,
$\mathscr{K}^{2}(T_{5})$ and
$\mathscr{K}^{2}(D_{2,3})\cong\mathscr{K}^{2}(I_{2,3})$. One can infer from
the examples the general cases.

\begin{overpic}[scale={0.160}]{Fig3} \put(8.0,63.0){$\mathscr{K}^{2}(L_{5})$}
\put(78.0,50.0){$\mathscr{K}^{2}(T_{5})$}
\put(6.0,12.0){$\mathscr{K}^{2}(D_{2,3})$} \end{overpic} Figure 3. Hasse
diagrams for some planar finite ch-lattices point generated.

######  Lemma 4.3.

$\mathscr{K}^{2}(V_{5})$ is infinite.

###### Proof.

Let $V_{5}=\\{a,b,c,d,e\\}$ be a set of points in the plane such that
$a{-}b{-}c$ and $a{-}d{-}e$, but $a,b,c,d,e$ are not collinear. Define the
transformation $T(a,b,c,d,e)=(a,b,c,d^{\prime},e^{\prime})$ where
$e^{\prime}=ad\wedge be$, and $d^{\prime}=e^{\prime}c\wedge bd$, see Figure
4(a). First, we see that it is well-defined. Since $a,b,d$ are not collinear
and $e\in\Theta(a,b,d)$, $ad\wedge eb\not=\emptyset$. Similarly, since $b,c,d$
are not collinear and $e^{\prime}\in\Theta(b,c,d)$, $e^{\prime}c\wedge
bd\not=\emptyset$. $a,b,c,d^{\prime},e^{\prime}$ is also a $V_{5}$
configuration. Clearly $a{-}b{-}c$ and $c{-}d^{\prime}{-}e^{\prime}$, by
definition. If $a,b,c,d^{\prime},e^{\prime}$ were collinear, $a,b,c,d,e$ would
be collinear. Finally notice that, $e{-}e^{\prime}{-}b$, with
$e^{\prime}\not=e$ and $e^{\prime}\not=b$. Applying the transformation as many
times as we want, we get
$e{-}e^{\prime}{-}e^{\prime\prime}{-}e^{\prime\prime\prime}{-}\cdots{-}b$,
which means that $\mathscr{K}^{2}(V_{5})$ is infinite. ∎

######  Definition 4.4.

We say that a configuration in the plane is in _B position_ if it has five
points $a,b,c,d,e$ such that $a{-}b{-}c$, $e\in H^{+}(ac,d)$ and that $d,e,b$
are not collinear.

######  Remark 4.5.

A planar five-point configuration is in B position iff it is equivalent to
$V_{5}$, $R$ or $R^{\prime}$.

######  Lemma 4.6.

If $X$ is in B position, then $\mathscr{K}^{2}(X)$ is infinite.

###### Proof.

Let $X=\\{a,b,c,d,e\\}$ be in B position. We show that we can always find a
$V_{5}$ configuration in $\mathscr{K}^{2}(X)$, which by Lemma 4.3, implies
directly that $\mathscr{K}^{2}(X)$ is infinite since we have a monomorphism of
lattices $\mathscr{K}^{2}(V_{5})\longrightarrow\mathscr{K}^{2}(X)$. Assume
that $ad\wedge ec=\emptyset$, otherwise we are done because $\\{a,b,c,ad\wedge
ec,d\\}\equiv V_{5}$, see Figure 4(b). That means that either $e\in
H^{-}(dc,a)\cap H^{-}(ad,c)$ or $d\in H^{-}(ec,a)\cap H^{-}(ae,c)$. Thus,
$e\in H^{-}(dc,a)\cap H^{-}(ad,c)$, and notice that by hypothesis $e,d,b$
cannot be collinear. Then either $ec\wedge ad\not=\emptyset,d$, and then
$\\{a,b,c,ec\wedge ad,d\\}\equiv V_{5}$, as in Figure 4(c), or $ec\wedge
dc\not=\emptyset,d$, and then $\\{a,b,c,ec\wedge dc,d\\}\equiv V_{5}$. The
second case $d\in H^{-}(ec,a)\cap H^{-}(ae,c)$ runs symmetrically. ∎

\begin{overpic}[scale={0.13}]{Fig4} \par\put(0.0,24.0){(a)}
\par\put(1.0,0.0){$a$} \put(11.5,0.0){$b$} \put(23.0,0.0){$c$}
\put(7.0,16.0){$e$} \put(11.0,25.0){$d$} \put(6.5,8.5){$e^{\prime}$}
\put(13.5,12.0){$d^{\prime}$} \par\put(38.0,24.0){(b)} \par\put(38.5,0.0){$a$}
\put(49.0,0.0){$b$} \put(61.0,0.0){$c$} \put(45.0,16.0){$d$}
\put(52.0,18.0){$e$} \par\put(75.0,24.0){(c)} \par\put(76.0,0.0){$a$}
\put(86.0,0.0){$b$} \put(98.0,0.0){$c$} \par\put(89.0,24.0){$e$}
\put(91.0,16.0){$d$} \end{overpic} Figure 4. Schemes for Lemmas 4.3 and 4.6.

######  Lemma 4.7.

If $X$ is a configuration of five points in the plane in general position,
then $\mathscr{K}^{2}(X)$ is infinite.

###### Proof.

Let $X=\\{a,b,c,d,e\\}$. A classical result asserts that any set of five
points in the plane in general position has a subset of four points that form
the vertices of a convex quadrilateral, [15, p. 27], [9]. Let the vertices
$a,b,c,d$ be in anticlockwise order of the convex quadrilateral in $X$. The
diagonals necessarily intersect, $ac\wedge bd=x\not=\emptyset$. Since points
in $P$ are in general position $e,a,c$ are not collinear and $x\not\in ac$. If
$e\in H^{+}(ac,d)$ or $e\in H^{-}(ac,d)$, then $a,x,c,b,e$ are in B position
and $\mathscr{K}^{2}(X)$ is infinite. ∎

######  Remark 4.8.

Notice that all the configurations from Definition 4.1 are subconfigurations
of the diamond configuration, except for the sporadic configuration:

$\displaystyle L_{p+q+1}\equiv D_{p,q}\setminus\\{s,s^{\prime}\\},\quad
T_{p+q+1}\equiv D_{p,q}\setminus\\{s^{\prime}\\},\quad I_{p,q}\equiv
D_{p,q}\setminus\\{c\\}.$

Any subconfiguration of the diamond configuration is equivalent to only one of
the configurations $L_{n},T_{n},I_{p,q},D_{p,q}$.

######  Theorem 4.9.

Let $X$ a subset of the plane. The lattice $\mathscr{K}^{2}(X)$ is finite iff
$X$ is a subconfiguration of $D_{p,q}$ for some integers $p,q$, or $X$ is
equivalent to $S_{6}$.

###### Proof.

We only need to prove the necessary condition for which we proceed by
induction on the size of $X$, $|X|=n$. Let $n\leq 5$. Then, the statement can
be manually checked as follows. Consider Figure 1. By Lemma 4.3, $V_{5}$ is
not finitely completable. $R$ and $R^{\prime}$ are in B position, and
$G,G^{\prime}$ and $P_{5}$ are in general position. By Lemmas 4.6 and 4.7,
they are not finitely completable. Therefore, the unique configurations
finitely completable are $L_{1}$, $L_{2}$, $L_{3}$, $L_{4}$, $L_{5}$, $T_{2}$,
$T_{3}$, $T_{4}$, $D_{0,2}$, $D_{1,1}$, $I_{0,2}$, $I_{0,3}$, $I_{1,1}$, and
$I_{1,2}$. By Remark 4.8, all of them are subconfigurations of the diamond
configuration.

Assume that the statement is true for any set $X$ with $|X|=n\geq 5$, and let
$X^{\prime}=X\cup\\{x\\}$ where $x\not\in X$. We distinguish the two cases of
the main statement.

  1. (1)

Suppose $X\subseteq D_{p,q}$, with $p+q+3=n$. If we add a point, then
$X\cup\\{x\\}\subseteq D_{p,q}\cup\\{x\\}$. If $x\in D_{p,q}$, we are done.
Therefore, we can suppose that $x\not\in D_{p,q}$. In this case, if we prove
that $D_{p,q}\cup\\{x\\}\subseteq D_{p^{\prime},q^{\prime}}$ or that
$D_{p,q}\cup\\{x\\}\equiv S_{6}$, we are done. Notice that in the first case,
$X^{\prime}\subseteq D_{p^{\prime},q^{\prime}}$ and $X^{\prime}\subseteq
S_{6}$ in the second case. However, all the proper configurations of $S_{6}$
are subconfigurations of $D_{p,q}$. Therefore, $X^{\prime}$ is a
subconfiguration of a diamond configuration or equivalent to the sporadic
configuration.

For brevity we will write $Y=D_{p,q}$ and $Y^{\prime}=D_{p,q}\cup\\{x\\}$. Let
$r$ be the axis and $s,s^{\prime}$ the star points. We need to distinguish
between $n=5$ and $n>5$. Consider first $n>5$. Suppose $x\in H^{+}(r,s)$. If
$x$ and $s$ are not collinear with any point in the axis $r\cap X$, then we
have a configuration in B position. By Lemma 4.6, the ch-lattice is infinite,
which is impossible by hypothesis. Suppose that there is some point $a$ in the
axis such that $\overline{xsa}$. Since $n>5$, the axis is conformed at least
by four points. We can ignore the point $a$, and then the axis has at least
three points and now $s$ and $x$ are not collinear with any point in the axis.
Therefore, we have again a configuration in B position. If we suppose that
$y\in H^{-}(r,s)$, then by symmetry, it is also impossible. If we want a
finite lattice, $y$ must be collinear with $r$. Then, $Y^{\prime}\equiv
D_{p+1,q}$ or $D_{p,q+1}$.

Now let $n=5$. Then, $Y\equiv D_{1,1}$ or $Y\equiv D_{0,2}$. If $Y\equiv
D_{1,1}$, $Y$ has symmetry $\operatorname{Aut}(Y)\cong\mathcal{D}_{4}$ and we
can consider that it has two axis. If $x$ is not collinear with any axis, then
we have a configuration in B position. So $x$ must be collinear with some axis
and then $Y^{\prime}\equiv D_{1,2}$.

Consider that $Y\equiv D_{0,2}$, and let $a_{1}{-}a_{2}{-}a_{3}$ the axis. As
in the above cases, if $x,s,a_{2}$ are not collinear, we will obtain a
configuration in B position. Thus, $\overline{ysa_{2}}$. First, we notice that
$y\not\in\triangle a_{1}sa_{3}$ because we could intersect the axis as
$x^{\prime}=sx\wedge r\not=\emptyset$, with which we would obtain a $V_{5}$
configuration $\\{s,x,a_{2},a_{1},x^{\prime}\\}$. That means that $x$ cannot
be between $s$ and $a_{2}$. And now, we get the sporadic configuration
$Y^{\prime}\equiv S_{6}$ given by $a_{1}{-}a_{2}{-}a_{3}$,
$s{-}a_{1}{-}s^{\prime}$, and $x{-}s{-}a_{2}$.

  2. (2)

$X\equiv S_{6}$. In this case, we will show that we cannot add an extra point
$x$ to get a finitely completable configuration. Due to the ternary symmetry
of $S_{6}$, $\operatorname{Aut}(S_{6})\cong\mathbb{Z}/(3)$, we only need to
check “one-third” of the total cases. Let
$a,b,c,a^{\prime},b^{\prime},c^{\prime}$ be the vertices of $S_{6}$, with the
collinear sets according to the Definition 4.1. Notice that
$H^{+}(ac^{\prime},c)\cup H^{+}(cb^{\prime},b)\cup
H^{+}(ba^{\prime},a)=\mathbb{R}^{2}$, and we only need to check that
$\mathscr{K}^{2}(Y)$ is infinite if $x\in H^{+}(ac^{\prime},c)$, and the rest
of half planes are similar. If $a^{\prime},c,x$ are not collinear, then we
have a configuration in B position. But if $\overline{a^{\prime}cx}$, then
$x^{\prime}=xc^{\prime}\wedge cb\not=\emptyset$, and then
$x^{\prime},a,a^{\prime},c^{\prime},c$ is also a configuration in B position.

∎

######  Corollary 4.10.

Let $X$ a subset of the plane. The lattice $\mathscr{K}^{2}(X)$ is finite iff
$X$ is equivalent to $L_{n},T_{n},D_{p,q},I_{p,q}$ or $S_{6}$ for some
integers $n,p,q$.

###### Proof.

By Theorem 4.9, $X$ is a subconfiguration of $D_{p,q}$ for some intgers or it
is equivalent to $S_{6}$. If we remove any number of points in $D_{p,q}$, we
obtain $L_{n},T_{n}$ or $I_{p,q}$. The reciprocal is trivial since all those
configurations are finitely completable. ∎

Even we have another characterisation of finiteness in the form of forbidden
configurations. Let the infinite linear configuration
$L_{\infty}=\\{(n,0)\in\mathbb{R}^{2}\mid n\in\mathbb{N}\\}$.

######  Theorem 4.11.

Let $X$ be a complete subset of the plane. The lattice $\mathscr{K}^{2}(X)$ is
finite iff $X$ does not contain the $V_{5}$ and $L_{\infty}$
subconfigurations.

###### Proof.

The direction $(\Rightarrow)$ is trivial from Corollary 4.10. For the
direction $(\Leftarrow)$, we sketch the argument and leave the details for the
reader, which essentially follow the same techniques and lemmas of the above
results. It suffices to see that if we want to build a finite and complete
configuration, there cannot appear a $L_{\infty}$ configuration nor $V_{5}$
configuration, which implies that neither can appear points in B position nor
in general position, because we are assuming that $X$ is complete. The
statement can be checked manually if $|X|<5$. If $|X|\geq 5$, we can assume
there are at least three collinear points; otherwise, we have five points in
general position, leading to an infinite configuration. Of course, if the
lattice is finite, then $L_{\infty}$ cannot be a subconfiguration. In
particular, a greatest maximal collinear set of $X$ is finite. Let $r\geq 3$
be the size of this set. If $r=3$, then it can be proved that $X$ is
equivalent to $L_{3},T_{3},A_{3},D_{0,2},D_{1,1}$ or $S_{6}$. By induction on
$r$, suppose that $X$ has a maximal collinear set of size $r$ with no
$L_{\infty}$ nor $V_{5}$ configurations. Suppose $X^{\prime}$ with a maximal
collinear set of size $r+1$. $X^{\prime}$ is obtained from $X$ by adding a
point in that collinear set. If $V_{5}$ is a subconfiguration of $X^{\prime}$,
it can be proved that $V_{5}$ is a subconfiguration of $X$ or that $X$
contains a subconfiguration in B position, which also entails a $V_{5}$
subconfiguration. ∎

##  5\. An insight on the spatial and the general case

\begin{overpic}[scale={0.18}]{Fig5} \put(1.0,87.0){(a)} \put(41.0,87.0){(b)}
\put(83.0,87.0){(c)} \par\put(31.0,58.0){(d)} \put(57.0,58.0){(e)}
\par\put(5.0,18.0){(f)} \put(38.0,18.0){(j)} \put(73.0,18.0){(g)}
\end{overpic} Figure 5. Some examples of spatial configurations derived from
cross-operator and taking subconfigurations: (a) $D_{2,3}*c$, (b) $D_{2,2}*c$,
(c) $D_{0,4}*c$. (d), (e), (j), (g) are of the form $S_{6}*c$, where white
dots are collinear or coplanar points. (f) is a subconfiguration of $S_{6}*c$.
\begin{overpic}[scale={0.13}]{Fig6} \put(10.0,36.0){(a)} \put(61.0,36.0){(b)}
\end{overpic} Figure 6. Spatial configurations $L_{4}+L_{5}$ and
$T_{5}+L_{3}$.

The natural continuation tackles the characterisation of finite lattices
generated by points in space and the general case in any dimension. For
example, the tetrahedron and the octahedron generate finite lattices, while
the cube, dodecahedron, and icosahedron generate infinite lattices.

Consider the following operator from configurations in $\mathbb{R}^{d}$ into
configurations in $\mathbb{R}^{d+1}$, which we will call _cross-operator_. Let
$\pi$ be an hyperplane in $\mathbb{R}^{d+1}$ and let $X\subseteq\pi$ a
configuration. Let $a$ and $b$ be two points on the opposite sides of $\pi$.
Notice that $ab\cap\pi\not=\emptyset$ and that $\dim ab\cap\pi=0$, i.e., the
intersection is a point, say $c$. Assume implicitly that $c\in X$ or
$c\not\in\operatorname{ch}(X)$, and define $X*c=X\cup\\{a,b\\}$. Then, the
cross-operator is well-defined. Let $a,a^{\prime}$ and $b,b^{\prime}$ be
points on the opposite sides of the hyperplane $\pi$, $X$ a configuration in
$\pi$ such that $ab\cap\pi=a^{\prime}b^{\prime}\cap\pi$. We can define the
isomorphism of configurations as $f(a)=a^{\prime}$, $f(b)=b^{\prime}$, and
$f(x)=x$ for all $x\in X$. Then, $X\cup\\{a,b\\}\equiv
X\cup\\{a^{\prime},b^{\prime}\\}$, and $X*c$ only depends on the chosen $c$,
not on the points $a,b$.

######  Lemma 5.1.

$\mathscr{K}^{d}(X)$ is finite iff $\mathscr{K}^{d+1}(X*c)$ is.

###### Proof.

$(\Leftarrow)$ Trivial since
$\mathscr{K}^{d}(X)\subseteq\mathscr{K}^{d+1}(X*c)$. $(\Rightarrow)$ We can
construct a surjective mapping
$\mathcal{C}_{2}\times\mathcal{C}_{2}\times\mathscr{K}^{d}(X)\longrightarrow\mathscr{K}^{d+1}(X*c)$,
although it is not a lattice epimorphism. Consider the two element lattices
$\mathcal{L}_{a}=\\{\emptyset,\\{a\\}\\}$ and
$\mathcal{L}_{b}=\\{\emptyset,\\{b\\}\\}$, where $ab\wedge\pi=\\{c\\}$, and
$X\subseteq\pi$. Clearly
$\mathcal{L}_{a}\cong\mathcal{L}_{b}\cong\mathcal{C}_{2}$. Notice that any
element $Z\in\mathscr{K}^{d+1}(X*c)$ is of one of the forms:

$Z=Z^{\prime},\quad Z=\operatorname{ch}(Z^{\prime}\cup\\{a\\}),\quad
Z=\operatorname{ch}(Z^{\prime}\cup\\{b\\}),\quad
Z=\operatorname{ch}(Z^{\prime}\cup\\{a,b\\}),$

for some $Z^{\prime}\in\mathscr{K}^{d}(X)$. This fact follows from the fact
that either $c\in X$ or $c\not\in\operatorname{ch}(X)$. Then, consider the
mapping:

$\displaystyle\mathcal{L}_{a}\times\mathcal{L}_{b}\times\mathscr{K}^{d}(X)$
$\displaystyle\longrightarrow\mathscr{K}^{d+1}(X*c)$ $\displaystyle(A,B,C)$
$\displaystyle\longmapsto\operatorname{ch}(A\cup B\cup C)$

which by the above comment is surjective, and then $|\mathscr{K}^{d}(X*c)|\leq
4|\mathscr{K}^{d}(X)|$. ∎

Except for the sporadic configuration $S_{6}$, any planar configuration is a
subconfiguration of $L_{n}*c$ for some point $c$. The cross operator and
taking subconfiguration are productive forms to obtain higher dimensional
configurations. We get many configurations in the space by this process; see
some examples in Figure 5. However, not all spatial configurations come from
planar configurations. For example, let $L_{p}+L_{q}$ be the union of $L_{p}$
and $L_{q}$, such that these linear configurations lie over two non-
intersecting and non-parallel lines. The intersection of any pair of convex
envelopments does not generate a new point, whereby $L_{p}+L_{q}$ is finite
and complete; Figure 6(a). Similarly, denote by $T_{p}+L_{q}$ the union of
$T_{p}$ and $L_{q}$ such that $L_{q}$ is in one of the semi-spaces delimited
by the plane $\pi$ where lies $T_{p}$, and such that $L_{q}$ is not parallel
to the axis of $T_{p}$; Figure 6(b). Clearly, $L_{p}+L_{q}$ is a
subconfiguration of $T_{p}+L_{q}$: remove the star point of $T_{p}$. If the
family $T_{p}+L_{q}$ were the only one not derived from planar configurations,
then $\mathscr{K}^{3}(X)$ would be finite iff

  1. (i)

$X$ is a subconfiguration of $D_{p,q}*c$, or

  2. (ii)

$X$ is a subconfiguration of $S_{6}*c$, or

  3. (iii)

$X$ is a subconfiguration of $T_{p}+L_{q}$

for some integers $p,q$, and some point $c$. Unfortunately, we have not
succeeded in proving it; many cases are under consideration. We aim that some
ideas in this paper open that and other questions, such as how the notion of
point configuration is related to other problems in discrete geometry. Some
combinatorial enumerative questions arise naturally, such as counting
configurations or isomorphy classes of ch-lattices.

There is another generalisation of ch-lattices. Denote by
$\mathscr{K}_{k}^{d}(X)$ the least ch-lattice containing the set of bounded
polytopes $X$ of dimension as much $k$. In particular,
$\mathscr{K}_{0}^{d}(X)=\mathscr{K}^{d}(X)$. For example, given three non-
collinear points $a,b,c$, the lattice $\mathscr{K}_{1}^{2}(\\{ab,c\\})$ is a
sublattice of $\mathscr{K}_{0}^{2}(\\{a,b,c\\})$. More in general
$\mathscr{K}_{k}^{d}(X)$ is always a sublattice of
$\mathscr{K}_{0}^{d}(\operatorname{ext}(X))$. Whereby the finiteness of the
latter implies the finiteness of the former. However, if
$V_{5}=\\{a,b,c,d,e\\}$, as in Lemma 4.3, $\mathscr{K}_{1}^{2}(\\{ad,e,bc\\})$
is finite, but
$\mathscr{K}_{1}^{2}(\operatorname{ext}(\\{ad,e,bc\\}))=\mathscr{K}_{0}^{d}(V_{5})$
is infinite. A last example from [2] considers the consecutive vertices of a
regular hexagon $H=\\{a_{1},\ldots,a_{6}\\}$.
$\mathscr{K}_{1}^{2}(\\{a_{1}a_{4},a_{2}a_{5},a_{3}a_{6}\\})$ if infinite. The
problem of discerning when $\mathscr{K}_{k}^{d}(X)$ is finite seems, at first
glance, more challenging than the point generated case, even in the planar
case.

## Declarations

  * •

Funding. Not applicable

  * •

Conflict of interest/Competing interests. No conflict of interest exists.

  * •

Ethics approval and consent to participate. Not applicable.

  * •

Consent for publication. Not applicable.

  * •

Data availability. Not applicable.

  * •

Materials availability. Not applicable.

  * •

Code availability. Not applicable.

  * •

Author contribution. Not applicable.

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First Author
Departament de Ciències Bàsiques, Universitat Internacional de Catalunya, c/
Josep Trueta s/n, Sant Cugat del Vallès, 08195, Spain.
Email<EMAIL_ADDRESS>


11institutetext: Department of Computer Science, School of Science,
Loughborough University, UK 22institutetext: School of Design and Creating
Arts, Loughborough University, UK 33institutetext: Health Services Safety
Investigations Body (HSSIB), United Kingdom 33email<EMAIL_ADDRESS>

# Intelligent Multi-Document Summarisation for Extracting Insights on Racial
Inequalities from Maternity Incident Investigation Reports

Georgina Cosma 11 Mohit Kumar 11 Patrick Waterson 22 Gyuchan Thomas Jun 22
Jonathan Back 33

###### Abstract

In healthcare, thousands of safety incidents occur every year, but learning
from these incidents is not effectively aggregated. Analysing incident reports
using AI could uncover critical insights to prevent harm by identifying
recurring patterns and contributing factors. To aggregate and extract valuable
information, natural language processing (NLP) and machine learning techniques
can be employed to summarise and mine unstructured data, potentially surfacing
systemic issues and priority areas for improvement. This paper presents
I-SIRch:CS, a framework designed to facilitate the aggregation and analysis of
safety incident reports while ensuring traceability throughout the process.
The framework integrates concept annotation using the Safety Intelligence
Research (SIRch) taxonomy with clustering, summarisation, and analysis
capabilities. Utilising a dataset of 188 anonymised maternity investigation
reports annotated with 27 SIRch human factors concepts, I-SIRch:CS groups the
annotated sentences into clusters using sentence embeddings and k-means
clustering, maintaining traceability via file and sentence IDs. Summaries are
generated for each cluster using offline state-of-the-art abstractive
summarisation models (BART, DistilBART, T5), which are evaluated and compared
using metrics assessing summary quality attributes. The generated summaries
are linked back to the original file and sentence IDs, ensuring traceability
and allowing for verification of the summarised information. Results
demonstrate BART’s strengths in creating informative and concise summaries.

###### Keywords:

Dynamic clustering Abstractive summarisation healthcare

##  1 Introduction

In recent years, deep learning (DL) and natural language processing (NLP)
models have demonstrated immense potential for automatically analysing and
summarising textual data across application areas. Within the healthcare
domain, key use cases include generating concise summaries of electronic
health records (EHRs) to accelerate access to patient histories, extracting
key findings from the latest medical literature to keep professionals informed
of advancements, and simplifying health education content into more readable
materials for patient consumption. However, applying DL in sensitive domains
like healthcare also introduces confidentiality, privacy, and ethical
considerations which demand accurate yet traceable model behaviours that
respect patient data sensitivity. To address the ethical considerations and
data privacy requirements when applying deep learning models to sensitive
healthcare data, it is crucial to ensure that any datasets used are fully
anonymised before analysis or modelling, with all personally identifiable
information removed from the outset.

Our prior work saw the development of the Intelligence-Safety Intelligence
Research (I-SIRch) framework/tool [10], which utilises multi-label text
annotation with the Safety Intelligence Research (SIRch) human factors
taxonomy to systematically categorise contributing factors in adverse maternal
care incidents. Trained on a corpus of real (anonymised) and tested on real
and synthetically generated maternity investigation reports from UK cases,
I-SIRch demonstrated robust performance across numerous evaluation metrics,
including recall, precision, and balanced accuracy [10]. Our research
represented an important step towards utilising advanced NLP techniques to
enhance patient safety and address care quality gaps, especially within
maternal health across different demographics. This paper aims to advance the
I-SIRch framework [10] by proposing an extended
I-SIRch:CS111https://github.com/gcosma/I-SIRchpapers system with robust multi-
document summarisation capabilities to effectively analyse maternal incident
investigation reports across various dimensions, including generating concise
yet sufficiently informative summaries, ensuring complete traceability back to
source data, and facilitating comparative equity evaluations by constructing
separate summaries across ethnic groups to identify any care quality
disparities underlying adverse incidents.

##  2 Related work

Models including Long Short-Term Memory (LSTM) networks and Transformer-based
architectures have been at the forefront, leveraging their ability to
understand context and sequence in texts. Specifically, the use of pretrained
models such as BERT (Bidirectional Encoder Representations from Transformers)
and its adaptations in the medical domain (e.g., BioBERT, ClinicalBERT) have
demonstrated promising results in improving the accuracy and relevance of
summaries. For instance, LSTM and Transformer models have been applied to
summarise patient Electronic Health Records (EHRs), extracting critical
information that can aid in diagnosis, treatment planning, and patient
monitoring [5], [14], [6], [13], [11]. These summaries provide a comprehensive
view of a patient’s medical history, current condition, and potential risks,
distilled into a format that is easily accessible to healthcare providers. The
introduction of models such as BERT (Bidirectional Encoder Representations
from Transformers) [1] and GPT (Generative Pretrained Transformer) [7] marked
a significant milestone in NLP. These models have shown remarkable success in
various NLP tasks, including text summarisation. Their ability to capture deep
contextual relationships within text makes them particularly suited for
summarising complex healthcare documents. One notable advancement is the
adaptation of these models for domain-specific tasks. For instance, BioBERT
[3] and ClinicalBERT [2] have been fine-tuned from BERT for biomedical and
clinical text processing, respectively, showing improved performance in tasks
such as disease classification and patient information summarisation.
Moreover, transformer-based models like BART (Bidirectional and Auto-
Regressive Transformers) [4] and T5 (Text-to-Text Transfer Transformer) [8]
have been utilised for both extractive and abstractive summarisation. BART, in
particular, has been effective in generating coherent summaries of medical
research articles by rephrasing and condensing the original text. While the
effectiveness of these models is well-documented, their computational demands
pose challenges, particularly in real-time healthcare applications. This has
led to the development of more efficient variants, such as DistilBERT [9] and
MiniLM [12], which maintain a balance between performance and computational
efficiency. These models enable the deployment of advanced NLP techniques in
resource-constrained environments, such as mobile health applications and low-
resource organisations.

##  3 I-SIRch:CS Framework

We propose an automated system for summarising text data, capable of
clustering sentences by similarity, determining the best number of clusters,
and generating concise summaries and keywords for each cluster. This process
transforms raw text into structured, actionable insights. We utilised offline
generative models to ensure patient confidentiality by anonymising and
synthesising data, preserving the integrity of insights while safeguarding
personal information.
Algorithm (1) provides the text clustering and summarisation process.

Algorithm 1 I-SIRch: CS Text clustering and summarisation

1:Input:

2: Set of sentences $S$ with labels $L$

3: Clustering model $M_{c}$

4: Summarisation model $M_{s}$

5:for each label $l\in L$ do

6: Filter $S$ to get subset $S_{l}$ with label $l$

7: Embed sentences in $S_{l}$ using $M_{c}$ getting embeddings $E_{l}$

8: Determine optimal clusters $k$ using elbow method on $E_{l}$

9: Cluster $S_{l}$ into $k$ clusters $\\{C_{1},\ldots,C_{k}\\}$ using
$k$-means on $E_{l}$

10:end for

11:for each cluster $C_{i}$ do

12: Concatenate sentences in $C_{i}$ into text $T_{i}$

13: Generate summary $S_{i}$ by applying $M_{s}$ to $T_{i}$

14: Generate heading $H_{i}$ by applying $M_{s}$ to $T_{i}$

15: Store sentence IDs used in $S_{i}$ as $ID_{i}$

16:end for

17:Outputs:

18:Summaries $S_{i}$ and corresponding Headings $H_{i}$ for each summary

19:Sentence IDs $ID_{i}$ used in each summary

20:Evaluation metrics for each summary

The algorithm takes as input a set of sentences, denoted mathematically as set
$\mathcal{S}$, along with an associated set of labels (a.k.a concepts)
$\mathcal{L}$ that categorises the sentences. It also requires two machine
learning models - a clustering model $M_{c}$ and a pretrained summarisation
model $M_{s}$. In the first stage, the algorithm groups the input sentences by
their labels (i.e. concepts) in $\mathcal{L}$. For each label
$l\in\mathcal{L}$, it extracts the subset of sentences $\mathcal{S}_{l}$ that
have that label value $l$ assigned. Next, it applies the clustering model
$M_{c}$ to embed the sentences in a vector space and determine an optimal
number of clusters based on the elbow method. The sentences with the same
label $l$ are then clustered into these topics, denoted by
$\\{C_{1},C_{2},\dotsc,C_{k}\\}$. The second stage is a summary and evaluation
loop over the clusters. For each cluster $C_{i}$ covering a topic, the
sentences are concatenated into a single text excerpt $T_{i}$. This is passed
into the summarisation model $M_{s}$ to generate a summary text $S_{i}$.
Additional metadata including a heading $H_{i}$ and the sentence IDs used are
also extracted for summary $S_{i}$. Evaluation metrics are then calculated for
summary $S_{i}$ \- including quantification of the diversity, relevance,
coverage, coherence, conciseness, and readability. These metrics are appended
to a master output list. In the end, the full pipeline outputs: the set of
summaries $S_{i}$ for each topic cluster; the headings $H_{i}$; sentence IDs
$ID_{i}$ used; and evaluation metrics on the different quality attributes for
each summary. The algorithm thus performs multi-document summarisation guided
by data labels, with integrated quantitative assessment.

##  4 Methodology

###  4.1 Dataset of maternity incident investigation reports

The Healthcare Services Safety Investigation Branch (HSSIB) provided a random
set of 188 anonymised investigation reports describing adverse maternity
incidents. The reports were written between 2019 to 2022. The number of
reports for each year is as follows: 4 reports in 2019, 115 reports in 2020,
42 reports in 2021, and 27 reports in 2022. Ethnicity was provided for 76
reports.

###  4.2 Experiment methodology

The I-SIRch:CS framework is presented in Fig. 1 and its components are
described as follows. Healthcare reports repository: The I-SIRch tool is used
for loading, cleaning, and preparing reports. During processing, the tool
generates a text file in CSV format that includes File IDs, Sentence IDs,
sentences, and concepts derived from the SIRch taxonomy. This phase involves
cleaning the files by selecting particular sentences from each report,
specifically targeting those with negative connotations, references to
physical characteristics, and mentions of medication names associated with
dispensing. These selected sentences are annotated according to the SIRch
taxonomy and aggregated into a CSV file for subsequent clustering and
summarisation tasks. Table 1 shows the list of SIRch concepts with the number
of sentences per concept shown in brackets. Dynamic clustering: Employs the
all-MiniLM-L6-v2 SentenceTransformer model for sentence embeddings and the
elbow method with KMeans for clustering. This process groups text data into
semantically similar clusters, maintaining traceability via file and sentence
IDs. Generative summarisation and heading generation: Selects from predefined
offline models (e.g., BART, T5, DistilBART) for summarisation. Keywords
extracted by KeyBERT for headings ensure summaries are informative and
traceable to original file and sentence IDs. Analytics engine (Qualitative
results): Outputs include the textual summaries and their corresponding topic
headings, where each summary is linked to each file and sentence ID. Analytics
engine (Quantitative results): Outputs include a detailed CSV report with
analysis on summarisation models, clustering, generated headings, summaries,
and evaluation metrics, ensuring explainability through file and sentence ID
traceability. Batch processing and process automation: The framework’s
scalable design allows for automated batch processing, enhancing the system’s
ability to handle large datasets efficiently, with a focus on traceability and
organisation.

Figure 1: I-SIRch:CS pipeline. Shows how the I-SIRch tool has been extended to include clustering and summarisation capabilities analysing intelligence from maternity incidence investigation reports. Table 1: ID, concept and number of sentences per concept. Total sentences: 3760. ID |  Concept | Index |  Label
---|---|---|---
1 |  Acuity (54) | 2 |  Antenatal (69)
3 |  Assessment, investigation, testing, screening (381) | 4 |  COVID (142)
5 |  Care Planning (132) | 6 |  Communication factor (477)
7 |  Decision error (169) | 8 |  Dispensing, administering (62)
9 |  Documentation (168) | 10 |  Escalation/referral factor (158)
11 |  Guidance factor (42) | 12 |  Interpretation (197)
13 |  Language barrier (30) | 14 |  Local guidance (88)
15 |  Monitoring (118) | 16 |  National and local guidance (80)
17 |  National guidance (169) | 18 |  Obstetric review (147)
19 |  Physical characteristics (320) | 20 |  Physical layout and Environment (35)
21 |  Psychological characteristics (54) | 22 |  Risk assessment (94)
23 |  Situation awareness (77) | 24 |  Slip or lapse (188)
25 |  Teamworking (530) | 26 |  Technologies and Tools-issues (112)
27 |  Training and education (40) |  |

###  4.3 Methods

We employed the following transformer-based architectures that have been fine-
tuned for summarisation, and which have demonstrated their capability to
address complex NLP challenges, particularly in healthcare. Sentence
Transformers (‘all-MiniLM-L6-v2’): A compact model that generates semantically
meaningful sentence embeddings suitable for semantic search, clustering and
summarisation, balancing performance with computational efficiency. BART
(‘facebook/bart-large-cnn’): A transformer model pretrained for generation
tasks, including summarisation. The ‘bart-large-cnn’ variant excels at
creating coherent and concise summaries, optimised for tasks requiring
detailed summary outputs. T5 (‘t5-small’): Adapts the text-to-text approach,
treating all NLP tasks as such. The ‘t5-small’ version offers a balance
between size, speed, and quality, ideal for diverse summarisation needs.
DistilBART (‘sshleifer /distilbart-cnn-12-6’): A lighter, faster distilled
version of BART, maintaining robust summarisation capabilities. Optimised for
efficiency, it provides a viable alternative to its parent model for
summarisation tasks.

###  4.4 Evaluation metrics

Several metrics have been used to assess different aspects of summarisation
quality, including diversity, relevance, coverage, coherence, conciseness, and
readability. Each metric provides insights into how well the summarisation
process captures and conveys the essential information from the original text.
Higher values across these metrics indicate better summarisation model
performance in producing summaries that are diverse, relevant, comprehensive,
logical, concise and readable.
Diversity measures the variety of vocabulary used in the summary. It is
calculated as the ratio of unique words to the total number of words in the
summary.

$\text{Diversity}=\frac{\text{Number of Unique Words}}{\text{Total Number of
Words}}$ (1)

Relevance assesses the similarity between the original text and its summary.
Cosine similarity between the vector representations of the original text and
the summary is often used for this purpose.

$\text{Relevance}=\cos(\theta)=\frac{\mathbf{A}\cdot\mathbf{B}}{\|\mathbf{A}\|\|\mathbf{B}\|}$
(2)

where $\mathbf{A}$ and $\mathbf{B}$ are the vector representations of the
original text and the summary, respectively.
Coverage evaluates the extent to which the summary captures the key concepts
of the original text. It can be quantified by the proportion of original text
tokens (or concepts) that appear in the summary.

$\text{Coverage}=\frac{\text{Number of Unique Tokens in Both Summary and
Original Text}}{\text{Number of Unique Tokens in Original Text}}$ (3)

Coherence measures the logical flow and connection between sentences within
the summary. While more challenging to quantify, coherence can be assessed by
evaluating sentence embeddings’ similarity in sequence.

$\text{Coherence}=\frac{1}{N-1}\sum_{i=1}^{N-1}\cos(\theta_{i,i+1})$ (4)

where $N$ is the number of sentences in the summary, and
$\cos(\theta_{i,i+1})$ is the cosine similarity between consecutive sentence
embeddings.
Conciseness indicates the brevity of the summary. It can be inversely related
to the summary’s length, encouraging summaries that convey information
efficiently.

$\text{Conciseness}=\frac{1}{\text{Word Count of Summary}}$ (5)

Readability measures how easy it is to understand the summary. The Flesch
Reading Ease score is a common metric, calculated as follows:

$\text{Readability}=206.835-1.015\left(\frac{\text{Total Words}}{\text{Total
Sentences}}\right)-84.6\left(\frac{\text{Total Syllables}}{\text{Total
Words}}\right)$ (6)

##  5 Results

The IDs of sentences assigned to each cluster are tracked and used for
generating the summary of that cluster. Hence, all sentences grouped into a
specific cluster are considered in the summarisation process. However, the
actual summary may not incorporate these sentences verbatim. Instead, the
summary provides a condensed version that captures the main points from these
sentences. An analysis of the summaries is presented below.

###  5.1 Performance comparison of summarisation methods

Table 2 provides the results when comparing the summarisation models using the
evaluation metrics defined in section 4.4. Fig. 2 provides an overview of the
performance and variability of each model across concepts. Among the models,
BART achieves the best overall performance and generalisation. BART creates
relevant and coherent summaries that maximise vocabulary diversity, and it
attains this reliable performance as evidenced by low standard deviations in
key metrics like diversity, relevance, and coherence. Below is a summary of
the results shown in Table 2. Diversity: BART’s diversity score is $0.806\pm
0.058$ standard deviation, indicating its summaries incorporate the richest
unique vocabulary with low variability run-to-run. Relevance: BART has highest
relevance at $0.922\pm 0.015$. This shows its summaries strongly preserve
source semantics with minimal fluctuation. Coverage: All models exhibit low
coverage of $\sim 0.07\pm 0.03$ standard deviations because summaries
significantly condense full text. Coherence: BART’s scores $0.794\pm 0.022$
coherence, meaning summary sentences interrelate accurately. This combination
enables easy comprehension. Conciseness: The models achieve similar
conciseness ($\sim 0.02\pm 0.004$), quantifying the summary brevity.
Readability: T5-small tops at $190.288\pm 5.799$ in linguistic simplicity. But
BART nearly matches at $189.148\pm 5.863$.

Table 2: Evaluation metrics for summarisation models Metric | BART | DistilBART | T5-small
---|---|---|---
Diversity Score | 0.806 $\pm$ 0.058 | 0.770 $\pm$ 0.057 | 0.771 $\pm$ 0.059
Relevance | 0.922 $\pm$ 0.015 | 0.709 $\pm$ 0.028 | 0.818 $\pm$ 0.035
Coverage | 0.070 $\pm$ 0.034 | 0.072 $\pm$ 0.034 | 0.064 $\pm$ 0.032
Coherence | 0.794 $\pm$ 0.022 | 0.674 $\pm$ 0.038 | 0.670 $\pm$ 0.030
Conciseness | 0.021 $\pm$ 0.004 | 0.019 $\pm$ 0.003 | 0.022 $\pm$ 0.004
Readability | 189.148 $\pm$ 5.863 | 186.896 $\pm$ 6.253 | 190.288 $\pm$ 5.799

(a) Diversity score boxplot

(b) Relevance boxplot

(c) Coverage boxplot

(d) Coherence boxplot

(e) Conciseness boxplot

(f) Readability boxplot

Figure 2: Overall comparison of evaluation metrics.

###  5.2 BART’s summarisation performance across concepts

Fig. 3 depicts the performance of BART’s summarisation model across each
concept. The x-axes show the concept numbers, and these correspond to the IDs
found in Table 1. Hence number 1 on the x-axis of Fig. 3(a) corresponds to the
Acuity concept that has ID 1 in Table 1.
The areas of strength are: Relevance: Across all labels, the model
consistently showed high relevance scores, with means above 0.9. This
indicates that the model is adept at generating content that is closely
related to the given topics. Low standard deviations in relevance scores, such
as 0.012 for Acuity and 0.007 for Assessment, investigation, testing,
screening, suggest that the model maintains this relevance across different
instances reliably. Coherence: The model also performed well in terms of
coherence, particularly for labels such as COVID, with a mean of 0.836 and
Acuity with a mean of 0.809. These scores suggest that the generated text
logically flows from one sentence to the next, making it easier for readers to
follow the narrative.
The areas for improvement are: Coverage: The model struggled with coverage
across several labels, notably so for Assessment, investigation, testing,
screening, with a mean of 0.03 and a very low standard deviation, indicating
consistently narrow topic coverage. This suggests that the model may not fully
address all relevant aspects of a topic, possibly omitting important details.
Diversity: While not as critical as coverage, diversity scores varied more
widely across labels, with COVID showing a notably lower mean (0.702) and
higher standard deviation (0.079). This indicates that the model’s ability to
generate varied text differs significantly across topics, which could limit
its effectiveness in engaging readers with novel content or perspectives.
Readability: The readability scores were consistently high across labels, with
means around 187 to 192, suggesting the text may be complex and potentially
challenging for a general audience to understand. While not a direct measure
of performance like relevance or coherence, high readability scores indicate
room for improvement in making the content more accessible to a broader
audience.

(a) Diversity Score

(b) Relevance

(c) Coverage

(d) Coherence

(e) Conciseness

(f) Readability

Figure 3: Summary of model evaluation metrics for BART.

###  5.3 Evaluating equity in summaries of investigation reports

Table 3 and Fig. 4 present a summary of various evaluation metrics across
different ethnic groups: Asian, Black, Data not received (DNR), Mixed
Background (MB), Other White (OW), and White British (WB) when using BART.

Figure 4: Results per ethnic group when using BART Table 3: Average and std. values of metrics by ethnicity when using BART Metric | Asian | Black | DNR | MB | OW | WB
---|---|---|---|---|---|---
Diversity | $0.67\pm 0.08$ | $0.64\pm 0.07$ | $0.67\pm 0.06$ | $0.69\pm 0.08$ | $0.65\pm 0.04$ | $0.68\pm 0.04$
Relevance | $0.89\pm 0.18$ | $0.87\pm 0.17$ | $0.95\pm 0.09$ | $0.97\pm 0.04$ | $0.93\pm 0.10$ | $0.40\pm 0.22$
Coverage | $0.89\pm 0.18$ | $0.87\pm 0.17$ | $0.95\pm 0.09$ | $0.97\pm 0.04$ | $0.93\pm 0.10$ | $0.40\pm 0.22$
Coherence | $0.46\pm 0.32$ | $0.54\pm 0.29$ | $0.65\pm 0.26$ | $0.63\pm 0.18$ | $0.42\pm 0.27$ | $0.18\pm 0.19$
Conciseness | $0.01\pm 0.00$ | $0.01\pm 0.00$ | $0.01\pm 0.00$ | $0.01\pm 0.00$ | $0.01\pm 0.00$ | $0.01\pm 0.00$
Readability | $21.54\pm 5.12$ | $17.53\pm 2.54$ | $19.23\pm 3.37$ | $16.08\pm 0.59$ | $19.54\pm 3.63$ | $19.62\pm 3.58$

Diversity: The diversity scores are relatively similar across all ethnicities,
ranging from 0.64 to 0.69, with small standard deviations (0.04 to 0.08). This
suggests that the BART model generates diverse outputs for all ethnicities.
Relevance and Coverage: The relevance and coverage scores are identical for
each ethnicity, indicating that these metrics are closely related. Asian,
Black, DNR, MB, and OW have high scores (0.87 to 0.97) with small standard
deviations (0.04 to 0.18), suggesting that the model generates relevant and
comprehensive outputs for these ethnicities. However, WB has a lower score
(0.40) with a larger standard deviation (0.22), indicating that the model’s
outputs for this ethnicity may be less relevant and comprehensive. Coherence:
The coherence scores vary across ethnicities, with DNR and MB having the
highest scores (0.65 and 0.63) and moderate standard deviations (0.26 and
0.18). Asian, Black, and OW have lower scores (0.42 to 0.54) with larger
standard deviations (0.27 to 0.32), suggesting that the model’s outputs for
these ethnicities may be less coherent. WB has the lowest coherence score
(0.18) with a small standard deviation (0.19), indicating that the model’s
outputs for this ethnicity are the least coherent. Conciseness: All
ethnicities have the same conciseness score (0.01) with no standard deviation,
suggesting that the model generates equally concise outputs for all
ethnicities. Readability: The readability scores range from 16.08 to 21.54,
with standard deviations ranging from 0.59 to 5.12. MB has the lowest
readability score (16.08) with the smallest standard deviation (0.59),
indicating that the model’s outputs for this ethnicity are the most readable
and consistent. Asian has the highest readability score (21.54) with the
largest standard deviation (5.12), suggesting that the model’s outputs for
this ethnicity are the least readable and have the most variability. In
summary, the BART model generates diverse, relevant, and concise outputs for
most ethnicities, with some variations in coherence and readability. The
model’s performance appears to be the weakest for the WB ethnicity, with lower
relevance, coverage, and coherence scores. These low scores could be
attributed to the sample size and variability of the WB sentences. While the
WB group has the most sentences, this does not guarantee better performance. A
larger dataset can introduce more variability, making it harder for the model
to learn consistent patterns. Table 4 presents two sample summaries (from
reports of different ethnic groups) generated by the BART summarisation model.

Table 4: Sample summaries from multiple reports for two ethnic groups. Ethnicities cannot be disclosed. Each summary is generated from multiple reports. Concept |  Summary
---|---
Communication |  Staff not heard. Staff voiced their concerns about this decision to the senior obstetrician. They were left feeling that their concerns had not been heard. There was no formal debriefing afterwards, which staff would have valued. The opportunity to share reflections and learning was not completed with all staff involved. The incremental delays caused by finding and allocating staff to the concurrent theatre cases, and communication breakdown within the team further impacted on the DDI.
Acuity |  Reviews with seniors did not occur. The ambulance Trust was experiencing high volumes of 999 calls at the time of a Mother’s call. Due to the high acuity on the labour ward the initial decisions were not discussed with the senior clinician. A senior face to face review did not occur until 16:05 hours, 2 hours and 50 minutes after the initial recognition of abnormalities of the Baby’s heart rate. The abnormal CTG trace from the IOL suite was not reviewed by the senior obstetrician.

##  6 Ethical risks of abstractive summarisation models

  * •

Risk of information hallucination and bias amplification: Abstractive
summarisation models like BART, T5, and DistilBART risk generating content not
found in the original data and amplifying biases from their training data.
Mitigation: Diversifying training datasets, implementing debiasing techniques,
and cross-referencing with original texts. Regular bias audits and integrating
ethical guidelines directly into the training process can enhance accuracy,
transparency, and fairness.

  * •

Risk of inadequate control over content: Unlike extractive methods,
abstractive models generate summaries based on learnt patterns, which can
result in misrepresentation or underrepresentation of certain groups or
incidents. Mitigation: Developing more sophisticated models with embedded
ethical and fairness constraints, and employing precise source mapping
mechanisms like sentence IDs, can strengthen control and ensure verifiability
and trustworthiness of the content generated.

  * •

Risk in processing sensitive data: Utilising online LLMs for sensitive data
can pose risks of data breaches and privacy violations. Mitigation: Switching
to offline LLMs lessens these risks but may reduce performance, as it depends
on local computing resources. Organisations need to balance security against
performance to find an appropriate balance.

  * •

Sample Size and Variability: Large datasets, whilst rich in information,
present challenges due to variability and the potential inclusion of low-
quality data. This complexity hinders a model’s ability to learn consistent
patterns, adversely affecting summarisation metrics. Mitigation: Explore
robust data preprocessing techniques to enhance data quality. Utilising
machine learning algorithms customised for high variability will enhance
consistency and equity in summarisation outcomes across diverse groups.

##  7 Conclusion

The I-SIRch:CS framework automates the analysis and summarisation of textual
data in maternity incident reports, holding significant potential for
uncovering critical insights and contributing factors to preventable harm.
Future work will prioritise enhancing the framework’s traceability
capabilities by implementing mechanisms to provide clear links between
generated summaries and original reports, allowing for easy verification of
accuracy and context. Explainable AI techniques will also be explored to offer
insights into how models generate summaries. A key limitation of the
I-SIRch:CS framework is the lack of Patient and Public Involvement (PPI) in
assessing its outputs. PPI is crucial as it ensures solutions are relevant to
patient needs and experiences, potentially improving the framework’s summaries
and their real-world applicability. Engaging with patients and the public
could also increase the model’s transparency and trustworthiness. Future
development will aim to integrate PPI feedback, enhancing the framework’s
effectiveness and its contributions to patient safety and care quality.

####  7.0.1 Acknowledgements

This report is independent research funded by NHSX and The Health Foundation
and it is managed by the National Institute for Health Research (AI_HI200006).
The views expressed in this publication are those of the author(s) and not
necessarily those of the NHSX, The Health Foundation, National Institute for
Health Research, or the Department of Health and Social Care.

####  7.0.2

The authors have no competing interests to declare that are relevant to the
content of this article.

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