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\begin{center}
\vskip 1cm{\LARGE\bf Generation of Union-Closed Sets and \\
\vskip 0.1in
Moore Families} \\
\vskip 1cm 
\large
Gunnar~Brinkmann and Robin Deklerck\\
Applied Mathematics, Computer Science and Statistics\\
Ghent University\\
Krijgslaan 281 S9\\
B9000 Ghent\\
Belgium\\
\href{mailto:gunnar.brinkmann@ugent.be}{\tt gunnar.brinkmann@ugent.be}\\
\href{mailto: robindeklerck@skynet.be}{\tt robindeklerck@skynet.be}.
\end{center}
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\begin{abstract}
We describe an algorithm to 
constructively enumerate 
non-isomorphic union-closed sets and Moore sets. We
confirm the number of isomorphism classes of union-closed sets and Moore sets
on $n\le 6$ elements presented by other authors and give the 
number of isomorphism classes of union-closed sets and Moore sets
on $7$ elements. 
Due to the enormous growth of
the number of isomorphism classes, it seems unlikely that
constructive enumeration for $8$ or more elements will be possible
in the foreseeable future. 

\end{abstract}



\section{Introduction}

All sets in this article are finite.  A {\it union-closed set\/}
is a set ${\cal U}$ of sets
with the property that for all $A,B\in {\cal U}$ we have $A\cup
B\in {\cal U}$. We call $\Omega_{{\cal U}}=\bigcup_{A\in {\cal U}}A$ the 
{\it universe\/}
of ${\cal U}$. Two union-closed sets with universe $\Omega_{{\cal U}}$, resp., $\Omega_{{\cal U}'}$
are defined to be {\it isomorphic\/} if there is a bijective mapping 
$\Omega_{{\cal U}} \to \Omega_{{\cal U}'}$ inducing a bijection between the
union-closed sets.

As we are mainly interested in isomorphism classes, we may assume
$\Omega_U= \Omega_n=\{1,\ldots ,n\}$ for some $n$.  While the whole
universe
$\Omega_U$ is by definition an element of a union-closed set ${\cal U}$,
this is not the case for the empty set.
As the empty set has no
impact on the structure of a union-closed set, one often either requires the
empty set to be an element of each union-closed set or forbids it to be an
element. We choose for the first convention, so our union-closed sets contain $\Omega_n$
as well as the empty set. We denote a set containing one representative
of each isomorphism class of union-closed sets with universe  $\Omega_n$ as ${\cal R}_n$.

The famous {\em union-closed sets conjecture} (or {\em Frankl's conjecture}) states that
for each union-closed set there is an element that is present in at least half of the sets.
Unfortunately, the number of union-closed sets grows so quickly that complete enumeration
is not a suitable approach for testing this conjecture.
A lot is known about the structure of possible counterexamples to 
the union-closed sets conjecture (see \cite{UCSsurvey} for a survey), so any 
approach to extend the knowledge on 
the smallest size of a possible counterexample by constructive enumeration
must focus on the subset of union-closed sets with those additional structural properties
(e.g., with small average size of the sets, without some subconfigurations 
like singletons, etc.).

Union-closed sets are closely related to {\em Moore families}. A Moore family
for a universe $\Omega_n$ is a set  of subsets of $\Omega_n$
that is closed under intersection and contains $\Omega_n$. It is easy to
see that for a union-closed set ${\cal U}$ the set ${\cal U}^c=\{\Omega_n\setminus A | A \in {\cal U}\}$
is a Moore family. For a Moore family ${\cal M}$ the set ${\cal M}^c=\{\Omega_n\setminus A | A \in {\cal M}\}$
is closed under union, but as the empty set is not necessarily contained in ${\cal M}$, it is a
union-closed set for $\Omega_n\setminus \bigcap_{A\in \cal M} A$, which is isomorphic to a union-closed set for some
$\Omega_{n'}$ with $n'\le n$.

A set ${\cal M}_n$ of representatives of Moore families (with the
canonical definition of isomorphism) for the universe $\Omega_n$ can be
obtained from sets $ {\cal R}_0, \ldots , {\cal R}_n$ of representatives
of union-closed sets containing the empty set as follows:
${\cal M}_n=\bigcup_{i=0}^n \{ {\cal U}^c  |  {\cal U}\in {\cal R}_i\}$
if the complement is in each case taken in the universe $\Omega_n$.

So union-closed sets correspond to Moore sets,
which again characterize closure operators,
and have many applications in topology, algebra and logic.

For $n<7$, several authors developed enumeration algorithms for
Moore families \cite{Moorefamilies7, Moorefamilies6,Moorefamilies5}. 
In the most advanced article, 
Colomb, Irlande, and Raynaud \cite{Moorefamilies7}
not only counted Moore families for $n\le 6$,
but they also generated representatives of 
isomorphism classes. For $n=7$, the approach is not suitable for
generating a set of representatives, and by clever use of the structure
of representatives of Moore families for $n=6$ they only determined 
the number of labeled Moore families
--- that is, without considering isomorphisms. In our algorithm we determine the number of
labeled union-closed sets resp., labeled Moore families for $n= 7$ from representatives and 
their automorphism groups of the union-closed sets for $n= 7$, resp., $n\le 7$.
This gives a very good independent test
for the implementation in \cite{Moorefamilies7} as well as for our implementation.
When computing the number of labeled Moore families for $\Omega_7$ from the
number of labeled union-closed sets with $n\le 7$, for those union-closed sets with $n< 7$, 
one must take into account that for $\Omega_7$ isomorphic copies not only occur for
the universe $\{1,\ldots ,n\}$, but for each $n$-element subset of $\{1,\ldots ,7\}$.


\section{The algorithm}

A subset $A \subseteq \Omega_n$ is represented as a number 
$b(A)$ given as the binary number $b_{n-1}\cdots b_0$ --- possibly with leading zeros --- with $b_i=1$
if $(i+1)\in A$ and $b_i=0$ otherwise.

We use an ordering of the subsets of $\Omega_n$. For $A,B \subseteq \Omega_n$
we define $A > B$ if $|A|<|B|$ (so sets with more elements are considered smaller in this order)
or $|A|=|B|$ and $b(A)>b(B)$. Whenever we refer to {\em larger} or {\em smaller} sets, we refer
to this ordering. 

The construction algorithm generates union-closed sets recursively based on
the following easy lemma:

\begin{lemma}
If ${\cal U} \not= \{\Omega_n\}$ is a union-closed set and $A$ is the largest 
non-empty element
in ${\cal U}$, then ${\cal U}\setminus \{A\}$ is also a union-closed set.
\end{lemma}

This implies that union-closed sets for universe  $\Omega_n$ can be recursively constructed
from the union-closed set $\{\Omega_n,\emptyset\}$ of smallest size by successively adding subsets of $\Omega_n$
that are larger than the largest non-empty set already present.
Of course it is not assured that adding a smaller set to a union-closed set does not violate
the condition that the set must be closed under unions.

In order to turn this into an efficient algorithm, two tests that are 
(in principle) applied to each structure generated must be very fast:

\begin{description}
\item[(i)] The test whether the set that has been constructed by adding a new set
is closed under union.

\item[(ii)] The test for isomorphisms.
\end{description}

We first discuss (i). A straightforward way to test (i) for a union-closed set ${\cal U}$ to which a new
set $A$ is added would be to form all unions $A\cup B$ with $B\in {\cal U}$ and test whether 
they are in ${\cal U}\cup \{A\}$. Although all these steps can be
implemented as efficient bit-operations, their number would slow down
the program. We make the following definition.

\begin{definition}
For a union-closed set ${\cal U}$ we 
define the reduced set $\red({\cal U})$ as follows:
$$\red({\cal U})=\{ A\in {\cal U} | A\not= \emptyset \mbox{ and there is no }A_1\not= \emptyset \mbox{ in } {\cal U}, A_1\subsetneq A\}.$$
\end{definition}

\begin{lemma}
Let ${\cal U}$ be a union-closed set for a universe $\Omega_n$ and let $A\subset \Omega_n$,
that is larger than any non-empty set in ${\cal U}$.
Then ${\cal U}\cup \{A\}$ is closed under union if and only if
$A\cup B\in {\cal U}$ for all $B\in \red({\cal U})$.

\end{lemma}

\begin{proof}
First assume that ${\cal U}\cup \{A\}$ is closed under union
and let $B\in \red({\cal U})$.
Then $A\cup B\in ({\cal U}\cup \{A\})$ and 
as $A$ is larger than $B$, we have $A\cup B \not= A$, so $A\cup B\in {\cal U}$.

For the other direction assume that 
$A\cup B\in {\cal U}$ for all $B\in \red({\cal U})$ and let 
$D\in {\cal U}$.

Choose any $D'\subset D$ so that $D'\in \red({\cal U})$. Then $A'=A\cup D'\in {\cal U}$
and therefore also $A'\cup D \in {\cal U}$ as ${\cal U}$ is closed under union, but 
$A'\cup D = A\cup D$ --- so $A\cup D\in {\cal U}\cup \{A\}$ and 
${\cal U}\cup \{A\}$ is closed under union.
\end{proof}

It would be inefficient to compute $\red({\cal U})$ each time a new union-closed set is constructed,
but as a new union-closed set  ${\cal U}'$ is constructed by adding a new smallest element $A$ to
${\cal U}$, the set  $\red({\cal U}')$ can easily be constructed from $\red({\cal U})$
by adding $A$ and removing elements that contain $A$. Nevertheless the few lines
of code testing whether the potential union-closed set is closed under union take more than 50\%
of the total running time when computing union-closed sets for $\Omega_6$, which is the largest case that can be profiled.

In order to solve problem (ii) efficiently --- that is, avoid the generation of isomorphic copies ---
we use a combination of Read/Farad{\v z}ev type orderly generation 
\cite{Fa76, Read78} and the homomorphism principle (see, e.g., \cite{B98}).

Our first aim is to define a unique representative for each isomorphism class
--- called the canonical representative --- 
and then only construct union-closed sets that are the canonical representatives of their class. We
represent a union-closed set ${\cal U}$ with $k+1$ elements $A_1<A_2<\cdots <A_k<\emptyset$ as the string 
$s({\cal U})=b(A_1),\ldots ,b(A_k)$ of numbers.
For a given isomorphism class of union-closed sets for a universe $\Omega_n$
we choose the union-closed set with the lexicographically smallest string as the representative.

It is {\em in principle} easy to test whether a given union-closed set ${\cal U}$ is the representative 
of its class by applying all
$n!$ possible permutations to ${\cal U}$ and comparing the strings. 
As $n\le 7$ this would not be extremely expensive, but due to the large number of
times that this test has to be applied, it is far too expensive to construct the union-closed sets for
$\Omega_7$. In the sequel we describe a way how this can be optimized.

In order to increase the efficiency by making it an orderly algorithm of Read/Farad{\v z}ev type, we 
use the canonicity test not only for structures we output, but also during the construction:
non-canonical structures are neither output nor used in the construction. This 
only leads
to a correct algorithm if we can prove that canonical representatives are constructed
from canonical representatives:

\begin{theorem}\label{thm:orderly}

Let ${\cal U} \not= \{\Omega_n,\emptyset\}$ be a union-closed set for the universe $\Omega_n$ that is the 
canonical representative for its isomorphism class. If ${\cal U}=\{A_1,A_2,\ldots ,A_k,\emptyset\}$
with $A_1<A_2<\cdots <A_k$ and $1\le m \le k$, then $\{A_1,A_2,\ldots ,A_m,\emptyset\}$
is also the canonical representative of its class.

\end{theorem}

\begin{proof}
We prove the result for $m=k-1$. For $k=m$ it is the assumption and for 
$m<k-1$ it then follows by induction.

Let $s({\cal U})=(s_1,\ldots ,s_{k})$. For a permutation $\Pi$ of $\Omega_n$ and a union-closed set
${\cal U}$ we write $\Pi({\cal U})$ for the union-closed set obtained by replacing each element
of a set in ${\cal U}$ by its image under $\Pi$. 
Assume that ${\cal U}'=\{A_1,A_2,\ldots ,A_{k-1},\emptyset\}$ is not the canonical representative
of its class. So there is a permutation $\Pi$ of $\Omega_n$ with $s(\Pi({\cal U}'))<s({\cal U}')$.
Let $s(\Pi({\cal U}'))=(p_1, \ldots ,p_{k-1})$ and we have $s({\cal U}')=(s_1,\ldots ,s_{k-1})$.
Let $j$ be the first position so that $p_j<s_j$. 
Let us now look at $s(\Pi({\cal U}))=(p'_1,\ldots ,p'_k)$ and let $r$ be the position
of $\Pi(A_k)$ in this string. If $r>j$ then $p'_i=p_i=s_i$ for $1\le i <j$ and
$p'_j=p_j<s_j$ --- so there is a smaller representative for the isomorphism class of
${\cal U}$. If  $r\le j$ then $p'_i=p_i=s_i$ for $1\le i <r$ and  
$p'_r<p_r\le s_r$ and again we have found a smaller representative contradicting the minimality of
$s({\cal U})$.

\end{proof}

This theorem proves that the algorithm can reject non-canonical union-closed sets and is a correct
orderly algorithm, but the cost of the canonicity test would make it impossible to determine
the number of union-closed sets for $\Omega_7$. 

For a given union-closed set ${\cal U}$ with universe $\Omega_n$
and $1\le m \le n$ we write ${\cal U}_m$ for the subset containing all sets of
size $k\ge m$ and $\Pi_m({\cal U})$ for all permutations $\Pi$ of $\Omega_n$
with the property that $\Pi({\cal U}_m)={\cal U}_m$.


\begin{lemma}\label{lem:homomorph}
Let ${\cal U}\not=\{\Omega_n,\emptyset\}$ be a non-canonical union-closed set 
for the universe $\Omega_n$ with sets $A_1<A_2<\cdots <A_k<\emptyset$, so that
$\{A_1,A_2,\ldots ,A_{k-1},\emptyset \}$ is canonical and $|A_k|=m$. Then all permutations
$\Pi$ with $s(\Pi({\cal U}))<s({\cal U})$ are in $\Pi_{m+1}({\cal U})$.
\end{lemma} 

\begin{proof}
Any permutation $\Pi$ not in $\Pi_{m+1}({\cal U})$ would by definition give
an isomorphic but different union-closed set $\Pi({\cal U}_{m+1})$. As due to 
Theorem~\ref{thm:orderly}
$s({\cal U}_{m+1})$ is minimal, $s(\Pi({\cal U}_{m+1}))$ would be larger
and therefore also the part of the string of $s(\Pi({\cal U}))$ 
describing sets of size at least $m+1$ would imply $s(\Pi({\cal U}))>s({\cal U})$.
\end{proof}

Lemma~\ref{lem:homomorph} speeds up the canonicity test dramatically:
We start with a list of all $n!$ permutations as $\Pi_n({\cal U})$.
When testing canonicity of a union-closed set with the smallest set of size $k<n$, we
only apply permutations from $\Pi_{k+1}({\cal U})$. During this application, we can 
already compute $\Pi_{k}({\cal U})$ by simply adding exactly those
permutations to the initially empty set  $\Pi_{k}({\cal U})$ that fix ${\cal U}$.
As we work with small sets, it is no problem to store and use the set of all
group elements instead of just a set of generators.

The impact is immediately clear: the number of permutations that has to be 
computed is much smaller and as soon as some $\Pi_{k}({\cal U})$ contains only the identity
--- which happens very fast --- no canonicity tests have to be performed, so that 
the total time for isomorphism checking is only about 7\%
of the total running time when computing union-closed sets for $\Omega_6$.

\subsection{The implementation}

The algorithm was implemented in the computer language $C$. Next to an efficient algorithm, of course also
implementation details are of crucial importance to be able to compute the results
for $\Omega_7$. We precomputed the action of all permutations on all sets, so that
they could be applied very fast and used data structures that allow to check
whether a set is contained in a union-closed set in constant time. 
Special functions were written that add sets with
only one element. As no sets of smaller size are added, no updates of the automorphism
groups are necessary and it turned out that at this stage it is also not efficient any more
to remove sets from $\red(\cal U)$ that are a superset of another element. Details
on the implementation can best be seen in the code which can be obtained from the 
authors or the {\it Journal of Integer Sequences}.

\subsection{Results}

Tables \ref{tab:ucs} and \ref{tab:moore} give the numbers of isomorphism classes of
union-closed sets and Moore families as well as the numbers of labeled structures.
Up to $5$ elements the running times are less than $0.01$ seconds. 
For $n=6$
it is $8.2$ seconds on a Xeon(R) CPU E5-2690 0 running with $2.90$ GHz
and a high load (which can make a difference for these processors). For
$n=7$ the jobs were run in parallel on different machines and some 
parts had to be divided further in order to finish, so it is hard to give precise times.
Estimating the total running time from those parts that were run on the same machine used for $n=6$,
the total time on this type of machine should be around $10$ to $12$ CPU-years.

\bigskip

\begin{table}[H]
\begin{center}
\begin{tabular}{l|c|c}
$n$ & union-closed sets & labeled union-closed sets \\
\hline
 1 & 1  & 1  \\
 2 & 3  & 4  \\
 3 & 14  & 45  \\
 4 & 165  & 2,271  \\
 5 &  14,480 & 1,373,701  \\
 6 &  108,281,182 & 75,965,474,236  \\
 7 & 2,796,163,091,470,050 & 14,087,647,703,920,103,947 \\
\end{tabular}
\caption{\label{tab:ucs} The number of union-closed sets  (sequence \seqnum{A108798})
and labeled union-closed sets (sequence \seqnum{A102894}).}
\end{center}
\end{table}

\bigskip

\begin{table}[H]
\begin{center}
\begin{tabular}{l|c|c}
$n$ & Moore families & labeled Moore families\\
\hline
 1 &  2 &  2  \\
 2 &  5 &  7  \\
 3 &  19 &  61  \\
 4 &  184 &  2480  \\
 5 &  14,664 &  1,385,552  \\
 6 & 108,295,846  &  75,973,751,474   \\
 7 & 2,796,163,199,765,896 & 14,087,648,235,707,352,472\\
\end{tabular}
\caption{\label{tab:moore} The number of Moore families (sequence \seqnum{A193674}) and labeled Moore families (sequence \seqnum{A102896}).}
\end{center}
\end{table}

\bigskip

A union-closed set on $n$ elements is called {\em sparse} if the average number of
elements in a set --- not counting the empty set --- is at most
$\frac{n}{2}$. For union-closed sets that are not sparse, the union-closed sets conjecture is
trivially true. Table~\ref{tab:ucssparse} gives the number of sparse union-closed sets.
These numbers were computed once by filtering all union-closed sets and once by
an independent implementation using special bounding criteria described in \cite{ucsthesis}.
Bruhn and Schaudt \cite{UCSasymptotic} give indications that sparse union-closed sets 
make up only a vanishingly small part of all union-closed sets. The numbers
of sparse union-closed sets for $n\le 7$ support this. Nevertheless also for sparse
union closed sets the numbers seem to grow so fast that even with specialized algorithms
complete enumeration will not be possible for sizes of the universe that are
interesting for the union-closed sets conjecture.

\bigskip

\begin{table}[H]
\begin{center}
\begin{tabular}{l|c}
$n$ & sparse union-closed sets \\
\hline
 1 &  0 \\
 2 &  0 \\
 3 &  0 \\
 4 &  2 \\
 5 &  27 \\
 6 &  3,133 \\
 7 & 5,777,931 \\
\end{tabular}
\caption{\label{tab:ucssparse} The number of sparse union-closed sets (sequence \seqnum{A299116}).}
\end{center}
\end{table}

\bigskip

\subsection{Testing}

The second author \cite{ucsthesis} developed an independent implementation of the algorithm together
with special bounding criteria for sparse union-closed sets. The implementation
was used to generate all isomorphism classes 
of union-closed sets for $\Omega_1, \ldots ,\Omega_6$, and --- using special bounding
criteria --- to confirm the numbers of  isomorphism classes 
of sparse union-closed sets  for $\Omega_7$.

A further and independent confirmation of the numbers for $\Omega_1, \ldots ,\Omega_6$
and also an independent confirmation for $\Omega_7$ was obtained by computing the
number of labeled structures corresponding to each representative from the size
of the automorphism group. Note that as the size of the automorphism group is
known in the algorithm anyway, the additional cost for this test can be neglected. From this
we computed the number of (labeled) Moore families and got complete agreement
with \cite{Moorefamilies7} for $\Omega_1, \ldots ,\Omega_7$. Due to the completely 
different approaches this makes implementation errors leading to the same incorrect results
in both cases extremely unlikely.





\section{Acknowledgments}

We thank Craig Larson for introducing us to these interesting structures
and for interesting discussions.
Furthermore, we thank Anastasiia Zharkova and Mikhail Abrosimov.
The first version of this algorithm was intended as a hands-on example to illustrate
isomorphism rejection techniques in a course they attended
at Ghent University. Without their visit, this algorithm might not have been developed.


\begin{thebibliography}{99}

\bibitem{B98}
G.~Brinkmann, Isomorphism rejection in structure generation programs,
in P.~Hansen, P.W. Fowler, and M.~Zheng, editors, \emph{Discrete
  Mathematical Chemistry}, \emph{ DIMACS Series on Discrete
  Mathematics and Theoretical Computer Science} \textbf{51} (2000), 25--38. 

\bibitem{UCSsurvey}
H.~Bruhn and O.~Schaudt, The journey of the union closed sets conjecture.
\emph{Graphs and Combinatorics} \textbf{31(6)} (2015), 2043--2074.

\bibitem{UCSasymptotic}
H.~Bruhn and O.~Schaudt, The union-closed sets conjecture almost holds
for almost all random bipartite graphs. \emph{European J. Combinatorics}
\textbf{59} (2017), 129--149.


\bibitem{Moorefamilies7}
P.~Colomb, A.~Irlande, and O.~Raynaud, Counting of {M}oore families for $n=7$,
in \emph{Formal Concept Analysis}, Vol.~5986 of \emph{Lecture Notes
  in Computer Science},  Springer, 2010, pp.~72--87. 

\bibitem{ucsthesis}
R.~Deklerck, Een constructief algoritme voor union closed sets,
Master's thesis, Ghent University, 2016.

\bibitem{Fa76}
I. A. Farad{\v z}ev, Constructive enumeration of combinatorial objects,
\emph{Colloques Internationaux C.N.R.S. No.~260 --- Probl\`emes
  Combinatoires et Th\'eorie des Graphes, Orsay}, 1976, pp.~131--135.

\bibitem{Moorefamilies6}
M.~Habib and L.~Nourine, The number of {Moore} families on $n=6$,
\emph{Discrete Math.} \textbf{294} (2005), 291--296.

\bibitem{Moorefamilies5}
A.~Higuchi, Lattices of closure operators, 
\emph{Discrete Math.} \textbf{ 179} (1998), 267--272.

\bibitem{Read78}
R. C. Read, Every one a winner, \emph{Ann. Discrete Math.}
\textbf{2} (1978), 107--120.

\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}: Primary
05-04; Secondary 05A15

\noindent \emph{Keywords:} enumeration, Moore set, union-closed set.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A102894},
\seqnum{A102896},
\seqnum{A108798},
\seqnum{A193674}, and
\seqnum{A299116}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received November 20 2017;
revised version received  February 5 2018.
Published in {\it Journal of Integer Sequences}, February 8 2018.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in


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