GROUPOID CARDINALITY AND EGYPTIAN
FRACTIONS
JULIA E. BERGNER AND CHRISTOPHER D. WALKER
How do mathematicians count? For a finite set, the idea is straightforward, and many mathematical objects, especially algebraic ones, are
just sets with extra structure. Given a finite group, the first thing we
want to know is how many elements it has. However, often we need
to move beyond simple counting. When sets are infinite, we can still
distinguish between different cardinalities, but sometimes we want an
alternate way of measuring them. We are more interested in the dimension of a vector space than its cardinality; for topological spaces
we use invariants such as the Euler characteristic.
However we assign a number to a mathematical object, the next
natural question is one of realization. Can we find an example of our
mathematical structure which attains a given number? For any positive
integer n, we know that there is at least one group of order n, for
example, the cyclic group with n elements.
The structure we look at in this paper is that of a groupoid, a generalization of a group. Taking the order of a groupoid in the same way
Date: April 2, 2013.
1
2
J.E. BERGNER AND C.D. WALKER
that we take the order of a group is not the most interesting way to
count it. An alternative definition of the cardinality of a groupoid, recently given by Baez and Dolan [1], assigns a positive rational number
to any finite groupoid. In nice cases, infinite groupoids also have this
kind of cardinality, given by a positive real number.
What about realization? Given any positive real number, is there a
groupoid with that cardinality? We show that a positive answer to this
question can be obtained by reducing it to another realization question
in number theory, whether any positive real number has an Egyptian
fraction decomposition.
Groups and groupoids
We can think of a group via a picture such as the one in Figure 1(a),
where the elements of the group are represented as arrows starting and
ending at a center dot. The arrows go in both directions, indicating
that each element has an inverse. We can think of these arrows as
functions with a common domain and range, so they compose in any
order.
GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS
$½ ¥z
:•
D Zd
½ ¥
*½ ¥
4•
D Z
•D Z tj
(a) A group.
3
(b) A groupoid.
Figure 1.
Why, however, do the arrows have to start and end in the same
place? Instead, we can draw more general pictures, as in Figure 1(b),
to visualize groupoids. Arrows compose whenever the range dot of one
arrow is the domain dot of the other, and we require this operation
to be associative. Such a structure on the set of arrows is a partially
defined operation, since not every arrow can be composed with any
other arrow. Each dot has an identity arrow, and all arrows have
inverses with respect to these identities. (For simplicity, we do not draw
identity arrows.) A groupoid may have many different components,
or collections of dots which are not connected to one another by any
arrows.
For example, an equivalence relation on a set defines a groupoid.
There is a dot for each element of the set, and there is a unique arrow
from a dot to any equivalent dot. The components of this groupoid are
exactly the equivalence classes.
Given a dot • in a groupoid, its automorphisms (arrows starting
and ending at that dot) form a group Aut(•). A basic fact about
4
J.E. BERGNER AND C.D. WALKER
groupoids is that any two dots in the same component have the same
automorphism group. Two groupoids are equivalent if they have the
same number of components and if the automorphism group of each
component of one groupoid agrees with the automorphism group of the
respective component of the other. Notice that equivalent groupoids
need not have the same number of objects, only the same number of
components.
Groupoid cardinality
Recall that the order of a finite group G is the number of elements
of G as a set, denoted #G. We could define the order of a groupoid
similarly, by counting the number of arrows, but then we could have
equivalent groupoids with different order. In particular, a groupoid
with a finite number of arrows can be equivalent to one with infinitely
many arrows, obtained by adding more isomorphic objects to a given
component. To remedy this problem, Baez and Dolan [1] provide a new
method for counting a groupoid. They define the groupoid cardinality
of a groupoid G to be
|G| =
X
[•]∈G
1
,
#Aut(•)
whenever it is defined, where [•] denotes a component of G.
GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS
5
There are many groupoids to which this definition does not apply, for
example if at least one of the automorphism groups is infinite. However,
if a groupoid has infinitely many components and all the automorphism
groups are finite, the cardinality is still defined if the resulting infinite
series converges.
Let’s look at some examples. If G is a group, then |G| =
1
. Two
#G
more interesting examples are depicted in Figure 2.
½ ¥
•D Z
$½ ¥z
:•
D Zd
ª
(a)G q H.
©
¾ ¤
¹
¸
• •
•
• •
(b) A more complicated example.
Figure 2.
If G and H are groups, then taking them together gives a two-component
groupoid (see Figure 2(a)), called their disjoint union, and denoted by
G q H, and we can compute |G q H| =
1
1
+
.
#G #H
Still more interesting is the groupoid G in Figure 2(b). Recall that
we only draw non-identity automorphisms, so the two-dot components
each have a trivial automorphism group, and the single-dot component
in the middle has an automorphism group of order 2. Therefore, the
groupoid cardinality is 1 + 1 +
1
5
= .
2
2
The cardinality of a groupoid must always be a positive number,
since groups have positive order. But how interesting can these numbers be? Let E be the groupoid with objects all finite sets and arrows
6
J.E. BERGNER AND C.D. WALKER
all the possible the isomorphisms between them. Since two finite sets
are isomorphic if they have the same number of elements, this groupoid
has one component for each natural number n ≥ 1. Since the automorphism group of a set with n elements is the symmetric group Sn ,
the cardinality of E is
|E| =
X
[•]∈E
X 1
1
=
= e.
#Aut(•) n∈N n!
So, we see that more interesting numbers result from groupoid cardinality than from group order. Hence we ask: is every positive real
number the cardinality of some groupoid?
We can obtain whole number cardinalities by taking n dots with no
non-identity arrows. For a rational number of the form
Z/n. Any rational number of the form
1
we can use
n
1
1
+
is also easy, since we can
n m
take a disjoint union of groups such as (Z/n) q (Z/m). In fact, for any
rational number
m
, we can take (Z/n) q · · · q (Z/n), with m copies
n
of Z/n. Finally, for a real number, consider the convergent series of
the fractions in its decimal expansion; take the disjoint union of all the
groupoids giving these fractions. So, the answer to our question is yes.
We find, however, the case of
m
to be unsatisfying, since we have
n
merely repeated the same group over and over again. Can we instead
find a groupoid with a given cardinality, such that no two components
GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS
7
of the groupoid have the same cardinality? In other words, can we
write any rational number between 0 and 1 in the form
m X 1
=
,
n
ni
i
where the positive integers ni are distinct? This question leads directly
to the study of Egyptian fractions.
Egyptian fraction decompositions
The ancient Egyptians had a curious way of working with rational
numbers [14]. They had notation for unit fractions, those with numerator 1, but they did not have notation for more general ones. When
they needed to represent such a number, they did so by taking sums
of unit fractions with no repeated summands. Such a representation
of a rational number is called an Egyptian fraction decomposition. For
example,
5
1 1
= + . This notation might seem cumbersome, but sup8
2 8
pose you have 5 muffins to divide among 8 people. You could measure
each muffin carefully and give each person exactly
5
of one, but it is
8
far easier to give each person half a muffin, then divide the remaining
muffin into 8 pieces, one for each person. Everyone has now received
5
1 1
= + of a muffin.
8
2 8
Were the Egyptians limited by this method? Can any rational number be written as the sum of finitely many distinct unit fractions? The
8
J.E. BERGNER AND C.D. WALKER
answer is yes. In fact, we can say something even stronger: every
rational number has infinitely many distinct Egyptian fraction decompositions.
Let us take existence for granted first, and show that, given any
Egyptian fraction decomposition, we can find another. For example,
1 1
3
1 1
1
3
= + but also = + + . We get the second decomposition
4
2 4
4
2 5 20
1
1
1
by applying the splitting algorithm: =
+
.
n
n + 1 n(n + 1)
More generally, let
q=
1
1
+ ··· +
d1
dn
be an Egyptian fraction decomposition, with d1 < · · · < dn and apply
the splitting algorithm to the last term. For dn > 1, we have dn <
dn + 1 < dn (dn + 1), so the terms of the sum
q=
1
1
1
1
+ ··· +
+
+
d1
dn−1 dn + 1 dn (dn + 1)
are all distinct and hence we have obtained a new Egyptian fraction
decomposition for q. If dn = 1, then no previous terms in the sum are
possible, by our assumption about the denominators, so q = 1. Apply1 1
+ , which is not an Egyptian
2 2
1 1 1
fraction decomposition, but applying it once more gives 1 = + + ,
2 3 6
ing the splitting algorithm gives 1 =
which is.
GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS
9
But do Egyptian fraction decompositions even exist? Following [10],
suppose that
m
is a fraction with 1 < m < n. The strategy, which
n
goes back to Fibonacci [13], is to find the largest possible unit fraction
smaller than
m
, subtract, and repeat. As we show below, this process
n
eventually terminates. Since we always take the largest possible unit
fraction less than
m
, this procedure an example of what is commonly
n
called a greedy algorithm.
For example, consider
4
. We first notice that
13
1
4
4
4
1
=
<
<
= ,
4
16
13
12
3
so
1
4
is the smallest unit fraction less than
. We then obtain
4
13
4
1
3
= + .
13
4 52
Since
3
is not a unit fraction, we repeat the procedure and see that
52
1
3
3
3
1
=
<
<
= ,
18
54
52
51
17
and subtract to see that
1
1
1
4
= +
+
.
13
4 18 468
To prove that this process must terminate for any
m
1
1
<
<
.
d1
n
d1 − 1
m
, write
n
10
J.E. BERGNER AND C.D. WALKER
Then, we know that
m
1
=
+
n
d1
The inequality
µ
m
1
−
n
d1
¶
=
1
md1 − n
+
.
d1
nd1
m
1
n
<
implies that m <
, or m(d1 − 1) <
n
d1 − 1
d1 − 1
n. We can multiply to get md1 − m < n and rearrange to see that
md1 − n < m. In other words, the numerator md1 − n is smaller
than our original numerator m. We can then find a positive integer
d2 such that
1
md1 − n
1
<
<
which gives a remainder whose
d2
nd1
d2 − 1
numerator is strictly smaller than md1 − n. Since each numerator
is a positive integer strictly smaller than the last, the process must
eventually give a numerator of 1, terminating the algorithm.
Notice, however, that the greedy decomposition is not necessarily
the shortest Egyptian fraction decomposition. For example, applying
the greedy algorithm to
83
yields
140
83
1
1
1
1
= +
+
+
.
140
2 11 514 395780
However, this fraction can also be decomposed as
1 1 1
83
= + + .
140
4 5 7
A calculator for computing shortest Egyptian fraction decompositions
can be found at [10].
We also want to consider positive irrational numbers, which do not
have a finite Egyptian fraction decomposition. We use the fact that,
GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS
given any positive real number x0 , the sequence {an } given by an =
∞
X
has a subsequence ani such that
ani = x0 .
11
1
n
i=1
How can such a subsequence be found? Given x0 , let an1 be the unit
1
1
< x0 ≤
. Let x1 = x0 − an1 . Inducn1
n1 − 1
1
1
< xi−1 ≤
so that ani =
and xi = xi−1 − ani .
ni
ni − 1
fraction such that an1 =
tively, choose ani
Then using that ani = xi−1 − xi and taking the sum, we get
∞
X
i=1
ani =
∞
X
(xi−1 − xi ) = x0
i=1
which gives the necessary subsequence.
Notice that if x0 is rational, it is preferable to use the original greedy
algorithm to get an Egyptian fraction decomposition of finite length.
However, if x0 is irrational, then we obtain an infinite series composed
of distinct unit fractions but which converges to x0 . In any case we
have answered our question: any positive real number can be written
as the sum of distinct unit fractions, possibly infinitely many.
Implications for groupoid cardinality
The results of the previous section immediately imply two facts about
groupoid cardinality. First, any positive real number is the cardinality
of a groupoid with no two components having the same cardinality.
Second, any positive rational number is the cardinality of infinitely
12
J.E. BERGNER AND C.D. WALKER
many non-equivalent groupoids, each of which has no two components
of the same cardinality. Both results have interesting implications.
First, when working with the concept of groupoid cardinality, it is
typical to have a desired cardinality in mind, then work backward to
find a groupoid with this cardinality. As an example, we already saw
that the groupoid E of finite sets and bijections has groupoid cardinality e. We got this cardinality by considering the power series expansion
X xn
ex =
with x = 1. What if, instead, we took other values for x?
n!
n
For example, we might want a groupoid with cardinality e2 , and the
methods of this paper provide one. But, as with the case of e, using
X 2n
the power series e2 =
could suggest an alternative approach; we
n!
n
2n
could try to identify groupoids of cardinality
which have a useful
n!
description. Since the summands are no longer unit fractions, such a
groupoid would likely be more complicated than E.
Our second result presents a stark contrast between the theory of
groupoids and that of finite groups. The classification of finite simple
groups is very difficult and a significant mathematical achievement.
Because there are so many groupoids with a given cardinality, we expect
that any classification of finite groupoids would be still more intricate,
and therefore probably intractable.
GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS
13
Other approaches
In proving the existence of Egyptian fraction decompositions, we
used a particular strategy, the greedy algorithm. However, there are
many other such algorithms, and the resulting decompositions often
look quite different. Some examples are the greedy odd algorithm [12],
the Yokota algorithm [16] and the Engel series method [5]. The splitting algorithm can also be used for existence, not just for finding new
decompositions [2], and there is a method making use of continued
fractions [3]. There has been a good deal of research on each of these
methods, and some relevant results include finding decompositions of a
given length [8], [9], [15]. Other topics of interest include finding Egyptian fraction decompositions where the denominators fall within certain
intervals [7], or are bounded by some fixed number [6], or where the
decompositions are particularly short or long in length [4], [11]. These
results facilitate finding groupoids with further restrictions on the number of components or on the order of their automorphism groups.
Acknowledgments. The authors thank John Baez and the participants in the Groupoid Seminar at UCR for discussions which led to
this paper, as well as the referees and editor for numerous suggestions.
14
J.E. BERGNER AND C.D. WALKER
The first-named author was partially supported by NSF grants DMS0805951 and DMS-1105766, and by a UCR Regents Fellowship.
summary
Two very new questions about the cardinality of groupoids reduce to
very old questions concerning the ancient Egyptians’ method for writing fractions. First, the question of whether any positive real number
is the groupoid cardinality of some groupoid reduces to the question of
whether any positive rational number has an Egyptian fraction decomposition. Second, the question of how many non-equivalent groupoids
have a given cardinality can be answered via the number of distinct
Egyptian fraction decompositions.
References
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GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS
15
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J.E. BERGNER AND C.D. WALKER
Department of Mathematics, University of California, Riverside,
CA 92521
E-mail address: [email protected]
Odessa College, 201 W University, Odessa,TX 79764
E-mail address: [email protected]