THE HIGGS FACTORIZATION OF A
GEOMETRIC STRONG f4AP
Douglas G. Kellyl
and
Daniel Kennedy2
Institute of Statistics Mimeo Series #1008
May, 1975
1
2
Department of Mathematics, Department of Statistics, and Curriculurn in
Operations Research and Systems Analysis, University of North Carolina at
Chapel Hill
Department of Mathematics, The Baylor School, Chattanooga, Tennessee
ABSTRACT
The Higgs faotopization of a strong map between matroids on a fixed set is
that factorization into elementary naps in which each matroid is the Higgs lift
of its successor.
This factorization is characterized by properties of the
modular filters which induce the elementary maps of the factorizations in two
different ways.
It is also
sho~m
to be minimal in a natural order on factori-
zations arising from the weak-map partial order on matroids.
The notion of essentiaZ nulZity of flats of a matroid is introduced; this
quantity is nonzero precisely for the cyclic flats, and is shown to be related
to the minimal flats of the modular filters inducing the maps of the Higgs
factorization.
AMS Subject Classification:
Key Words and Phrases:
0535
liatroid, combinatorial geometry, strong map, elementary
strong map, strong map factorization, Higgs lift.
1.
Introduction.
The factorization
-+
between matroids on the same set
G :: G of a strong map
n
H
-+
G
X into elementary maps, or strong maps that
reduce the rank by one. was first discovered by Higgs ([9]; see also [2])
using a construction we call the Higgs lift of the map.
T. Brown [1] later
studied a special type of elementary map, called'11F-products" by him and
principal maps in a subsequent paper of Dowling and Kelly [7].
canonical strong maps
B -+ G (B
is the free matroid on
Matroids whose
X) can be factored
into principal maps are shown to be the duals of transversal rnatroids.
In a
second paper [8]. Dowling and Kelly investigated general elementary strong
maps between matroids on the same set, using extensively the notions of modular
cuts and modular filters introduced by CTapo [3], as well as the dual notion of
modular ideals.
Kennedy [11] later studied strong map factorizations, intro-
ducing the notion of the major of a factorization, and proving several results
about the Higgs factorization. in which every matroid
the strong map
H -+ G. 1 .
J+
G.
J
is the Higgs lift of
This factorization was studied also in [8].
The work in [7]. [8]. [10], and [11] vas partly motivated by Higgs I tantalizing notion of the essential flats of a matroid.
These are the flats which,
as submatroids, are truncations of matroids of higher rank, and thus whose
existence as flats cannot be predicted from the flats they contain.
The
essential flats of a matroid, together with their ranks. determine the matroid;
this was first noticed by Higgs, according to Crapo [4]; for a proof see [8].
Dowling and Kelly [8] noticed an apparent connection between factorizations
of the canonical strong map
B -+ G and essential flats of G. and conjectured
-2that every matroid admits a "proper factorization".
This is one in which the
minimal flats of the modular cuts determining the elementary maps are precisely
the essential flats of
flat, where
e(A)
G, each flat
A appearing
e(A)
times as a minimal
is the largest difference in rank between
whose truncation is
A and a matroid
A.
This conjecture was shown by Kennedy [10] to be false; his counterexample
is the rank-4 geometry on eight points
(rank-3 flats) are
a, b, c, d, e, f, g, h , whose copoints
abcd, crlef, efgh, abgh, cdgh,
contained in one of these.
It appears as
and all 3-subsets not
Ill, 8, 28, 41w,'/1 in Crapo's
catalog [5], and in affine 3-space is "pictured" as follows (although in fact
it is not representable):
a
f
b
In this paper we introduce the notion of the essential nullity
N(A)
of
a flat, a quantity that is positive if and only if the flat is cyclic, i.e.
isthmus-free.
(All the essential flats of a matroid are cyclic.)
We show that
"proper factorizations 9i exist fCir these flats; specifically, in the Higgs
factorization of
B + G , the minimal flats of the modular cuts are exactly
the cyclic flats of
minimal flat.
G, and. each such flat
A appears
N(A)
times as a
(The result is actually proved in greater generality for an
arbitrary strong map
f: H + G , lvith essential nullity generalized to
"essential f-nullity.")
-3-
In addition, we prove some other results about the Higgs factorization of
an arbitrary strong map, including its characterization as the only factorization in which the modular cuts are nested.
Many of these results can be found
in a slightly different context in [8].
The next section concludes with a more precise statement of our main
results.
2.
Definitions and Statements of Results.
In this paper we will consider only matroids on a fixed finite set
size
k
contains
A matroid
G will be viewed as a family of subsets of
X which
X itself and which forms a geometric lattice under inclusion.
members of this lattice are the flats of
which every subset of
G is denoted
The
G; we will use the term G-fZats when
there are more than one matroid under consideration.
of
X of
The free matroid, in
X is a flat, will be denoted by
r G • and closure by A ~
B.
The rank function
A.
The reader is assumed to be familiar with the following notions for
matroids (combinatorial pregeometries): geometric lattice, rank, nullity,
closure, circuit, spanning set, dual matroid.
An introduction to the subject
of matroids can be found in [6]; further elaboration of the results collected
below can be found in [8].
He define a relation anong matroids on
strong map if every G-flat is an H-flat.
X by saying that
H -+ G is a
(This curious way of referring to a
relation comes from the fact that every G-flat is an H-flat if and only if the
identity on
of Hand
X induces a strong map (see [9]) between the geometric lattices
G.) If H -+ G is a strong map, then
reH)
~
reG) , with equality
-4if and only if H = G.
If
= reG)
r(H)
+
1 , we then say that
H + G is an
elementary strong map. or simply an elementary map.
Subsets
rH(A) + rH(B)
A
B of X are called a modular pair in the matroid H if
a~d
= rH(AnB)
+
A family P of subsets of
rH(AuB) .
H-flats) is an order filter if whenever
H-flats) and
B 2
A
€
P •
then
B
A and
B are subsets of
Crespo
X
F •
€
H is an order filter
A modular out of a matroid
A n B € H whenever
A and
X (resp.
M of H-flats such that
B are a modular pair in M.
H is an order filter of subsets of
A modular filter of
X with the same property.
We call a
X
modular filter proper if it is nonempty and does not contain all subsets of
The connection between modular cuts and modular filters is
a modular cut
H • the family
F
M
filter; given a modular filter
F
of sets spanning members of
the family
MF of flats in
Mp are inverses.
FM and F t-+There is one-one correspondence between matroids
modular cut; and the maps
MI-l-
G
sin~le:
l~
given
is a modular
is a
F
for which H+ G is
an elementary map and proper modular filters of H. as follows.
If
F
is a
proper modular filter. then the rank function of G is given by
if A
€
F
(1)
if A ¢ F .
Similarly, if H -+ G is elementary. then the sets
rG(A)
= rH(A) -
1
forms a modular filter in
modular filter assooiated with the map
F
writing H-+-G.
We say that
F is the
H -+ G , and we denote the situation by
as in the previous paragraph. then it follows
that the flats of H that are not flats of
but covered by flats in
H
A for which
M
G are precisely those not in
M
-5H + G is a strong (resp. elementary) map if and only if G*
strong (resp. elementary) map.
+
H*
is a
The relations between the modular filters
associated with these maps can be found in [8], but need not concern us here.
We will also need the notion of weak maps between matroids of the same
rank on
X.
We say that there is a weak map from
in the weak-map partiaZ order, if
X..
rH(A)
~
rH,(A)
X, then H
Let
of the map
~
HI
if and only if H*
H + G be a strong map with
r(H)
H + G if H + L is strong and
~
H
~
HI
are matroids of the same
H'* .
reG)
>
If' , or that
for all subsets of A of
A result of Lucas [12] is that if Hand H'
rank on
H to
L is a Zift
A matroid
L + G is elementary.
Lifts
always exist; the Higgs lift is that matroid whose flats are the flats of G
together with those flats
A of H for which
rH(A)
= rG(A)
.
We shall deal
enough with the above situation to give i t a name: we will say" H + L
a lifted (or Higgs- lifted) strong map" to signal that
£...
G is
H + L is strong and
L + G is the elementary map associated to the modular filter
F of
L
It
follows from results in Section 7 of [8] that if H + L ~ G is a lifted
strong map, then it is a Higgs-lifted strong map if and only if
Another situation we will deal with is that of a factorization of a strong
map
H + G , namely, a sequence
F
~ G = G
(2)
n
of elementary maps.
Higgs lift of the map
The Higgs factorization is the one in which
for
H + G.
nullity of the map H + G.
J
It
j
= 1,
... ,n
The integer
G.J- I
n
is the
is the
will sometimes be convenient to denote the map
-6-
H + G by f. in order to use the notation nfCA)
quantity is called the f-nullity of A ; when
x • it reduces to the nullity of
G).
f
nfCA)
nfCA)
~
rHCA) - rGCA) .
H is the free matroid
Cand n
This
B on
is the nullity of the matroid
It is easy to check that for any strong map
CAS B implies
of
A
for
f,
is nondecreasing
f
nf(B) ) and that because of CI), for any factorizatio~
TI
equals the number of modular filters among
FI , ... ,Fn of
which A is a member.
We will also denote the map
convenience we will write
n.
J
H + G.
J
instead of
in a factorization by
11
and
f.
= nfCA)
Because n f
J
J
We define the essential f-nuUity of a set
NfCA)
r.
f. ; and for
J
instead of r G.
.
J
A .s X by
- max { B: B is a G-flat properly contained in A} .
is nondecreasing, the maximum is attained at some flat
B covered
by A:
Vfuen
NCA)
H = B.
To say that
NfCA)
NCA)
is called the essential nullity of A and denoted
= 0 for a flat A is the same as saying that A
covers a flat of the same nullity, i.e. that
rank
A has an isthmus (a subflat of
1 \..hose points are in no circuit of A).
So the flats of positive
essential nullity are just the cyclic flats Ci.e. flats that are unions of
circuits).
At last the main results of this paper can be stated.
1.
NfCA)
is the number of modular filters among
Higgs factorization in which A is a minimal flat.
FI ,
They are
,Fn
CCorollary 2)
in. the
-7-
2.
A factorization (2) is the Higgs factorization if and only if
Fl 5: F2
3.
.s ...
.s Fn'
(Theorem 3)
In the Higgs factorization,
F
= {A:
j
nf(A) > n-j} ,
j
= 1,
... ,n .
(Theorem 4)
4.
G
n
Let (2) be the Higgs factorization and
=g
any other factorization of H + G
G. s G' .
J
5.
G* + H*
3.
= Gn *
G*
+
j
= 1,
n-l
+
... ,n ,
(Theorem 8)
G _ * + •.. + Gl * + GO*
n l
= H*
is the Higgs factorization of
if (2) is the Higgs factorization of H + G
Properties and Construction of the Higgs Factorization.
Proposition 1.
by
Then for
in the weak-map partial order.
J
H = GO + G'l + •.. + G'
f
Let
H+ L
H + L by
and
F
"-+
fi .
G be a Higgs-lifted strong map; denote
Then for any subset
+ 1
if
A
A of
H+ G
X,
F
is a minimal flat of
if not .
Proof.
It suffices to prove the assertion for the case in which
G-flat, so that
Case 1.
because
nfCB)
all
Let
A is in
= n fl
(B)
NfCA)
= n.cCA)
.l-
- max infeR): B ~ A}
A be a minimal flat of
F
and if
Thus
nf(A)
B i A , then
B
.
Then
F
A is a
nf(A)
is not in
= n f , (A)
F
+ 1
so
This is true for
B ~ A , so
Case 2.
Suppose
is true for any flat
A is not in
B.s A
F
Then
nf(A)
= Nfl (A)
= nf,(A)
, and the same
-3-
Case 3.
Suppose A is in
B ~ A with
there is a flat
A is in
F.
if
B in
F.
We have
max {nfCB): B ~ A}
Moreover,
B in
both attained at flats
F but is not a minimal flat of
F
and
nfCA)
+
Then
I
because
max {nfICB): B ~ A}
are
Cnot necessarily at the same flat), because
= nf,(B) = O.
B is not in F, then nf(B)
(This is the only place in
the proof where properties of the Higgs lift are used.)
Corollary 2.
= nflCA)
F.
And if
B
is in
F
If
F
~ G = G
(3)
n
is the Higgs factorization of the strong map
FI ,
number of modular filters among
Theorem 3.
Proof.
The factorization
But
J
J+
is the Higgs factorization of
(3)
<
~
r. ICA)
J+
PI
G. I
So
Then
A is in
and so if A is in
r·CA)
J
rH(A) } which implies that
Conversely, suppose
Nf(A)
is in
A
is the
in which A is a minimal flat.
Suppose (3) is the Higgs factorization.
r. (A) < fH(A).
r. I (A)
, Fn
f: H + G , then
P.J+ I
Thus
f:
H + G
F.
iff
J
if
F. , then
J
F.
c
J - F.J+ 1
F2 So ••• So Pn ; we show that A is in F. iff
J
If A is
is the Higgs lift of f.: H+G.
J
J
JAnd if A is not in
in F. , then r.(A) = r. leA) - 1 < r j _1 C1\) ~ rH(A)
J
J
JFj , then A is not in FI , F2 ,
,F j , so that rj(A) = rHeA) .
Theorem 4.
If (3) is the Higgs factorization of
F
j
= {A: n£CA) > n-j},
j
F: H +
= 1, ... ,n .
G , then
-9-
Proof.
= r j _1 (A)
rG(A)
F.
If A is in
then
J
is in
A
F. ,
J
• ••
,F
so
n
Thus
(n - j + 1) •
nf(A) > n - j .
On the other hand, if A is not in
F1 ,
...
, F.J- l' F. ; so
J
r n (A) + n-j , Le.
4.
rH(A)
= r.J (A)
s r. leA)
J+
+
1 s r. 2(A)
+
J+
2 s
S
.
nf(A) s n - j
Minimality of the Higgs
Fj , then A is not in
Fa~torization.
The results of this section appear in a slightly different context in [8],
using modular ideals and free quotients, which are dual to modular filters and
Higgs (free) lifts.
Lemma 5.
Let
L
F
-+
F and
L'
pi
G be elementary strong maps.
~
in the weak-map partial order if and only if
Proof.
F
~
Then
L s L'
Fi .
For an arbitrary pair of elementary maps as given, the following are
true.
(1)
rL(A)
(2)
If A is in both
= r L, (A)
(3)
and
=0
F but not
thus
i.e. iff
L s L'
F
~
F
F' .
iff
TL(A)
rL(A) - TG(A)
F' • then
rL(A) >
Similarly, if A is in
Thus
or in neither of
Fl
and
Fi , then
.
If A is in
TLI(A) - rG(A)
F
Fi
T
L
I
while
L\ (A)
but not
~ T.
=1
(A)
F, then
for all
rL(A)
<
TLi(A)
A , Le. iff
F' - F = ~ ,
-10-
Proposition 6.
H ~ G , then
Proof.
If
L
thus
= rG(A)
L
~
be the associated modular filters in
pi
= {A:
F
rH(A)
+ 1 , so
rG(A)}.
>
Proof.
Now if A is in
rH(A) > rG(A) , so
A is in
LI
Land
,
F' , then
Thus
F.
FI SF, and
L' .
FZ
and H ~ LZ --+ GZ be Higgs-lifted strong
Proposition 7.
maps.
any lift of a strong map
Ll
L' .
F and
Let
respectively.
r LI (A)
~
L is the Higgs lift and
If
G
l
~
GZ , then
r G (A) - r (A)
Gz
I
Ll
1.
.
2
for all
0
~
~
r G. (A)
A
, and
F.1
if A is in
+ I
1.
=
r L . (A)
if not .
r . (A)
G
1.
1.
fIe need to show r L (A)
1
r
r L (A)
1
r L (A) =
2
r L (A)
~
for all
'{A}- r
G1
A for
A
implies
€
FZ - FI
hypothesis.
Theorem 8.
So
if
G (A)
2
But
A
E:
F n F2
I
if
A E: FI - F2
r,...\.1 (A) - r G (A) - I
Z
1
if
A
GI
(A)
r G (A)
2
~nlich
r
+
Gz
FI - F2
r
(A)
GZ
>
F2 - FI
(A) > rH(A)
r
Gl
and
F2 - F1
E
= ~.
r
Gl
But
(A)
A ;.. F
I
or
I
r
So the result will follow if we show
of all sets
A.
2
F2 - FI
If (3) is the Higgs factorization and
i
is the set
= rH(A) , so that
=~ .
n
F
2
.
(A) , which is impossible since
F I
n-l ~ G
U
=G
G
l
~
GZ by
-11-
is any other factorization of the strong map
weak-map partial order,
Proof. Certainly Gn
j
= 1,
H -+ G , then
~
G ; and
Suppose inductively that
Gk_l
Consider the following three lifted strong maps:
Gk_ ' .
1
I
in the
Gn-l <- Gn-l '
Proposition 6.
~
G.
J
... ,n .
Gnl , since both equal
~
G.
J
Gk
~
by
G ' . we show that
k
'
G I
k-l
.
Lk I is the Higgs lift of the map H -+ Gk I
We have G
k
,
Proposition 7, since Gk _l and L
k'
k are Higgs lifts. And L'
Proposition 6, since L I is the Higgs lift of H -+ Gk i
k
where
Corollary 9.
Let (3) be the Higgs factorization of H -+ G.
G*
= G*n
-+ G
*.:~ •.. -.-+ G ;\'., -+ G *
n-1
1
0
is the Higgs factorization of G*
Proof.
-+
H* .
-+
H*.
~
~
Gk I by
G _ I by
k l
Then
= H*
Let
be any other factorization of
H
= GO
I
-+
GIl
is a factorization of H -+ G .
G*
-+ ••• -+
Then
Gn-,I'
By Theorem S',
-+
Gn i
j = 1, ... ,n .
G. ::; G. i
J
J
the theorem of Lucas mentioned in Section 2 above,
The result follows.
=G
G.*
J
~
G.'* , j
J
= 1,
By
... ,n
-12-
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II]
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Missouri at Kansas City, Mo .• 64110.
[2]
Brylawski, T.H., A decomposition for combinatorial geometries, Trans. Am.
f1ath. Soc.~ Vol. 171, (1972), pp. 235-282.
(3]
Crapo, H. H., Single-element extensions of matroids, J. Res. NatZ. Bur.
Standards Vol. 69, Section B, (1965), pp. 55-65.
j
[4]
Crapo, H.H.', Erecting geometries, Proceedings of the Second Chapel, HiU
Conference on Combinatorial Mathematics and its AppZications,
Department of Statistics, University of North Carolina at Chapel
Hill, N.C., (1970).
[5]
Crapo, H.H., A catalog of tombinatorial geometries, Preprint, University
of Water~oo, Waterloo, Ontario, Canada, (1969).
[6]
Crapo, H.B., and G.C. Rota, On the Foundations of Combinatorial Theory:
Combinatorial Geometries (preliminary edition), M.I.T. Press,
Cambridge, Mass., (1970).
[7]
Dowling, T.A. and D.G. Kelly, Elementary strong maps between combinatorial
geometries, Rendiconti del. CoUoquio Internaziona'le su'l Tema Corribinatorie, Rome, Italy, (1973).
[8]
Dowling, T.A. and D.G. Kelly, Elementary strong maps and transversal
geometries, Disc. Math, Vol. 7, (1974), pp. 209-224.
[9]
Higgs, D.A., Strong maps of geometries, J. Comb. Theory, Vol. 5, (1968),
pp. 185-191.
[10]
Kennedy, D., Factorizations and majors of geometric strong maps, Ph.D.
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[11]
Kennedy, D., Majors of geometric strong maps, to appear in Disc. Math.
[12]
Lucas, T.D., Properties of rank-preserving weak maps, Bull,. Amer. Math.
Soc." Vol. 80, (1974), pp. 127-131.
'