Variations on the Higman’s Lemma
JAIST Logic Seminar
Dr M Benini
Università degli Studi dell’Insubria
marco.benini@uninsubria.it
August 8th, 2016

Well quasi orders
Quasi orders can be identified with the small categories having at
most one arrow between each pair of objects. In turn, quasi
orders and monotone maps form a category QOrd.
A well quasi order can be characterised in at least three different
ways, which are equivalent in classical logic with a sufficiently
strong notion of set:
Every proper descending chain is finite and every antichain is
finite;
Every infinite sequence contains an increasing pair;
Every infinite sequence contains an infinite ascending chain.
Well quasi orders identify a full subcategory of QOrd.
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Quasi orders
In QOrd:
0 = 〈 ; 〉;
1 is the discrete category with one element;
A ×B = 〈A×B; ((a ,b ),(a ,b )) : a ≤A a & b ≤B b 〉;
A +B is the disjoint union of A and B;
the equaliser of f ,g : A → B is 〈 x ∈ A: f (x) = g(x) ;≤A〉;
the coequaliser of f ,g : A → B is B/ ≈, with ≈ the minimal
equivalence relation containing (f (x),g(x)) : x ∈ A , ordered
by the reflexive and transitive closure of ([x]≈,[y]≈) : x ≤B y ;
the exponential object BA
is the set of monotone maps
A → B, where f ≤ g exactly when f (x) ≤B g(x) for all x ∈ A;
it is easy to find a counterexample showing that QOrd does
not have a subobject classifier.
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Well quasi orders and related categories
AQOrd is the full subcategory of QOrd whose objects are the
quasi orders with finite antichains;
WFQOrd is the full subcategory of QOrd whose objects are
the well-founded quasi orders, that is, those having finite
proper descending chains;
WQO is the full subcategory of QOrd whose objects are the
quasi orders having both the properties above, so this is the
category of well quasi orders.
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Well quasi orders and related categories
Being finite quasi orders, the initial and terminal objects of
QOrd lie in all these subcategories, thus they have initial and
terminal objects, too.
Also, it is immediate to see that the construction of coproducts
and equalisers in QOrd can be replicated in all these
subcategories.
On the contrary, products and coequalisers do not follow
immediately. And exponentiation is not evident at all. . .
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Products
Products in the subcategories of interest, when they exist, must
have the same shape as in QOrd because they arise from the
right adjoint of the restriction of the diagonal functor.
In WQO, finite products exist thanks to Dickson’s Lemma.
The same holds in WFQOrd, by considering each proper
descending chain in the candidate product, and constructing out
of it two descending chains in the components, whose proper
kernel must be finite, forcing the original chain to be finite too.
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Products
But, AQOrd does not have all the binary products: consider
〈N;≤〉 and the (total) lexicographic order O on the free monoid
over 0 < 1. Let xn = 0...01 be the word in O having n zeroes.
Then {xn}n∈ω is an infinite descending chain in O, so (n,xn) n∈ω
defines an infinite antichain in the product.
This counterexample is perfect: pairing elements from an
infinite ascending chain with elements from an infinite descending
chain is the only way to construct a counterexample in AQOrd.
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Coequalisers
It is immediate to see that the shape of coequalisers in AQOrd,
WFQOrd, and WQO is the same as in QOrd because the
forgetful functor to Set has a right adjoint.
Since every antichain in the coequaliser of f ,g : A → B in
AQOrd induces an antichain in B, it must be finite. So,
AQOrd has coequalisers.
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Coequalisers
But WFQOrd does not. In fact, consider 〈N;=〉, the set of
naturals with the discrete order, and the disjoint union of ω
copies of a < b < c. Let f (n) = cn+1 and g(n) = an, for all n ∈ N.
Calculating the coequaliser:
c0 c1 c2
b0
OO
b1
OO
b2
OO
···
a0
OO
=
a1
OO
=
a2
OO
Thus, {bi}i∈ω is an infinite proper descending chain in WFQOrd.
As before, combining an infinite set of finite proper descending
chains along an infinite antichain is the only way to construct a
counterexample in WFQOrd.
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Coequalisers
In fact, WQO has coequalisers. The proof is not immediate: first
one has to notice that each descending chain [ei]≈ i∈ω in the
coequaliser object of f ,g : A → B can be embedded into another
descending chain such that either g(xn) ≥Q en+1 ≥Q f (xn+1) for
each n ∈ ω, or f (xn) ≥Q en+1 ≥Q g(xn+1) for each n ∈ ω. That is,
there is sequence {xi}i in A which bounds the elements of {ei}i in
B by means of f and g.
Since {xi}i is infinite, there is a subsequence which forms an
infinite ascending chain, and it has the form {xi}i>m for some
m ∈ ω. Thus, using the fact that f , g, and the candidate
coequaliser map z → [z]≈ are monotone. . .
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Coequalisers
. . . one gets that
[g(xm)]≈
[em+1]≈
oo [f (xm+1)]≈
oo
[g(xm+1)]≈
=
[em+2]≈
oo [f (xm+2)]≈
oo
=
...
...
...
[g(xn−1)]≈
=
[en]≈
oo [f (xn)]≈
oo
=
[g(xn)]≈
=
[en+1]≈
oo [f (xn+1)]≈
oo
=
...
...
...
Then, for n m+1, [en]≈ and [en+1]≈ are equivalent. So, every
descending chain of length ω is not proper.
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Exponentiation
AQOrd, WFQOrd, and WQO do not have exponentiation.
In fact, AQOrd does not have products, so it cannot have
exponentiation.
And, in WQO (or WFQOrd), consider the set of natural
numbers with the standard ordering, the order 2 given by 0 1,
and the family {fi : 〈N;≤〉 → 2}i∈N, defined by
fi(x) =
1 if x ≥ i
0 otherwise .
Whenever i j, fi(x) fj(x) for all x ∈ N, that is fi fj, since
if x ≥ j i, fi(x) = 1 = fj(x);
if x i j, fi(x) = 0 = fj(x);
if i ≤ x j, fi(x) = 1 0 = fj(x).
Thus f0 f1 f2 ··· is an infinite proper descending chain.
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Summary
0 1 × + ⊃ eq coeq
QOrd
AQOrd - -
WFQOrd - -
WQO -
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The Higman’s Lemma
Let A∗
be the set of finite sequences over the alphabet A, and let
≤∗ be the ordering given by embedding, that is
[a1,...,an] ≤∗ [b1,...,bm]
if and only if there is an injective and monotone map
e : {1,...,n} → {1,...,m} such that ai ≤A be(i) for all 1 ≤ i ≤ n.
Lemma 1 (Higman)
〈A∗
;≤∗〉 is a well quasi order if and only if 〈A;≤A〉 is so.
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The Higman’s Lemma
One direction is obvious: when 〈A∗
;≤∗〉 is a well quasi order, so
is 〈A;≤A〉.
Categorically, consider
A∗
⊥, = 〈A∗
{⊥, };≤∗ (⊥,x),(x, ) : x ∈ A∗
{⊥, } 〉 ,
the well quasi order obtained by adding a global maximum and
minimum. Let f : 〈A∗
;≤∗〉 → A∗
⊥, be
f ([x1,...,xn]) =



[x1] if n = 1
⊥ if n = 0
otherwise.
The equaliser of f and the canonical inclusion 〈A∗
;≤∗〉 → A∗
⊥, is
isomorphic to 〈A;≤A〉. Thus, 〈A;≤A〉 is a well quasi order.
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The Higman’s Lemma
The second step is to show that
Proposition 2
〈A∗
;≤∗〉 is a well founded quasi order if and only if 〈A;≤A〉 is a
well founded quasi order.
The proof goes by proving that a suitable coequaliser can be
constructed such that it is isomorphic to 〈A∗
;≤∗〉.
We add one isolated element ∗ to 〈A;≤A〉, and then we consider
the collection of sequences of length n over this augmented quasi
order. By ordering them pointwise, we get another quasi order
〈An
∗;≤×〉. And, when 〈A;≤A〉 is well-founded, 〈An
∗;≤×〉 is
well-founded, too.
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The Higman’s Lemma
Thus, 〈P;≤P〉 = i∈ω〈Ai
∗;≤×〉 is a well-founded quasi order, since
each proper descending chain lies in a single component of the
disjoint union, so it is finite.
Consider f ([x1,...,xn]) = [y1,...,ym] where [y1,...,ym] is the
maximal subsequence of [x1,...,xn] containing no ∗’s. It is clear
that f is monotone.
Calculating the coequaliser of f and the identity of 〈P;≤P〉 in
QOrd, we see that it is well-founded.
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The Higman’s Lemma
For the third step, showing that if 〈A;≤A〉 is a well-quasi order,
then 〈A∗
;≤∗〉 has finite antichains, we may follow the standard
proof by Nash-Williams.
In fact, a bad sequence is a ‘relaxed form’ of infinite antichain.
The third step is not satisfactory: we replicate the kernel of the
Nash-Williams’ proof to get the result, which suffices to prove
the whole Higman’s Lemma, not just the property on antichains.
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Proper descending chains
Let A be a quasi order. Define
D(A ) = {xi}i∈I : {xi}i∈I is a proper descending chain in A ,
and {xi}i∈I ≤D(A ) yi i∈J if and only if there is an injective and
monotone map η: I → J from the ordinal I to the ordinal J such
that, for each i ∈ I, xi ≤A yη(i).
Fact 3
The structure D(A ) = 〈D(A );≤D(A )〉 is a quasi order.
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Proper descending chains
Proposition 4
If A is a well founded quasi order, so is D(A ).
Proof.
Let S = x1
i i∈I1
≥D(A ) ··· ≥D(A ) xn
i i∈In
≥D(A ) ··· be a
descending chain in D(A ). Then, fixed the embedding maps η,
each element x1
i , i ∈ I1, is the starting point of a descending
chain in A , by definition of ≤D(A ). But, by hypothesis, each
descending chain in A is either proper and finite, with length mi,
or it has a finite prefix of length mi followed by a possibly infinite
tail of equivalent elements. Thus, since I1 is finite, being x1
i i∈I1
a proper descending chain in A , there is m = max{mi : i ∈ I1}.
Then, necessarily, xn
i i∈In
is equivalent to xk
i i∈Ik
, for every n,k
greater than m.
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Proper descending chains
Corollary 5
A is a well founded quasi order, if and only if D(A ) is.
Proof.
One direction is the previous proposition. The other direction is
immediate, considering the embedding A → D(A ), A → {A}.
We notice that the property that proper descending chains in A
are finite, has been used only twice: (i) to establish that I1 is
finite; and (ii) to get the mi’s.
Thus, if we substitute D(A ) with the collection of finite
sequences over A in the same proof, we get the Higman’s
Lemma on well founded quasi orders, for free.
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Proper descending chains
Proposition 6
If A is a well quasi order, so is D(A ).
Proof. (i)
It suffices to show that D(A ) has the finite antichain property.
So, let S = xi
j j∈Ji i∈I
be an antichain in D(A ), and suppose,
with no loss of generality, that Ji ≤ Jk whenever i ≤ k.
Then, for each i,k ∈ I, i = k, xi
j j∈Ji
xk
j j∈Jk
, that is, for all
i,k ∈ I, i = k, for every ηi,k : Ji → Jk injective and monotone, for
some j∗
∈ Ji, xi
j∗ ≤ xk
ηi,k (j∗)
, and for every ηk,i : Jk → Ji injective
and monotone, for some j∗∗
∈ Jk, xk
j∗∗ ≤ xi
ηk,i (j∗∗)
. →
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Proper descending chains
→ Proof. (ii)
Fix any sequence ηi,i+1 i∈I, eventually excluding the last element
if I is a limit ordinal. Then, there is ξ mapping each i ∈ I to Ji
such that xi
ξ(i)
≤ xi+1
ηi,i+1(ξ(i))
, obtained by choosing the j∗
above.
Consider the sequence C defined as
x0
ξ(0),x0
ξ(0) ∪
x
2(i+1)
ξ(2(i+1))
,x
2(i+1)
η2i+1,2(i+1)(ξ(2i+1))
i∈I
∪
x2i+1
η2i,2i+1(ξ(2i)),x2i+1
ξ(2i+1) i∈I
.
→
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Proper descending chains
→ Proof. (iii)
By induction on the initial prefix of I up to ω:
x0
ξ(0)
≤ x1
η0,1(ξ(0))
, so x0
ξ(0)
,x0
ξ(0)
≤ x1
η0,1(ξ(0))
,x1
ξ(1)
;
if i = 2k +1, x2k+1
ξ(2k+1)
≤ x
2(k+1)
η2k+1,2(k+1)(ξ(2k+1))
, so
x2k+1
η2k,2k+1(ξ(2k)),x2k+1
ξ(2k+1) ≤ x
2(k+1)
ξ(2(k+1))
,x
2(k+1)
η2k+1,2(k+1)(ξ(2k+1))
,
that is,
xi
ηi−1,i (ξ(i−1)),xi
ξ(i) ≤ xi+1
ξ(i+1),xi+1
ηi,i+1(ξ(i)) ;
→
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Proper descending chains
→ Proof. (iv)
if i = 2k, k ≥ 1, then x2k
ξ(2k)
≤ x2k+1
η2k,2k+1(ξ(2k))
, so
x2k
ξ(2k),x2k
η2k−1,2k (ξ(2k−1)) ≤ x2k+1
η2k,2k+1(ξ(2k)),x2k+1
ξ(2k+1) ,
that is,
xi
ξ(i),xi
ηi−1,i (ξ(i−1)) ≤ xi+1
ηi,i+1(ξ(i)),xi+1
ξ(i+1) .
So, any two consecutive elements αi, αi+1 in C are such that
αi ≤ αi+1, for any i ∈ I ∩ω. →
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Proper descending chains
→ Proof. (v)
Thus, the sequence Cω obtained restricting C to I ∩ω, is a bad
sequence in A ×A . But A is a well quasi order, so by Dickson’s
Lemma, A ×A is a well quasi order, too, forcing any bad
sequence to be finite. Then, necessarily, Cω is finite, so I ∩ω ω,
i.e., I is finite. Thus, the antichain S is finite, being I its
length.
Corollary 7
A is a well quasi order if and only if D(A ) is.
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Higman’s Lemma, again
The same result can be derived from Higman’s Lemma by
noticing that the set D(A ) induces a full subcategory in the well
quasi order of finite sequences over A , which must be a well
quasi order, because it is the restriction of a well quasi order.
However, the proof of the previous proposition does not depend
on the fact that the elements are proper descending chains: in
fact, it just requires the elements to be in the well quasi order A .
Corollary 8
If A is a well quasi order, the collection of sequences over A
ordered by embedding has the finite antichain property.
And, of course, we get the Higman’s Lemma for free.
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Exponentiation, finally
Let Ω(A ) be the collection of arbitrary sequences over A .
Viewing each sequence as a function from an ordinal to the
indexed element, we get,
Proposition 9
The exponential object A I in QOrd, with I any ordinal, has
finite antichains whenever A is a well quasi order.
Proof.
The exponential object A I is isomorphic to a sub quasi order of
Ω(A ). Since Ω(A ) has the finite antichain property, the same
holds for A I.
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Exponentiation, finally
Assuming the Axiom of Choice,
Corollary 10
The exponential object A B
in QOrd, with B any quasi order,
has finite antichains whenever A is a well quasi order.
Hence, the counterexample to the existence of exponential
objects in WQO is, in fact, maximal, that is, any counterexample
will violate the finite descending chain property, whereas no
counterexample could be found which contradicts the finite
antichain property, as far as we operate within a sufficiently rich
set theory.
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The end
Questions?
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