WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
Well-quasi-ordering Hereditarily Finite Sets
Alberto Policritia , Alexandru I. Tomescua,b
a
b
Dip. Matematica e Informatica, Università di Udine
Fac. Mathematics and Computer Science, University of Bucharest
LATA 2011, Tarragona, Spain
May 27, 2011
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
O UTLINE
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
2 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
3 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
W ELL - QUASI - ORDERINGS
We say that (Q, 4) is a well-quasi-order if
I
4 is a reflexive and transitive binary relation
I
for every infinite sequence (qi )i=1,2,... of elements of Q,
there exist 1 ≤ i < j such that qi 4 qj
For example,
I
(N, ≤)
I
(Qk , 4k ), for every wqo (Q, 4), and every fixed k
I
(Q∗ , 4∗ ), for every wqo (Q, 4)
I
the class of all finite trees, under the topological minor
relation [Kruskal, 1960]
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
WQO’ S AND G RAPHS
In a series of over twenty papers (1983–2004), Robertson and
Seymour showed that
I
The class of all finite graphs is well-quasi-ordered by the
minor relation.
I
The class of all finite graphs is well-quasi-ordered by the
weak immersion relation.
An analog study for digraphs has only recently started:
1. in general, no sensible ‘immersion’ relation is a wqo
2. some results for Eulerian digraphs
3. the so called strong immersion is a wqo on the class of all
finite tournaments [Chudnovsky, Seymour, 2011]
5 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
A DDING S EMANTICS TO D IGRAPHS : S ETS
I
We assume the axioms of Zermelo-Fraenkel, including
I
I
I
The standard model: von Neumann’s cumulative
hierarchy of sets. Its subclass of hereditarily finite sets, HF:
I
I
I
I
Axiom of Extensionality: two sets are equal iff they have the
same elements
Axiom of Foundation: there are no membership cycles or
infinite descending membership chains
S
HF = i Vi , over all i ∈ N,
V0 = ø,
Vi+1 = P(Vi ).
For example,
I
I
I
V1 = {ø}
V2 = nø, {ø}
o
V3 = ø, {ø}, {{ø}}, {ø, {ø}} , . . .
6 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
R EPRESENTING S ET BY D IGRAPHS
The transitive closure of a set x is TrCl(x) =def x ∪
D EFINITION
S
y∈x TrCl(y).
The membership digraph of x is GTrCl(x) =def (TrCl(x), Ex ),
Ex =def {u → v | u, v ∈ TrCl(x) ∧ u 3 v}.
7 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
R EPRESENTING S ET BY D IGRAPHS
The transitive closure of a set x is TrCl(x) =def x ∪
D EFINITION
S
y∈x TrCl(y).
The membership digraph of x is GTrCl(x) =def (TrCl(x), Ex ),
Ex =def {u → v | u, v ∈ TrCl(x) ∧ u 3 v}.
D EFINITION
An acyclic digraph D is extensional if ∀u, v ∈ V(G),
N+ (u) 6= N+ (v).
P ROPOSITION
There is a 1-1 correspondence between membership digraphs and
extensional acyclic digraphs.
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S TRONG I MMERSION
We will work with strong immersion, as considered in
[Chudnovsky, Seymour, 2011].
A strong immersion of a digraph H into G is a map η such that:
I
for every v ∈ V(H), η(v) ∈ V(G); η is injective on V(H);
I
for each arc uv ∈ E(H), η(uv) is a simple directed path in G
from η(u) to η(v);
I
if e, f ∈ E(H) are distinct, then η(e) and η(f ) have no arcs in
common, although they may share vertices;
• the vertices of η(e) are not images of V(H) (except for its
two end-points).
9 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION in
AND E XTENSIONAL A CYCLIC D IGRAPHS
A Sa
TRUCTURE T HEOREM FOR S LIM S ETS
As already observed
[5], weak immersion is not
wqo on the set of all
digraphs. Just consider the acyclic digraphs Dn formed by orienting the edges
a cycle of length 2n alternately clockwise and counterclockwise (see Fig. 3).
Iof
MMERSION
IS NOT A WQO ON ALL DIGRAPHS
That being so, the collection {Dn | n � 2} has the property that none of its
elements
can
be weakly is
immersed
intoon
another
one.of all finite digraphs
Strong
immersion
not a wqo
the class
b1
a1
bn−1
b2
a2
a3
... ...
an−1
bn
an
Fig. 3. Digraphs Dn , n � 2
For example,
Since the collection of transitive closures of sets is a collection of digraphs, it
s
is natural to ask
b1whether
b2 � w
i or �i form a wqo on such a collection. The answer
is no, since the collection of membership digraphs Fn , n � 3, constructed as in
Fig. 4 has, analogously to the corresponding Dn ’s, the property that no digraph
can be weaklyaimmersed
a2 into another one.
1
WQO’ S , G RAPHS , S ETS
I MMERSION in
AND E XTENSIONAL A CYCLIC D IGRAPHS
A Sa
TRUCTURE T HEOREM FOR S LIM S ETS
As already observed
[5], weak immersion is not
wqo on the set of all
digraphs. Just consider the acyclic digraphs Dn formed by orienting the edges
a cycle of length 2n alternately clockwise and counterclockwise (see Fig. 3).
Iof
MMERSION
IS NOT A WQO ON ALL DIGRAPHS
That being so, the collection {Dn | n � 2} has the property that none of its
elements
can
be weakly is
immersed
intoon
another
one.of all finite digraphs
Strong
immersion
not a wqo
the class
b1
a1
bn−1
b2
a2
a3
... ...
an−1
bn
an
Fig. 3. Digraphs Dn , n � 2
For example,
Since the collection of transitive closures of sets is a collection of digraphs, it
s
is natural to ask
b1whether
b2 � w
i or �i form a wqo on such a collection. The answer
η(b1 )
is no, since the collection of membership digraphs F
n , n � 3, constructed as in
Fig. 4 has, analogously to the corresponding Dn ’s, the property that no digraph
can be weaklyaimmersed
a2 into another one.
1
WQO’ S , G RAPHS , S ETS
I MMERSION in
AND E XTENSIONAL A CYCLIC D IGRAPHS
A Sa
TRUCTURE T HEOREM FOR S LIM S ETS
As already observed
[5], weak immersion is not
wqo on the set of all
digraphs. Just consider the acyclic digraphs Dn formed by orienting the edges
a cycle of length 2n alternately clockwise and counterclockwise (see Fig. 3).
Iof
MMERSION
IS NOT A WQO ON ALL DIGRAPHS
That being so, the collection {Dn | n � 2} has the property that none of its
elements
can
be weakly is
immersed
intoon
another
one.of all finite digraphs
Strong
immersion
not a wqo
the class
b1
a1
bn−1
b2
a2
a3
... ...
an−1
bn
an
Fig. 3. Digraphs Dn , n � 2
For example,
Since the collection of transitive closures of sets is a collection of digraphs, it
s
is natural to ask
b1whether
b2 � w
i or �i form a wqo on such a collection. The answer
η(b1 )
is no, since the collection of membership digraphs F
n , n � 3, constructed as in
Fig. 4 has, analogously to the corresponding Dn ’s, the property that no digraph
can be weaklyaimmersed
a2 into another one.
1
η(a1 )
WQO’ S , G RAPHS , S ETS
I MMERSION in
AND E XTENSIONAL A CYCLIC D IGRAPHS
A Sa
TRUCTURE T HEOREM FOR S LIM S ETS
As already observed
[5], weak immersion is not
wqo on the set of all
digraphs. Just consider the acyclic digraphs Dn formed by orienting the edges
a cycle of length 2n alternately clockwise and counterclockwise (see Fig. 3).
Iof
MMERSION
IS NOT A WQO ON ALL DIGRAPHS
That being so, the collection {Dn | n � 2} has the property that none of its
elements
can
be weakly is
immersed
intoon
another
one.of all finite digraphs
Strong
immersion
not a wqo
the class
b1
a1
bn−1
b2
a2
a3
... ...
an−1
bn
an
Fig. 3. Digraphs Dn , n � 2
For example,
Since the collection of transitive closures of sets is a collection of digraphs, it
s
is natural to ask
b1whether
b2 � w
i or �i form a wqo on such a collection. The answer
η(b1 )
is no, since the collection of membership digraphs F
n , n � 3, constructed as in
Fig. 4 has, analogously to the corresponding Dn ’s, the property that no digraph
can be weaklyaimmersed
a2 into another one.
1
η(a1 ) η(a2 )
WQO’ S , G RAPHS , S ETS
I MMERSION in
AND E XTENSIONAL A CYCLIC D IGRAPHS
A Sa
TRUCTURE T HEOREM FOR S LIM S ETS
As already observed
[5], weak immersion is not
wqo on the set of all
digraphs. Just consider the acyclic digraphs Dn formed by orienting the edges
a cycle of length 2n alternately clockwise and counterclockwise (see Fig. 3).
Iof
MMERSION
IS NOT A WQO ON ALL DIGRAPHS
That being so, the collection {Dn | n � 2} has the property that none of its
elements
can
be weakly is
immersed
intoon
another
one.of all finite digraphs
Strong
immersion
not a wqo
the class
b1
a1
bn−1
b2
a2
a3
... ...
an−1
bn
an
Fig. 3. Digraphs Dn , n � 2
For example,
Since the collection of transitive closures of sets is a collection
of digraphs, it
?η(b2 )?
s
is natural to ask
b1whether
b2 � w
i or �i form a wqo on such a collection. The answer
η(b1 )
is no, since the collection of membership digraphs F
n , n � 3, constructed as in
Fig. 4 has, analogously to the corresponding Dn ’s, the property that no digraph
can be weaklyaimmersed
a2 into another one.
1
η(a1 ) η(a2 )
10 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
A BAD SEQUENCE FOR HF SETS
Digraphs from the previous ‘bad sequence’ can be easily
extensional:
6 rendered
A. Policriti
and A.I. Tomescu
b1
a1
bn−1
b2
a2
a3
... ...
an−1
bn
an
b�n
Fig. 4. Membership digraphs Fn , n � 3; notice that Fn is acyclic and extensional
We need to control memberships, i.e., arcs.
Indeed,anomalies:
given Fn and Fm , with n < m, and supposing that η is a weak immerSome
sion ofIFn into Fm , observe that η(bn ) must be bm , since bn has 3 out-neighbors
unbounded number of sources
in Fn and bm is the only vertex of Fm having more than 2 out-neighbors. Then,
�
reach� too ‘deep’
η(a1 ) I
= aarcs
1 and η(bn ) = bm . Next, η(b1 ) = b1 , since otherwise the directed paths
η(b1 a1 ) and η(a2 a1 ) would have to share the arc a2 a1 , against the fact that η
is an immersion. This implies that η(a2 ) = a2 . By a similar argument, induc11 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S OME S ET- THEORETIC I NSIGHT
The rank of x:
rk(x) =def 1 + max {rk(y) | y ∈ x}.
D EFINITION
The skewness of a set x:
skewness(x) =def max {rk(y) − rk(z) | y, z ∈ TrCl(x) ∧ z ∈ y}.
Well-quasi-ordering hereditarily finite sets
rank: 0
1
2
3
4
5
5
6
skewness(x) = 5
Fig. 2. A membership digraph of skewness 5
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
T HE M ISSING L INK
D EFINITION
A set x is slim if the digraph obtained by removing any arc
from GTrCl(x) is not extensional.
Equivalently, x is slim if
I ∀y ∈ V(GTrCl(x) ) and for any y → z, ∃y0 of GTrCl(x) such that
N+ (y0 ) = N+ (y) \ {z} ⇔
I ∀y ∈ TrCl(x) and ∀z ∈ y, it holds that y \ {z} ∈ TrCl(x) ⇔
I ∀y ∈ TrCl(x), P(y) ⊆ TrCl(x).
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
T HE M ISSING L INK
D EFINITION
A set x is slim if the digraph obtained by removing any arc
from GTrCl(x) is not extensional.
Equivalently, x is slim if
I ∀y ∈ V(GTrCl(x) ) and for any y → z, ∃y0 of GTrCl(x) such that
N+ (y0 ) = N+ (y) \ {z} ⇔
I ∀y ∈ TrCl(x) and ∀z ∈ y, it holds that y \ {z} ∈ TrCl(x) ⇔
I ∀y ∈ TrCl(x), P(y) ⊆ TrCl(x).
O UR R ESULT
Strong immersion is a wqo on the class
Msh = {GTrCl(x) | x is slim ∧ |x| ≤ s ∧ skewness(x) ≤ h}.
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S LIM D ISCRIMINATION
D EFINITION
S
Let F be a finite family of sets. A z ∈ F is said to be
redundant if |F| = |{v \ {z} | vS∈ F}|. We say that F is
irredundant if no element of F is redundant.
15 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S LIM D ISCRIMINATION
D EFINITION
S
Let F be a finite family of sets. A z ∈ F is said to be
redundant if |F| = |{v \ {z} | vS∈ F}|. We say that F is
irredundant if no element of F is redundant.
L EMMA (S LIM DISCRIMINATION LEMMA )
Given an n-element slim and nonempty family
S F = {v1 , . . . , vn }
such that ∅ ∈
/ F, we can find z1 , . . . , zk ∈ F, with k 6 n so that:
(1) {vi ∩ {z1 , . . . , zk } | vi ∈ F} ∪ {∅} is irredundant, of cardinality
n + 1;
(2) if rk(F) = % + 2 and there is a 1 ≤ j ≤ n so that
|vj ∩ {z1 , . . . , zk }| > 1, then |{zi | rk(zi ) = %}| < n.
15 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S LIM S TRUCTURE
L EMMA (S LIM STRUCTURE LEMMA )
For any slim set x and for every 0 < r < rk(x), the following hold:
(1) |TrCl(x)=r−1 \ x| ≤ |TrCl(x)=r |;
(2) |TrCl(x)=r | ≤ |x≥r |.
16 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S LIM S TRUCTURE
L EMMA (S LIM STRUCTURE LEMMA )
For any slim set x and for every 0 < r < rk(x), the following hold:
(1) |TrCl(x)=r−1 \ x| ≤ |TrCl(x)=r |;
(2) |TrCl(x)=r | ≤ |x≥r |.
The structure of the digraphs in Msh :
I
at any rank of GTrCl(x) there are at most s vertices (from (2));
I
the number of vertices at any rank r of GTrCl(x) is
non-increasing (for decreasing r), save for at most |x| ≤ s
times (from (1));
• whenever x ∩ TrCl(x)=r = ∅ and |TrCl(x)=r+1 | = |TrCl(x)=r |,
each element of TrCl(x)=r+1 has cardinality 1.
16 / 22
WQO’SS,, G
GRAPHS
RAPHS,, SSETS
ETS
WQO’
MMERSION AND
AND EEXTENSIONAL
XTENSIONAL A
ACYCLIC
CYCLIC D
DIGRAPHS
IGRAPHS
IIMMERSION
A SSTRUCTURE
TRUCTURE TTHEOREM
HEOREM FOR
FOR SSLIM
LIM SSETS
ETS
A
AN
N EXAMPLE
EXAMPLE
A
10
A. Policriti and A.I. Tomescu
3
3
2
rk 2
rk 7
The membership digraph G
rk 12
rk 15
, where x is slim, |x| = 6, and
The
membership
digraph
, where
x is slim,6|x|
= skewness
6, and 3; we
TrCl(s)
Fig. 5.
The
membership digraph
of G
a TrCl(s)
slim set
x of cardinality
and
skewness(x)
=
3.
skewness(x) = 3.
have δ(x) = 1 3 3 2
σ(xi2 ) = . . .. This implies that between the membership digraphs corresponding
17//22
22
17
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
J UNCTURE RANKS
We say that r is a juncture rank of x if
I
there is an element of x at rank r in GTrCl(x) , or
I
there is a change in cardinality between TrCl(x)=r and
TrCl(x)=r−1 .
18 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
J UNCTURE RANKS
We say that r is a juncture rank of x if
I
there is an element of x at rank r in GTrCl(x) , or
I
there is a change in cardinality between TrCl(x)=r and
TrCl(x)=r−1 .
We need to encode
(1) the subdigraphs at juncture ranks
(2) the origin/destination of the directed paths between
juncture ranks
(3) the lengths of these paths
For (1)+(2) we use an alphabet Σs,h whose characters are all
possible digraphs appearing at juncture ranks, having labeled
sinks.
For (3) we use natural numbers.
18 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
T HE E NCODING
To every GTrCl(x) ∈ Msh we assign two encodings:
I σ(x) ∈ Σ∗ , a sequence of subdigraphs appearing at
s,h
juncture ranks;
I
I
choose those digraphs whose labels on the sinks comply
with Ackermann’s order between sets
δ(x) ∈ N∗ , a sequence of lengths of the paths between
juncture ranks.
19 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
T HE E NCODING
To every GTrCl(x) ∈ Msh we assign two encodings:
I σ(x) ∈ Σ∗ , a sequence of subdigraphs appearing at
s,h
juncture ranks;
I
I
choose those digraphs whose labels on the sinks comply
with Ackermann’s order between sets
δ(x) ∈ N∗ , a sequence of lengths of the paths between
juncture ranks.
L EMMA
For every s, h ≥ 1, Σs,h is finite.
L EMMA
For any s, h ≥ 1, there exists a computable function g(s, h) such that
for any slim set x, with |x| ≤ s and skewness(x) ≤ h, it holds
|σ(x)| ≤ g(s, h).
19 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S TRONG IMMERSION IS A WQO ON Msh
Let (GTrCl(xi ) )i=1,2,... be an infinite sequence of elements of Msh .
I since |Σs,h | is finite and |σ(xi )| ≤ g(s, h), there exists an
infinite subsequence of it, (GTrCl(xi ) )j=1,2,... , such that
σ(xi1 ) = σ(xi2 ) = · · · .
j
20 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S TRONG IMMERSION IS A WQO ON Msh
Let (GTrCl(xi ) )i=1,2,... be an infinite sequence of elements of Msh .
I since |Σs,h | is finite and |σ(xi )| ≤ g(s, h), there exists an
infinite subsequence of it, (GTrCl(xi ) )j=1,2,... , such that
j
I
σ(xi1 ) = σ(xi2 ) = · · · .
since |σ(xi1 )| = |σ(xi2 )| = · · · , we also have
|δ(xi1 )| = |δ(xi2 )| = · · · =def `, thus δ(xij ) ∈ N` .
20 / 22
WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S TRONG IMMERSION IS A WQO ON Msh
Let (GTrCl(xi ) )i=1,2,... be an infinite sequence of elements of Msh .
I since |Σs,h | is finite and |σ(xi )| ≤ g(s, h), there exists an
infinite subsequence of it, (GTrCl(xi ) )j=1,2,... , such that
j
I
I
σ(xi1 ) = σ(xi2 ) = · · · .
since |σ(xi1 )| = |σ(xi2 )| = · · · , we also have
|δ(xi1 )| = |δ(xi2 )| = · · · =def `, thus δ(xij ) ∈ N` .
since (N` , ≤` ) is a wqo, there exist 1 ≤ j < k so that for
y =def xij and z =def xik we have
I
I
σ(y) = σ(z);
δ(y) ≤` δ(z).
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
S TRONG IMMERSION IS A WQO ON Msh
Let (GTrCl(xi ) )i=1,2,... be an infinite sequence of elements of Msh .
I since |Σs,h | is finite and |σ(xi )| ≤ g(s, h), there exists an
infinite subsequence of it, (GTrCl(xi ) )j=1,2,... , such that
j
I
I
σ(xi1 ) = σ(xi2 ) = · · · .
since |σ(xi1 )| = |σ(xi2 )| = · · · , we also have
|δ(xi1 )| = |δ(xi2 )| = · · · =def `, thus δ(xij ) ∈ N` .
since (N` , ≤` ) is a wqo, there exist 1 ≤ j < k so that for
y =def xij and z =def xik we have
I
I
σ(y) = σ(z);
δ(y) ≤` δ(z).
Strongly immerse y into z by
I mapping digraphs at juncture ranks into corresponding
isomorphic digraphs;
I mapping directed paths into (longer) directed paths.
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
T O SUM UP...
!
!
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WQO’ S , G RAPHS , S ETS
I MMERSION AND E XTENSIONAL A CYCLIC D IGRAPHS
A S TRUCTURE T HEOREM FOR S LIM S ETS
C ONCLUDING R EMARKS
I
This is a first step in studying digraph immersions and
wqo’s for various classes of sets.
I
In particular, can some of the restriction considered here be
dropped?
I
I
The bounds on cardinality and skewness cannot be both
dropped at the same time.
Can the bound on cardinality be dropped?
I
The generalization of this result to cyclic digraphs (relating
to hypersets) is within reach.
I
WQO’s proved to be a key ingredient in generalizing and
unifying results concerning the decidability of verification
problems for infinite state transition systems; can this
set-theoretic framework improve matters?
22 / 22