In mathematics, Higman's lemma states that the set
of finite sequences over a finite alphabet
, as partially ordered by the subsequence relation, is well-quasi-ordered. That is, if
is an infinite sequence of words over a finite alphabet
, then there exist indices
such that
can be obtained from
by deleting some (possibly none) symbols. More generally this remains true when
is not necessarily finite, but is itself well-quasi-ordered, and the subsequence relation is generalized into an "embedding" relation that allows the replacement of symbols by earlier symbols in the well-quasi-ordering of
. This is a special case of the later Kruskal's tree theorem. It is named after Graham Higman, who published it in 1952.
Proof
Let
be a well-quasi-ordered alphabet of symbols (in particular,
could be finite and ordered by the identity relation). Suppose for a contradiction that there exist infinite bad sequences, i.e. infinite sequences of words
such that no
embeds into a later
. Then there exists an infinite bad sequence of words
that is minimal in the following sense:
is a word of minimum length from among all words that start infinite bad sequences;
is a word of minimum length from among all infinite bad sequences that start with
;
is a word of minimum length from among all infinite bad sequences that start with
; and so on. In general,
is a word of minimum length from among all infinite bad sequences that start with
.
Since no
can be the empty word, we can write
for
and
. Since
is well-quasi-ordered, the sequence of leading symbols
must contain an infinite increasing sequence
with
.
Now consider the sequence of words
![{\displaystyle w_{1},\ldots ,w_{{i_{1}}-1},z_{i_{1}},z_{i_{2}},z_{i_{3}},\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48a5b3f9920712da13bb5d764855b518ce7a5ec2)
Because
![{\displaystyle z_{i_{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f830c4a4264ae77a6753e248a7edb89860585a14)
is shorter than
![{\displaystyle w_{i_{1}}=a_{i_{1}}z_{i_{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bead4dab5fc086fa0a224eefacc2746bd26303fd)
, this sequence is "more minimal" than
![{\displaystyle W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7)
, and so it must contain a word
![{\displaystyle u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
that embeds into a later word
![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
. But
![{\displaystyle u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
and
![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
cannot both be
![{\displaystyle w_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/326f4828cd2d8b281d5837f977d435d47450a191)
's, because then the original sequence
![{\displaystyle W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7)
would not be bad. Similarly, it cannot be that
![{\displaystyle u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
is a
![{\displaystyle w_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/326f4828cd2d8b281d5837f977d435d47450a191)
and
![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
is a
![{\displaystyle z_{i_{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04cd2ad167216c8a1d66cd3af93a4a5843b9119e)
, because then
![{\displaystyle w_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/326f4828cd2d8b281d5837f977d435d47450a191)
would also embed into
![{\displaystyle w_{i_{k}}=a_{i_{k}}z_{i_{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf3e483656f5de18c99142c932c04cd0e0b2042)
. And similarly, it cannot be that
![{\displaystyle u=z_{i_{j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c23a22483684f59c7e157d9fac6260e842dcd90)
and
![{\displaystyle v=z_{i_{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/027a618efb75bbcc9b22cacb237f7c97a4e922c3)
,
![{\displaystyle j<k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bad1af0d1a37fef58095a8f4d0c21a5fa00cb73)
, because then
![{\displaystyle w_{i_{j}}=a_{i_{j}}z_{i_{j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74d7764da17fb27e7e0386cbde66f1a9140291fb)
would embed into
![{\displaystyle w_{i_{k}}=a_{i_{k}}z_{i_{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf3e483656f5de18c99142c932c04cd0e0b2042)
. In every case we arrive at a contradiction.
Ordinal type
The ordinal type of
is related to the ordinal type of
as follows:[1][2]
![{\displaystyle o(\Sigma ^{*})={\begin{cases}\omega ^{\omega ^{o(\Sigma )-1}},&o(\Sigma ){\text{ finite}};\\\omega ^{\omega ^{o(\Sigma )+1}},&o(\Sigma )=\varepsilon _{\alpha }+n{\text{ for some }}\alpha {\text{ and some finite }}n;\\\omega ^{\omega ^{o(\Sigma )}},&{\text{otherwise}}.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8905525b6f75739374d71c0f3991e290b33fc024)
Reverse-mathematical calibration
Higman's lemma has been reverse mathematically calibrated (in terms of subsystems of second-order arithmetic) as equivalent to
over the base theory
.[3]
References
- Higman, Graham (1952), "Ordering by divisibility in abstract algebras", Proceedings of the London Mathematical Society, (3), 2 (7): 326–336, doi:10.1112/plms/s3-2.1.326
Citations
- ^ de Jongh, Dick H. G.; Parikh, Rohit (1977). "Well-partial orderings and hierarchies". Indagationes Mathematicae (Proceedings). 80 (3): 195–207. doi:10.1016/1385-7258(77)90067-1.
- ^ Schmidt, Diana (1979). Well-partial orderings and their maximal order types (Habilitationsschrift). Heidelberg. Republished in: Schmidt, Diana (2020). "Well-partial orderings and their maximal order types". In Schuster, Peter M.; Seisenberger, Monika; Weiermann, Andreas (eds.). Well-Quasi Orders in Computation, Logic, Language and Reasoning. Trends in Logic. Vol. 53. Springer. pp. 351–391. doi:10.1007/978-3-030-30229-0_13.
- ^ J. van der Meeren, M. Rathjen, A. Weiermann, An order-theoretic characterization of the Howard-Bachmann-hierarchy (2015, p.41). Accessed 03 November 2022.