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Finite Diagram Semigroups: Extending theComputational Horizon
James East Attila Egri-Nagy Andrew R. FrancisJames D. Mitchell
Centre for Research and MathematicsSchool of Computing, Engineering and Mathematics
University of Western Sydney, Australia——–
School of Mathematics and StatisticsUniversity of St Andrews, United Kingdom
London Mathematical Society EPSRC Durham SymposiumPermutation groups and transformation semigroups
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I What is computable with n states?
I What is the minimal number of states required for a particularcomputation?
I What is the structure of these computations?
Here we use computational enumeration.
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Enumerating by Order: Abstract Semigroups
(1954 Gakugei, J., Notes on finite semigroups and determination ofsemigroups of order 4, Tokushima Univ. Math., 5 (1954), 17–27.)
1955 Forsythe, G. E., SWAC computes 126 distinct semigroups of order4, Proc. Amer. Math. Soc., 6 (1955), 443–447.
Tetsuya, K., Hashimoto, T., Akazawa, T., Shibata, R., Inui, T. andTamura, T., All semigroups of order at most 5, J. GakugeiTokushima Univ. Nat. Sci. Math., 6 (1955), 19–39.
1967 Plemmons, R. J., There are 15973 semigroups of order 6, Math.Algorithms, 2 (1967), 2–17.
1977 Jurgensen, H. and Wick, P., Die Halbgruppen der Ordnungen ≤ 7,Semigroup Forum, 14 (1) (1977), 69–79.
1994 Satoh, S., Yama, K. and Tokizawa, M., Semigroups of order 8,Semigroup Forum, 49 (1) (1994), 7–29.
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Number of semigroups of order n
order #groups #semigroups #3-nilpotent semigroups
1 1 1 02 1 4 03 1 18 14 2 126 85 1 1,160 846 2 15,973 2,6607 1 836,021 609,7978 5 1,843,120,128 1,831,687,0229 2 52,989,400,714,478 52,966,239,062,973
http://www-groups.mcs.st-andrews.ac.uk/~jamesm/smallsemi.php
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Flip-flop, the 1-bit memory semigroup
write 0 write 1 read
[11] [22] [12]
So these are computational devices... ≈ automata
With transformation semigroups, we get all semigroups. (Cayley’s theorem)
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Enumerating by Degree: Transformation SemigroupsEnumeration: subsemigroups of the full transformation semigroup
[11],[12],[21],[22]
[11],[12],[22]
[12],[22][11],[22] [11],[12] [12],[21]
[11][22] [12]
∅
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Data flood
Number of subsemigroups of full transformation semigroups.
#subsemigroups #conjugacy classes #isomorphism classes
T0 1 1 1
T1 2 2 2
T2 10 8 7
T3 1 299 283 267
T4 3 161 965 550 132 069 776 131 852 491
After discounting the state-relabelling symmetries the database of degree 4transformation semigroups is still around 9GB.
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Summary of Results
T1 T2 T3 T4#nilpotent 1 2 4 22
#commutative 1 4 18 158
#band 1 5 41 1503
#regular 1 7 116 33285
#subsemigroups 1 7 282 132 069 776
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0
500000
1e+06
1.5e+06
2e+06
2.5e+06
3e+06
3.5e+06
4e+06
4.5e+06
5e+06
0 50 100 150 200 250
frequency
size
subsemigroups of T4
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0
5
10
15
20
25
160 180 200 220 240
frequency
size
subsemigroups of T4 of size ≥ 157
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0
10
20
30
0 20 40 60 80 100 120 140 160 180 200 220 240
#D-classes
size of semigroups
SubS4(T4)
0
0.4
0.8
1.2
1.6
Figure: Size versus the number of D-classes for transformation semigroups up todegree 4. Frequency values in millions.
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How is it done?
Upper and lower torsos
S \ I
S
I
T
T
L
U
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Enumerating lower torsos
S \ I S
I
T
L
U
x1
x2
|T | << |〈T ∪ {x1}〉| << |〈T ∪ {x1, x2}〉|
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Partitioned Binary Relations
∈ PBn,
{(2, 1), (2, 3′), (5, 4′), (5, 5′), (1′, 1), (2′, 2′), (2′, 3), (3′, 4′), (4′, 3′), (5′, 5)
}
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Multiplying Partitioned Binary Relations
α
β
αβ
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Diagram Semigroups – Typical Elements
∈ PBn,
∈ Bn, ∈ Pn
∈ PTn, ∈ I∗n
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Diagram Semigroups – Typical Elements
∈ In, ∈ Bn
∈ Tn, ∈ TLn
∈ Sn, 1n
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Diagram Semigroup Land
Tn In I∗n Bn
Sn1n
TLn
PTn
Bn
PBn
Pn
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n = 1 n = 2 n = 3 n = 4 n = 5 n = 6
PBn 1262Bn 4 385Pn 4 272PTn 4 50 94232In 4 23 2963I∗n 2 6 795Tn 2 8 283 132069776Bn 2 6 42 10411TLn 2 4 12 232 12592 324835618Sn 1 2 4 11 19 56
Table: Summary table of the numbers of distinct (up to conjugacy) semigroups ofgiven diagram type and degree.
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0
2
4
6
8
10
12
14
16
0 5 10 15 20 25 30 35
#D-classes
size of semigroups
0
20
40
60
80
100
120
140
160
180
SubS3(I3)
Figure: Size versus the number of D-classes for partial permutation semigroups upto degree 3.
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0
5
10
15
20
25
0 10 20 30 40 50 60
#idem
potents
size of semigroups
SubS3(PT3)
02004006008001000120014001600180020002200
Figure: Size versus the number of idempotents for partial transformationsemigroups up to degree 3.
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0
10
20
30
40
0 20 40 60 80 100 120 140 160 180 200 220 240
#idem
potents
size of semigroups
SubS4(T4)0
0.4
0.8
1.2
1.6
Figure: Size versus the number of idempotents for transformation semigroups up todegree 4. Frequency values in millions.
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P1 ↪→ T2P2 ↪→ T5
B1∼= T1
B2 ↪→ T3
TL1 ∼= T1TL2 ↪→ T2TL3 ↪→ T4
P1 ↪→ B2
Tn In I∗n Bn
Sn1n
TLn
PTn
Bn
PBn
Pn
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n-generated embeddings
Sn↪→ T if S embeds into a subsemigroup of T that can be generated by n
elements
A simple algorithm:
1. Find all n-generated subsemigroups of T .
2. Filter this set by the property that size is ≥ |S |.3. Try constructing embeddings.
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P2, 3-generated
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P22↪→ P3?
The conjecture was: NO.
But, there are 4, err... 3 distinct ways of this embedding.
Generators of P2 Solution 1 Solution 2 Solution 3
Oooooh!!!!
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P22↪→ P3?
The conjecture was: NO.
But, there are 4, err... 3 distinct ways of this embedding.
Generators of P2 Solution 1 Solution 2 Solution 3
Oooooh!!!!
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B3, the Brauer monoid of degree 3
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B32↪→ B5? There are at most 21 distinct ways.
B32↪→ B4?, B4
2↪→ B5? Answers: NO. (as expected)
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Temperley-Lieb, Jones monoid
Catalan numbers, sequences of well-formed parentheses.
corresponds to (()(()))()
Applications in Physics: statistical mechanics, percolation problem.
=
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First glimpse through the usual eggbox diagrams, J9
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J16, top level
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J16, level 2
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J16, level 3
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J16, level 4
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J16, level 5
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J16, level 6
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J16, level 7
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J16, level 8
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J16, level 9
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Summary
We produce huge amount of data. =⇒ Open problems pop up everywhere.
In case you can use your data or know how to answer the questions, pleasetalk to us!
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Links
GAP www.gap-system.org
Semigroups
www-groups.mcs.st-andrews.ac.uk/~jamesm/semigroups.php
SubSemi github.com/egri-nagy/subsemi
VIZ bitbucket.org/james-d-mitchell/viz
Blog on computational semigroup theory:
compsemi.wordpress.com
Other software packages: GraphViz, Gnuplot, TEX
Thank You!