An algorithm for constructing minimal degree diagram representations of finite semigroups

Transcription

1 An algorithm for constructing minimal degree diagram representations of finite semigroups James East Attila Egri-Nagy Andrew R. Francis James D. Mitchell Centre for Research and Mathematics School of Computing, Engineering and Mathematics University of Western Sydney, Australia School of Mathematics and Statistics University of St Andrews, United Kingdom 8th Australia New Zealand Mathematics Convention 2014

2 Foundations of Computing leading questions What is computable with n states? What is the minimal number of states required for a particular computation? By computation we mean a semigroup.

3 Semigroups ABSTRACT Definition A semigroup is a set S with an associative binary operation S S S. Example (Flip-flop monoid) 1 a b 1 1 a b a a a b b b a b

4 Semigroups ABSTRACT Definition A semigroup is a set S with an associative binary operation S S S. Example (Flip-flop monoid) 1 a b 1 1 a b a a a b b b a b TRANSFORMATION Definition A transformation semigroup (X, S) is a set of states X and a set S of transformations s : X X closed under function composition. Example (Transformations) =

5 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)

6 Enumeration: subsemigroups of the full transformation semigroup [11],[12],[21],[22] [11],[12],[22] [11],[22] [11],[12] [12],[22] [12],[21] [22] [11] [12]

7 Data flood Number of subsemigroups of full transformation semigroups. #subsemigroups #conjugacy classes #isomorphism classes T T T T T After discounting the state-relabelling symmetries the database of degree 4 transformation semigroups is still around 9GB.

8 5e e+06 subsemigroups of T 4 frequency 4e e+06 3e e+06 2e e+06 1e size

9 25 subsemigroups of T 4 of size frequency size

10 Enumeration/classification gives a simple table-lookup algorithm for finding minimal degree representations. For up to degree 4 transformation representations...

11 Diagram Semigroups Typical Elements PB n, B n, P n PT n, I n

12 Diagram Semigroups Typical Elements I n, B n T n, TL n S n, 1 n

13 Diagram Semigroup Land PB n B n P n PT n T n I n I n B n S n 1 n TL n S D n where D {PB, B, PT, T, S, I, I, P, B, TL}

14 Temperley-Lieb, Jones monoid Catalan numbers, sequences of well-formed parentheses. corresponds to (()(()))() Applications in Physics: statistical mechanics, percolation problem. =

15 Direct construction of embedding Example Z T

16 Direct construction of embedding Example Z T , 2 3

17 How to search for the embedding map? Partial solutions exist (just put zero for an element not in the table), thus backtrack search can be used. Types of elements: the index-period of s the smallest values m 1, r 1 such that s m+r = s m. The semigroup analogue of the order of a group element. Ask about the story of this algorithm!

18 I M AN IDIOT

19 Minimum diagram representation degree Definition µ D (S) = min{n S D n } where D {PB, B, PT, T, S, I, I, P, B, TL} S = {( ) 0 0, 0 0 ( ) 0 0, 0 1 ( ) 0 0, 1 0 S ( ) 0 1, 0 1 µ T (S) µ P (S) µ B (S) µ TL (S) ( )}

20 PB n P 1 T 2 P 2 T 5 B 1 = T1 B 2 T 3 B n PT n P n TL 1 = T1 TL 2 T 2 TL 3 T 4 P 1 B 2 T n I n I n B n S n 1 n TL n

21 n-generated embeddings S n 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.

22 P 2, 3-generated

23 P 2 2 P 3? The conjecture was: NO.

24 P 2 2 P 3? The conjecture was: NO. But, there are 4, err... 3 distinct ways of this embedding. Generators of P 2 Solution 1 Solution 2 Solution 3 Oooooh!!!!

25 B 3, the Brauer monoid of degree 3

26 B 3 2 B5? There are at most 21 distinct ways. B 3 2 B4?, B 4 2 B5? Answers: NO. (as expected)

27 Summary Good news. We have an algorithm for constructing minimal degree representations for relatively small semigroups. Bad news. We have an algorithm for constructing minimal degree representations for relatively small semigroups.

28 Links GAP Semigroups www-groups.mcs.st-andrews.ac.uk/~jamesm/semigroups.php SubSemi VIZ bitbucket.org/egri-nagy/subsemi bitbucket.org/james-d-mitchell/viz Blog on computational semigroup theory: compsemi.wordpress.com Other software packages: GraphViz, Gnuplot, TEX

29 Thank You!