Computing with finite semigroups: prt I J. D. Mitchell School of Mthemtics nd Sttistics, University of St Andrews Novemer 20th, 2015 J. D. Mitchell (St Andrews) Novemer 20th, 2015 1 / 34
Wht is this tlk out? Given: finite semigroup S; nd question out S. Aim: to descrie how to nswer your question using computer descrie the stte of the rt. Why? perform low-level clcultions such s multipliction, inversion,... suggests new theoreticl results otin counter-exmples gin more detiled understnding perform more intricte clcultions. J. D. Mitchell (St Andrews) Novemer 20th, 2015 2 / 34
Insert semigroup into computer... numer 1 Cyley tles J. D. Mitchell (St Andrews) Novemer 20th, 2015 3 / 34
Insert semigroup into computer... numer 1 Cyley tles Resons not to: Too mny! 12 418 001 077 381 302 684 semigroups up to isomorphism nd nti-isomorphism with 10 elements (Distler-Kelsey 13); Complexity! O( S 3 ) just to verify ssocitivity; Hrd to input! A semigroup with 1000 elements hs 1 million entries in the Cyley tle; Requires nerly complete knowledge! J. D. Mitchell (St Andrews) Novemer 20th, 2015 4 / 34
Insert semigroup into computer... numer 2 Presenttions Words in genertors nd reltions:, 2 =, =, 2 =, 3 =, 2 =. Resons not to: Reltively difficult to find! given semigroup S it cn e difficult to find presenttion for S; Undecidility! lmost every meningful question is undecidle, i.e. word prolem, isomorphism prolem,... J. D. Mitchell (St Andrews) Novemer 20th, 2015 5 / 34
Insert semigroup into computer... numer 3 Genertors Specify genertors of prticulr type. Definition A trnsformtion is function f from {1,..., n} to itself written: ( ) 1 2 n f =. 1f 2f nf A trnsformtion semigroup is just semigroup consisting of set of trnsformtions under composition of functions. Theorem (Cyley s theorem) Every semigroup is isomorphic to permuttion trnsformtion semigroup. J. D. Mitchell (St Andrews) Novemer 20th, 2015 6 / 34
Fundmentl tsks Input: genertors A (trnsformtions, prtil perms, mtrices, inry reltions, prtitions,... ) for semigroup S. Output: the size of S memership in S fctorise elements over the genertors the numer of idempotents (x 2 = x) the mximl su(semi)groups the idel structurl of S (i.e. Green s reltions) is S group? n inverse semigroup? regulr semigroup? the utomorphism group of S the congruences of S... J. D. Mitchell (St Andrews) Novemer 20th, 2015 7 / 34
An lgorithm S cting on itself y right multipliction Input: set A of genertors (trnsformtions, prtil perms, mtrices, inry reltions, prtitions,... ) for semigroup S. Output: the elements X of S. Assumes: we cn multiply nd check equlity. Supposing the genertors re distinct. 1: X := A 2: for x X do 3: for A do 4: if x X then 5: ppend x to X 6: return X J. D. Mitchell (St Andrews) Novemer 20th, 2015 8 / 34
An exmple Let S e the semigroup generted y the trnsformtions ( ) ( ) 1 2 3 1 2 3 = nd =. 2 2 3 2 1 2 The grph of the ctions of nd :, 1 2 3 forth J. D. Mitchell (St Andrews) Novemer 20th, 2015 9 / 34
The elements nd the right Cyley grph Edges of the form: x y xy 1 2 3 2 2 3 2 1 2 1 1 2 2 2 2 2 1 2 1 2 2 2 1 1 1 1 2, 2 2 =, =, 2 =, 2 =, 3 =, 2 =, 3 =, =, 2 = ck forth J. D. Mitchell (St Andrews) Novemer 20th, 2015 10 / 34
The left Cyley grph Edges of the form x 1 2 3 2 2 3 2 1 2 1 1 2 2 2 2 2 1 2 1 2 2 2 1 1 1 1 y yx..., 2 2, 2 =, =, 2 =, 2 =, 3 =, 2 =, 3 =, =, 2 = ck forth J. D. Mitchell (St Andrews) Novemer 20th, 2015 11 / 34
R-clsses 2, 2 The R-clsses re the strongly connected components of the right Cyley grph. J. D. Mitchell (St Andrews) Novemer 20th, 2015 12 / 34
L -clsses, 2 2, The L -clsses re the strongly connected components of the left Cyley grph. J. D. Mitchell (St Andrews) Novemer 20th, 2015 13 / 34
The Green s structure The D-clsses re the strongly connected components of the union of the left nd right Cyley grphs. The prtil order of the D-clsses is the trnsitive reflexive closure of the quotient of the union of the left nd right Cyley grphs y its strongly connected components. J. D. Mitchell (St Andrews) Novemer 20th, 2015 14 / 34
Semigroupe V. Froidure nd J.-E. Pin, Algorithms for computing finite semigroups, in Foundtions of Computtionl Mthemtics, F. Cucker et M. Shu (eds), Berlin, 1997, pp. 112 126, Springer. J.-E. Pin, Algorithmic spects of finite semigroup theory, tutoril, www.lif.jussieu.fr/ jep/pdf/exposes/standrews.pdf J.-E. Pin, Semigroupe, C progrmme, ville t www.lif.jussieu.fr/ jep/logiciels/semigroupe2.0/semigroupe2.html The Semigroups pckge for GAP version 3.0 (not yet relesed) J. D. Mitchell (St Andrews) Novemer 20th, 2015 15 / 34
GAP nd Semigroupe J. D. Mitchell (St Andrews) Novemer 20th, 2015 16 / 34
Pros nd Cons Pros: only requires: equlity tester multipliction then we cn run the lgorithm! Does not use the representtion of the semigroup! Cons: hs complexity O( S A ) it cn e costly to multiply elements it cn e costly to check if we ve seen n element efore ll the elements re stored, which uses lots of memory Does not use the representtion of the semigroup! J. D. Mitchell (St Andrews) Novemer 20th, 2015 17 / 34
The limittions of exhustive enumertion n # trnsformtions memory unit 1 1 16 its 2 4 16 ytes 3 27 162 ytes 4 256 2 k 5 3 125 30 k 6 46 656 546 k 7 823 543 10 m 8 16 777 216 256 m 9 387 420 489 6 g 10 10 000 000 000 186 g 11 285 311 670 611 6 t 12 8 916 100 448 256 194 t.... n n n n n n 16 its Storing the elements of semigroup is imprcticl. J. D. Mitchell (St Andrews) Novemer 20th, 2015 18 / 34
Bck to semigroups... Suppose we wnt to compute the trnsformtion semigroup S generted y: ( ) 1 2 3 4 5 = (2 3), = (1 2 3)(4 5), c =. 1 3 3 2 2 We wnt to use lgorithms from computtionl group theory. We do not wnt to find or store the elements of S. J. D. Mitchell (St Andrews) Novemer 20th, 2015 19 / 34
Schreier s Lemm for semigroups Suppose tht S = A cts on the right on set Ω. If Σ Ω, then we denote y S Σ the group of permuttions of Σ induced y elements of the stiliser of Σ in S. If s S is such tht Σ s = Σ, then s induces permuttion of Σ, denote y s Σ. Proposition (Linton-Pfeiffer-Roertson-Ruškuc 98) Let {Σ 1,..., Σ n } e s.c.c. of the ction of S on P(Ω). Then: (i) for every i > 1, there exist u i, v i S such tht Σ 1 u i = Σ i, Σ i v i = Σ 1, (u i v i ) Σ1 = id Σ1 nd (v i u i ) Σi = id Σi (ii) S Σ1 = (u i v j ) Σ1 : 1 i, j n, A, Σ i = Σ j. forth J. D. Mitchell (St Andrews) Novemer 20th, 2015 20 / 34
Stilisers Let S e the semigroup generted y: α 1 1, 2, 3, 4, 5 α 2 1, 2, 3 α 3 1, 3 α 4 1, 2 α 5 2, 3 α 6 3 α 7 2 α 8 1 = (2 3), = (1 2 3)(4 5), c = α 1 α 2 ( ) 1 2 3 4 5. 1 3 3 2 2 α 4 α 3 α 5, c c, c c,, c α 7 α 6,, c J. D. Mitchell (St Andrews) Novemer 20th, 2015 21 / 34 c α 8, c
Stilisers Let S e the semigroup generted y: α 1 1, 2, 3, 4, 5 α 2 1, 2, 3 α 3 1, 3 α 6 3 = (2 3), = (1 2 3)(4 5), c = S {1,2,3,4,5} = (2 3), (1 2 3)(4 5) S {1,2,3} = (2 3), (1 2 3) S {1,3} = (1 3), c S {3} = id, c α 1 c α 2, α 3 ( ) 1 2 3 4 5. 1 3 3 2 2 α 4, c α 5 c α 6 c α 7,, c α 8 J. D. Mitchell (St Andrews) Novemer 20th, 2015 22 / 34, c
Relting the ction nd the R-clsses Proposition Let S e trnsformtion semigroup, let x S, nd let R e the R-clss of x in S. Then: (i) { im(y) : y R } is s.c.c. of the ction of S (ii) { y R : im(y) = im(x) } is group isomorphic to the stiliser S im(x) (iii) if im(y) elongs to the s.c.c. of im(x), then S im(x) = Sim(y). An R-clss R cn e represented y triple consisting of the representtive x the s.c.c. of im(x) the stiliser S im(x). forth J. D. Mitchell (St Andrews) Novemer 20th, 2015 23 / 34
The structure of n R-clss Proposition Let S e trnsformtion semigroup, let x S, nd let R e the R-clss of x in S. Then: (i) { im(y) : y R } is s.c.c. of the ction of S (ii) { y R : im(y) = im(x) } is group isomorphic to the stiliser S im(x) (iii) if im(y) elongs to the s.c.c. of im(x), then S im(x) = Sim(y). The R-clss R c 2 the representtive of c 2 cn e represented y the triple: c 2 = ( 1 2 3 4 ) 5 1 3 3 3 3 the s.c.c. {{1, 3}, {1, 2}, {2, 3}} of im(c 2 ) the stiliser S im(c 2 ) = S {1,3} = (1 3) J. D. Mitchell (St Andrews) Novemer 20th, 2015 24 / 34
The structure of n R-clss Proposition (i) { im(y) : y R } is s.c.c. of the ction of S (ii) { y R : im(y) = im(x) } is group isomorphic to the stiliser S im(x) (iii) if im(y) elongs to the s.c.c. of im(x), then S im(x) = Sim(y). {1, 3} {1, 2} {2, 3} R c 2 c 2 y y J. D. Mitchell (St Andrews) Novemer 20th, 2015 25 / 34
Finding the R-clsses... Input: set A of trnsformtions generting semigroup S. Output: the R-clsses of S. 1: find the ction of S on {1,..., n} the orit lgorithm 2: find the s.c.c.s of the ction stndrd grph lgorithms 3: R := {1} R-clss reps 4: for x R do 5: for A do 6: if (x, y) R for ny y R then see the next slide 7: ppend x to R 8: return R. J. D. Mitchell (St Andrews) Novemer 20th, 2015 26 / 34
Vlidity Suppose tht S =,. If s S, then write Then = 1 R s = min. length of word in nd equl to s. = 1 R if nd only if (, ) R... Suppose R = {r 1 =, r 2,..., r k } contins representtives of R-clsses of elements s S with s < N for some N (nd mye more elements). if s S nd s = N, then s = t or s = t for some t S with t = N 1. (t, r i ) R for some i, nd so (s, r i ) = (t, r i ) R (R is left congruence) The previous lgorithm is vlid! J. D. Mitchell (St Andrews) Novemer 20th, 2015 27 / 34
Testing memership in n R-clss - I If x, y S, then xry implies tht ker (x) = ker (y). For exmple, c 2 = since ( ) ( ) 1 2 3 4 5 1 2 3 4 5 2 3 1 5 4 1 3 3 3 3 = ( ) 1 2 3 4 5 R 3 3 1 3 3 c 2 ker (c 2 ) = {{1}, {2, 3, 4, 5}} {{1, 2, 4, 5}, {3}} = ker (c 2 ). forth J. D. Mitchell (St Andrews) Novemer 20th, 2015 28 / 34
Testing memership in n R-clss - II Is x = ( ) 1 2 3 4 5 R 3 2 2 2 2 c 2? {1, 3} {1, 2} {2, 3} c 2 c 2 c 2 2 R c 2 S {1,3} = (1 3) c 2 (1 3) c 2 (1 3) c 2 (1 3) 2 Every element of R c 2 is of the form: c 2 g i where g S {1,3}. ck forth J. D. Mitchell (St Andrews) Novemer 20th, 2015 29 / 34
Testing memership in n R-clss - III x = ( 1 2 3 4 ) 5 3 2 2 2 2 x R c 2 if nd only if x = c 2 g 2 for some g S {1,3} = (1 3) if nd only if x = c 2 g for some g S {1,3} = (1 3) {1, 3} {1, 2} {2, 3} R c 2 c 2 x x J. D. Mitchell (St Andrews) Novemer 20th, 2015 30 / 34
Testing memership in n R-clss - IV x R c 2 if nd only if x = c 2 g for some g S {1,3} = (1 3) if nd only if (c 2 ) 1 x = g S {1,3} = (1 3) = S {1,3} (c 2 ) 1 x (c 2 ) 1 x J. D. Mitchell (St Andrews) Novemer 20th, 2015 31 / 34 forth
( ) 1 2 3 4 5 = (2 3), = (1 2 3)(4 5), c = 1 3 3 2 2 r 1 12345 1 2 3 4 5 r 2 c 123 1 23 45 r 3 c 123 12 3 45 c r 4 c 2 13 1 2345 r 5 c 123 13 2 45 r 6 cc 13 145 23 r 4 r 7 c 2 13 1245 3 c r 8 cc 13 123 45 r 7 r 9 (c) 2 13 12 345, r 10 c 2 13 1345 2 r 11 c 2 c 3 12345 r 10 r 12 (c) 2 13 13 245 r 1 r 2 r 6 c c c c c, r 3 c r 12 r 9, c r 5 = id R = (1 ( 3)(4 5) R ) 1 2 3 4 5 c = 1 2 2 3 3 r 8 c r 11 ck forth J. D. Mitchell (St Andrews) Novemer 20th, 2015 32 / 34
Complexity In the worst cse the ove lgorithm hs the sme complexity s the Froidure-Pin Algorithm O( S A ) where S = A. The worst cse is relised when S is J -trivil. In the est cse the complexity is the sme s tht of the Schreier-Sims Algorithm. The est cse is relised when S hppens to e group (ut mye doesn t know it). If S = T n, i.e. S hs lots of lrge sugroups nd R-clsses, the complexity is O(2 n ) compred with O(n n ) for the Froidure-Pin Algorithm. J. D. Mitchell (St Andrews) Novemer 20th, 2015 33 / 34
More theory It is possile to generlize the technique descried ove to ritrry susemigroups of regulr semigroup. Exmples include: semigroups of mtrices over finite fields susemigroups of the prtition monoid semigroups nd inverse semigroups of prtil permuttions susemigroups of regulr Rees 0-mtrix semigroups.... The theory is descried in: J. Est, A. Egri-Ngy, J. D. Mitchell, nd Y. Péresse, Computing finite semigroups, http://rxiv.org/s/1510.01868, 45 pges. J. D. Mitchell (St Andrews) Novemer 20th, 2015 34 / 34