Generating finite transformation semigroups: SgpWin

Transcription

1 Generting finite trnsformtion semigroups: SgpWin Donld B. McAlister ( don@mth.niu.edu ) Deprtment of Mthemticl Sciences Northern Illinois University nd C.A.U.L. Septemer 5, 2006 Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd5, C.A.U.L.) / 34

2 SgpWin (Semigroup for Windows) Written in C++ s stndlone progrm Originlly written for PC under DOS ut rewritten for Windows Not prt of mjor project like GAP Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd5, C.A.U.L.) / 34

3 How mny semigroups re there? Numer of elements Numer of groups Numer of semigroups Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd5, C.A.U.L.) / 34

4 How ig re they? Size of set Mx. size of group Mx. size of semigroup Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd5, C.A.U.L.) / 34

5 Consequences for generting semigroups Storing the elements internlly quickly ecomes imprctile Solutions? Don t store the elements. Insted e stisfied with some structurl informtion Lllement nd McFdden On the determintion of Green s reltions in finite trnsformtion semigroups, J. Symolic Computtion (10) (1990), (5), Do store the elements Be stisfied with smll semigroups Froidure nd Pin, Algorithms for computing finite semigroups, Foundtions of Computtionl mthemtics (Rio de Jneiro), (1997), Springer, Berlin SgpWin Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd5, C.A.U.L.) / 34

6 Wht do we men y generte? Let X e finite set nd A (finite) set of distinct trnsformtions on X. By the universl properties of the free semigroup A + on A, there is unique homomorphism φ of A + onto A which mkes the digrm A A + φ A commute. (The un-leled mps re the ovious inclusions.) To generte A mens here 1 to specify set of representtives of the clsses of the congruence φ φ 1, with the memers of A s the representtives of their clsses. 2 to e le to sy how these representtives multiply. For this it suffices, given representtive x nd A, to e le to specify the representtive of the clss of x. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd5, C.A.U.L.) / 34

7 Generl strtegy 1 List, in some wy, the elements of A + nd, for ech element x, clculte the corresponding trnsformtion T (x) 2 If T (x) hs lredy ppered in the list of constructed trnsformtions, discrd x nd move on to the next memer of A +. If it hs not ppered dd x nd T (x) to the list nd move on to the next memer of A +. Prolems This isn t n lgorithm. It hs no stopping rule. Both clculting the trnsformtions from scrtch nd serching to see if we hve new trnsformtion require time nd effort. serching, in prticulr - Google notwithstnding - requires lot of work The process doesn t tke dvntge of work we hve lredy crried out, nd informtion we my hve gined. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd5, C.A.U.L.) / 34

8 The short-lex ordering This egins with totl ordering on A which is then extended to comptile totl ordering on A +, s follows { length(u) < length(v) u < v or length(u) = length(v) nd u = xu, v = xv where x, u v A,, A nd <. This ordering is comptile with mulipliction on oth sides in A + - in fct, it is comptile well ordering. Ech φ φ 1 -clss hs lest memer. We cll these the cnonicl words nd use them for the clss representtives in generting A. Lemm The set C of cnonicl words is fctoril in the sense tht w C implies tht ech fctor of w elongs to C. If Y is fctoril suset of A + nd w Y, A, then Y {w} is fctoril if nd only if w = v where v Y, A. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd5, C.A.U.L.) / 34

9 The lgorithm - Step 1 The lgorithm constructs the Cyley grph of A - ctully the Cyley grph of A with n extr identity djoined. This gurntees tht we hve root node. The set C denotes the set of nodes constructed so fr. It is fctoril set. The lgorithm proceeds y trversing the su-tree of A + corresponding to the cnonicl words using the short-lex ordering. 1 The genertors A re nodes of the Cyley grph. Thus we egin with C = A. There is n rrow from the root to for ech genertor. root Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd5, C.A.U.L.) / 34

10 The lgorithm - Step 2 Becuse the cnonicl words form fctoril set, new cnonicl word of length n + 1 must hve the form w where w is cnonicl word of length n. 2. Suppose tht w is cnonicl word (which we hve constructed) nd tht we hve delt with ll cnonicl words smller thn w nd ll genertors smller thn. This mens tht the rrows in the Cyley grph strting t nodes smller thn w hve een constructed nd tht the sme is true for ll rrows strting t w leled y letters smller thn. We need to construct the rrow w? in the Cyley grph strting t w with lel. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

11 Step 2 - continued Test to see if C {w} is fctoril set. By the lemm erlier, this occurs if nd only if w = v (in A + ) where A nd v C. 1 If not then w cnnot e cnonicl word, nd v lso is not cnonicl word. The cnonicl word u such tht uφ = vφ is smller thn v so u < v = w while u nd w hve the sme imge under φ, nd thus the sme representtive. By our induction hypothesis, we cn find this representtive z. Drw the rrow w z 2 If C {w} is fctoril, then w my or my not e cnonicl word. We cn t deduce this internlly; tht is using the prt of the Cyley grph we hve constructed so fr. We need some externl help. For this we use n orcle who cn nswer the question for us. 1 If w is new, drw n rrow from w to w with lel nd dd w to C. 2 If w = v, s trnsformtions, drw n rrow from w to v with lel. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

12 Step 3 - the next step 1 If is not the lst genertor, replce y the next genertor nd repet Step 2. 2 if is the lst genertor ut w is not the lst memer of C on its level, replce w y the next memer of C on tht level, replce y the first genertor nd repet Step 2. 3 If w is the lst memer of C on this level 1 if C contins n element on the next level, replce w y the first such memer nd y the first genertor nd repet Step 2. 2 otherwise QUIT. The Cyley grph is complete. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

13 Algorithm - comments In the first plce, it does not explin, in Step 2, how to determine whether or not C {w} is fctoril. Nor does it explin, in cse it is not, how to find the cnonicl word equivlent to w. Theorem In Step 2(), it does not explin how the orcle works. The lgorithm gives, for free, presenttion for S = A. There re two wys in which ckwrd rrows rise 1 those which rise from old informtion when C {w} is not fctoril; 2 those which rise from new informtion provided y the orcle; these give set P = {w = v} of reltions. S = A P. P is the Knuth-Bendix presenttion of S ssocited with the ordered lphet A. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

14 Algorithms + dt structures = progrms (N. Wirth) A semigroup element is considered s dt structure with two prts node dt rrows The node dt consists of things tht chrcterize the elements of the semigroup The rrows re pointers, one for ech genertor, to other elements. They represent the dynmics of the Cyley grph. The semigroup itself consists of n rry of pointers to elements. We could use list insted of n rry which would llow us, theoreticlly, to del with semigroups of ritrry size. I chose n rry when I first strted to write the progrm nd never chnged this. onld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

15 Node dt At its most sic level, the node dt might e considered to consist of word ction where the word is the cnonicl word equivlent to the element nd the ction represents the ction of the trnsformtion corresponding to the element. It is more convenient to include third item word ction structurl informtion where the structurl informtion contins dt which is useful for semigroup theoretic nlysis. onld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

16 Completing the lgorithm One of the questions not yet fully nswered in the lgorithm is the following. Given cnonicl word w nd A, with w = v how do you know if v is cnonicl word? 1 Strt t the root nd trvel the pth leled y v in the the Cyley grph to rech the node whose word is v. 2 If v = v then v is cnonicl word nd we consult the orcle out w. 3 If not w = v in S nd we need to get the cnonicl word for v. Since v < w we cn find this y mking second voyge on the Cyley grph to rech the pproprite node. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

17 Avoiding words Insted of storing the cnonicl word w = uc, store c nd pointer to u (the prefix of w). prefix (u) lel (c) In the grph we hve the rrow c u w Then w cn e recovered y ck-trcking to the root throught the predecessors. lels, prefixes, nd suffixes come for free in the lgorithm. If we store the suffixes s well s prefixes nd lels we cn cut down the numer of trips on the Cyley grph y one. This results in sustntil increse in speed especilly if we still store the cnonicl words. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

18 The orcle The orcle here is just inry tree whose nodes re pointers to the semigroup elements. It uses inry serch to determine whether or not given ction is tht of one of its memers. If it is, it returns the ddress of the semigroup element. If it isn t - the semigroup hs new element - it dds the dress of the new element to the tree. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

19 Anlyzing the structure The dt structure prt of ech element consists of L-clss Rep R-clss Rep D-clss Rep imge rnk type All of these re un-signed integers except for type which is chrcter. Given tht we use n rry to contin the elements of the semigroup, it is nturl tht integers - the indices of the elements - re used to locte the representtives of the vrious Green s clsses, Type is one of eight chrcters i h r n idempotent non-idempotent group element regulr non-group element non-regulr element The sme letter cpitlized indictes tht the element is lso genertor. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

20 Anlysing the structure - Green s reltions The lgorithm used is hyrid one. It uses 1 the fct tht it is esy to trvel on the Cyley grph to clculte the R-clsses; 2 the fct tht we re deling with trnsformtion semigroup to clculte L. The lgorithm is trditionl in tht it first clcultes the regulr Green s clsses - those tht contin idempotents - nd then uses this informtion to find the non-regulr ones. Lemm Let S e finite trnsformtion semigroup nd let e e n idempotent. Then S elongs to the R-clss of e if nd only if rnk() = rnk(e) nd e S. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

21 Regulr R-clsses In order to use this lemm, we need to e le to identify the idempotents. To do this we use itwise opertions to otin representtion of the imge of trnsformtion s n un-signed integer Set Action Imge = = 15 The rnk is just the numer of 1 s in the representtion. It is esy while finding this to determine whether or not the element is n idempotent - or group element. Bitwise opertions on the imge lso mke to esy to determine, for exmple, if n idempotent e is right identity for n element. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

22 Regulr R-clsses The R-clss of e consists of ll the elements of S, with the sme rnk s e, from which it is possile to trvel to e. To find these we construct the djoint grph Adj y turning the rrows round: there is n rrow in Adj from to if nd only if rnk() = rnk() nd there is n rrow in the Cyley grph from to. the R-clss of e consists of ll elements in Adj which cn e reched from e. Finding these is simple mtter. this sugrph of Adj is fmilir oject. It is wht Buck Stephen clls the Schützenerger grph of e. onld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

23 Regulr L-clsses Now tht we know the regulr elements, we re le to use the clssicl result Lemm Let S e finite trnsformtion semigroup nd let nd e regulr elements of S. Then L if nd only if imge() = imge(). to find the regulr L-clsses. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

24 D-clsses Once we know the regulr L nd R-clsses it ecomes stndrd mtter, using Green s reltion theory, to identify the regulr D-clsses. To find the non-regulr clsses we mke use of the fct tht the non-trivil non-regulr L nd R-clsses re trnsltions of the regulr ones. This is the process used y Lllement nd McFdden. A detiled description cn e found in their pper. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

25 Finl remrks The structurl informtion strored with the semigroup elements, llied with stndrd semigroup theory llows us to chrcterize mny semigroups efficiently in terms of their structure. For exmple, result of FitzGerld shows tht regulr semigrop is orthodox if nd only if the product of D- relted idempotents is idempotent. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

26 Trce congruences Let S e finite inverse semigroup nd ρ e congruence on S. Then the restriction π of ρ to the set E of idempotents of S is congruence on E with the dditionl property tht (e, f ) π ( 1 e, 1 f) π for ech S. Such congruence on E is clled trce congruence nd π = tr(ρ) is clled the trce of ρ. The reltion on the lttice of congruences on S defined y ρ τ tr(ρ) = tr(τ) is esily seen to e complete lttice congruence. (, ) π min ( 1, 1 ) π nd e = e for some e = e 2 with (e, 1 ) π (, ) π mx ( 1 e, 1 e) π for ll e E. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

27 Trce congruences Suppose tht π is trce congruence. Then, ecuse S is finite, ech π clss hs minimum memer. We cll these idempotents the π-miniml idempotents if e is n idempotent then ε(e) denotes the minimum memer of the π-clss contining e. In terms of these we cn give simpler description of π min : (, ) π min ε( 1 ) = ε( 1 ) = ε nd ε = ε. Furthermore, the π-miniml idempotents determine the idempotent nd Green s reltion structure of S/π min. For ech S denote y [] the π min -clss of. Let e e π-miniml idempotent. Then the mp [] is groupoid isomorphism of the D-clss of e onto the D-clss of [e] in S/π min. The prtilly ordered set of D-clsses tht contin π-miniml idempotents is isomorphic to the set of D-clsses of S/π min. The semiltice of idempotents of S/π min is isomorphic to the prtilly ordered set of π-miniml idempotents. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

28 Trce congruences It follows tht if we cn efficiently find the π-miniml idempotents we cn efficiently red off the structurl constnts of the quotient semigroup S/π min. Indeed, since every quotient of S/ρ, with tr(ρ) = π, is n idempotent seprting homomorphic imge of S/π min, we cn red off the the Green s reltion nd idempotent structure of S/ρ. Even if, in generl, we cnnot red off the sugoups. If S is trnsformtion semigroup on finite semigroup then there is unique idempotent of lowest rnk in ech π-clss. These elements re precisely the π-miniml idempotents. onld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

29 Trce congruences - the lgorithm We use wht is essentilly the sieve of Eristothsenes - the wy one finds the primes - twice. Order the idempotents of S in order of incresing rnk, so tht the lest idempotent z is the first in the list. Set curr = z. Go througth the list, strting t curr nd delete ny idempotent e which is π-equivlent to curr. At the end of the list, move curr to the next remining memer nd repet. If there is no remining memer end. At this stge we hve, up to isomorphism, the idempotents of the quotient semigroup. We re-sieve this time using D insted of π. Now, to give the D-clss structure, ll we hve to do is to print out the D-clss dt for ech idempotent in the list. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

30 Primitive trce congruences Let D e D-clss of S nd define π D y (u, v) π D {e = e 2 D : e u} = {e = e 2 D : e v} π D is trce congruence. We sy it is primitive trce congruence. Every trce congruence is n intersection of primitive trce congruences. The numer of trce congruences cn e very lrge in reltion to the numer of D-clsses. For exmple, if S = E is chin with n elements then S hs 2 n 1 (trce)-congruences. onld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

31 Finite Trnsformtion Semigroups Fim! Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

32 Exmple = ( ) nd = ( ) () onld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

33 Exmple = ( ) nd = ( ) () onld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34

34 Exmple - Presenttion () Presenttion: =, =, =, =, =. Donld B. McAlister ( don@mth.niu.edu ) (Deprtment Finite trnsformtion of Mthemticl semigroups Sciences Northern Illinois University Septemer nd 5, C.A.U.L.) / 34