On semigroups of matrices in the (max,+) algebra

Texte intégral

(1)On semigroups of matrices in the (max,+) algebra Stéphane Gaubert. To cite this version: Stéphane Gaubert. On semigroups of matrices in the (max,+) algebra. [Research Report] RR-2172, INRIA. 1994. �inria-00074501�. HAL Id: inria-00074501 https://hal.inria.fr/inria-00074501 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On Semigroups of Matrices in the (max,+) Algebra Ste´phane GAUBERT. N˚ 2172 Janvier 1994. PROGRAMME 5. Traitement du signal, automatique et productique. ISSN 0249-6399. apport de recherche. 1994.

(3)

(4) On Semigroups of Matrices in the (max,+) Algebra Ste´phane GAUBERT Programme 5 — Traitement du signal, automatique et productique Projet META2 Rapport de recherche n˚2172 — Janvier 1994 — 19 pages. Abstract: We show that the answer to the Burnside problem is positive for semigroups of matrices with

(5)

(6) entries in the -algebra (that is, the semiring ), and also for semigroups of

(7) -linear projective maps with rational entries. An application to the estimation of the Lyapunov exponent of certain products of random matrices is also discussed. Key-words: Semigroups, Burnside Problem,.

(8). algebra, Lyapunov Exponents, Projective Space.. . (Re´sume´ : tsvp). e-mail: Stephane.Gaubert@inria.fr, tel: (33 1) 39.63.52.58. Unite´ de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Te´le´phone : (33 1) 39 63 55 11 – Te´lećopie : (33 1) 39 63 53 30.

(9) Sur les semigroupes de matrices dans l’alge`bre (max,+) Re´sume´ : Nous montrons que la re´ponse au proble`me de Burnside est positive pour les semigroupes de

(10)

(11) matrices a` coefficients dans “l’alge`bre ” (c’est-a`-dire le semianneau ) ainsi que pour les semigroupes d’applications lineáires projectives a` coefficients rationnels dans la meˆme alge`bre.

(12) On donne une application a` l’estimation de l’exposant de Lyapunov de certains produits de matrices aleátoires. Mots-cle´ : Semigroupes, Proble`me de Burnside, alge`bre.

(13). , Exposants de Lyapunov, Espace projectif..

(14) On Semigroups of Matrices in the (max,+) Algebra. 1. 1 Introduction

(15). -algebra”, In this paper, we give some results for semigroups of matrices with entries in the “

(16) that is, the semiring , denoted in the sequel. This is a particular example of idempotent semiring (that is a semiring such that ), also known as dioid [18, 19, 2]. This algebraic structure has been popularized by its applications to Graph Theory and Operations Research [18, 9]. Linear operators in this algebra are central in Hamilton-Jacobi theory and in the study of certain asymptotics (see [29] for a recent overview). The study of automata and semigroups of matrices with entries in the analogous

(17)

(18) “tropical” semiring has been motivated by some decidability problems in language

(19) theory [28, 34, 26, 35, 36, 20, 21, 23, 24, 25]. From our point of view, the interest of the algebra arose from the study of Discrete Event Dynamic Systems [2, 13], where sequences driven by

(20) -linear equations represent synchronization and saturation phenomena.An account of the related -linear

(21) algebra are a natural extension of system theory can be found in [2, 8, 32]. Automata over the this formalism and have noticeable applications to Discrete Event Systems [15, 14]. In particular, certain finiteness results for semigroups of matrices can be used to compute some asymptotic performance measures (mean-case and optimal-case Lyapunov exponents [15, 14], the latest being essentially equivalent to the

(22) classical nondeterministic complexity [37]). We also mention that the theory of rational series in a single variable is well developed [13, 16, 25]. One basic theorem characterizes rational series by an ultimate periodicity property. This is intimately related with the cyclicity theorem for powers "of ! irreducible matrices [2, 11], which states that such a matrix satisfies for some # and where is the “Perron root” of . Some finiteness results presented here can be seen as an attempt to generalize this simple property to semigroups of matrices. . We first show that the answer to the Burnside problem for semigroups of matrices over $ is positive, which extends a theorem of Simon [34] for the tropical semiring. The main originality by comparison with

(23) Simon’s proof consists in using the -spectral theory. A different proof based on an adaptation of a combinatorial argument of Straubing can be provided in another special class of dioids. Later, we

(24) consider semigroups of -linear projective maps. In a previous paper [15], we showed that under certain coarse irreducibility assumptions, finitely generated semigroups of linear projective maps with rational entries are finite. Here, we extend this result, showing that the answer to the Burnside problem is also positive for semigroups of linear projective maps with rational entries. The rationality assumption is important: we provide a counter example which is based on a kind of system of addition of irrational vectors. The decidability of the limitedness problem for rational series over is also obtained as an easy consequence of the decidability result of Hashiguchi [21, 20] for the same problem over the tropical semiring. We conclude by giving an application where these finiteness results allow a simple computation

(25) automata. of the Lyapunov exponeint of some particular. 2 Statement of the Results Recall that a semigroup % is torsion if for all &(')% , there exists *' Consider the two following properties 1.. % is finite. 2.. % is finitely generated and torsion..

(26). and. ! '.

(27) ,+ . such that &-. /& 10. Obviously, (1) 2 (2). The well known “Burnside problem” consists in finding some classes of semigroups satisfying the converse implication. See [10] for a survey. The answer is positive for matrices with entries in commutative rings [30, 22, 38]. It is also positive in some more exotic semirings, such as the tropical semiring.

(28) 2 . Ste´phane GAUBERT.

(29). . . .

(30).

(31). . . .

(32). (Simon [34]), the “dual” –but non isomorphic– semiring and

(33)

(34) the semiring of rational languages in a single letter 0 (Mascle [28]). We show that this property also holds in the case of : . 2.0.1 Theorem A finitely generated torsion semigroup % . is finite.. Moreover, this result admits an effective translation: . 2.0.2 Theorem It is decidable if a finitely generated semigroup %

(35) . is torsion.. Indeed, the proof provides an algorithm which after some reduction coincides with Simon’s algorithm for the tropical semiring (whose complexity is essentially ). . The proof of Theorem 2.0.1 uses the spectral theory of together with the linearity of the order. In spite of certain generalizations of the spectral theory to others dioids [11, 16], the argument does not seem to extend easily. However, we mention another class of dioids for which the answer to the Burnside problem is also positive. We shall use the natural order of dioids, which can be defined by. 2 2.0.3 Theorem Let . 0. be a commutative dioid such that . . '. Then, a finitely generated torsion semigroup %

(36) !. . . . is finite. 0. (1). . is finite.. . The proof is an adaptation to the dioid’s case of a combinatorial argument of Straubing [38]. This yields

(37)

(38) another proof of Mascle’s finiteness result for matrices over the semiring (but not for the tropical semiring, for which the algebraic order is the opposite of the standard one). Let us give another #"

(39) "

(40) example of nontrivial dioid satisfying (1). Let be a commutative monoid. Then, the dioid of $&% " finite subsets of , equipped with (as addition) and sum of subsets (as product) satisfies the condition of Theorem 2.0.3. We next consider linear projective semigroups. We define the matrix projective space as the quotient of. '$ . by the parallelism relation ". '. ("*) +-,. . +. . /. 0". 1. "1). (we use the notation for the zero element of semirings, in particular, in , ). We write 2 $ . for the quotient semigroup (of “linear projective maps”), and 3 denotes the canonical morphism of multiplicative semigroups .. .. . . $ -7 6 % Let us introduce the subsemiring of $ : 8 :9# ; 8 354. . 2 . . $ . 0. .

(41). . We set 2<8. . % '$ . 9 ;. 3=8. (this. is the semigroup of linear projective maps with rational entries). 2.0.4 Theorem A finitely generated torsion semigroup % !2<8. ' is finite.. . Finally, extend to a theorem of Hashiguchi [21] for rational series over the semiring. we

(42) ?> . See also [20], Simon [36] and Krob [24, 23] for a first extension to. . @. Recall that a dioid is a semiring whose addition is idempotent: ACBAEDA ..

(43) .

(44). . ..

(45) On Semigroups of Matrices in the (max,+) Algebra. 3. 2.0.5 Theorem It is decidable if a rational series & with coefficients in the values of the coefficients of & ,. % . .

(46). &'. '. . . . is limited, that is, if the set of. . . is finite.. 3 Preliminary Results 3.1. Some Results from the. Matrix Theory.

(47) . We next recall the definition and basic properties of the “norm” of a matrix is defined by. . 3.1.1 Lemma Let 1.. . 2.. . 3. 4.. !. *. . '. . . . . . $. . .

(48) ' & '. . . . . . 0. - . The following quantities are equal: + /. ' , + /. '

(49). 0 0 0 ,. . $. spectral radius [17, 7, 2, 9]. First, the. . . '. . . '. #"$%". . .

(50). . . . !. !. . . . . '. &)(((. "$%". &. . . . . . & '. '. &. &. .. $. . This common value will be denoted by +.

(51). .. Of course, ' with the semiring notation of the -algebra means , in the usual algebra. In the $ following, it should be clear from the context whichever algebra is used. However, we shall sometimes write & .- ' to avoid ambiguities.

(52). &. We begin with a Lemma which is almost obvious. 3.1.2 Lemma For all . '. . and ,1# : /. . .

(53). Proof + 0$ 3.1.1,2. For if . . +. .

(54)

(55) . .

(56). . +. $. . +. .

(57)

(58). 0 $. (2). $ follows immediately from Lemma 3.1.1,3. The converse inequality follows from

(59)

(60)

(61) $ $ . 1 , then $ , hence + $ + .

(62). The most useful result of the -matrix theory is perhaps the following cyclicity theorem which

(63) is the exact counterpart of a well known asymptotic 5 property for usual nonnegative matrices. Let " " 32 )4 , .. $ us recall that the matrix is irreducible if / 1# 3.1.3 Theorem ([7, 2, 11]) If. ". ' ,8. where +. "

(64). 76. is irreducible, then " ! " ) 6. denotes the spectral radius of. ". ". 8. .. +.

(65)

(66) ". . 0. (3).

(67) 4. Ste´phane GAUBERT. It is very natural to look for generalizations of this cyclicity property to finitely generated semigroups of 6 6 matrices. To this end, we observe that (3) rewrites as follows in the projective linear semigroup 2 : . " 3.

(68). . " 3.

(69). 0. That is, an irreducible linear projective map is torsion. This suggests to consider finitely generated projective linear semigroups. We just recall here a first result taken from [15, 14], that we shall use and generalize . 6 . a morphism ( denotes the free 000 hereafter. Let be an alphabet of letters, 4 6 product). We consider semigroup on , that is, the set of nonempty words equipped with the concatenation

(70)

(71) 0-0 0 6 . Alternatively, the finitely generated semigroup % if , we shall write % for the semigroup generated by the matrices 0 0-0 6 . Obviously, each finitely generated semigroup %

(72) for some alphabet and morphism . We say that the semigroup % is primitive if can be written 8 there is an integer such that for all word ,. . . . . . . . 8. . 32 )4. 2. . . . . . .

(73). . . . (4). where denotes the length of the word . That is, we require every sufficiently long product of matrices to . be without entries. When % admits a unique generator, this reduces to the notion of primitivity well known in the theory of nonnegative matrices. We say that a set % of matrices is projectively finite if 3 % is finite. . . 3.1.4 Theorem Let finite.. . 000 . . ' 6. 8. . $ . . If . . . . 0 0-0 . . is a primitive semigroup, then it is projectively. 6. . Contrarily to the cyclicity theorem 3.1.3 which is essentially relative to the case # , we require the entries . to be rational (or equal to ). This restriction is important, as it will be shown in §7.1. For the sake of completeness, we include the proof, which exploits some bounding arguments and some norm properties which will be more intensively used hereafter in the study of the Burnside problem for projective linear semigroups..

(74) !. . Proof Let be the of the denominators of the entries of the matrices. Since 6 ( with classical notations) is an automorphism of 8 which maps all the entries to integers, we shall assume that 0 0-0 6 ' > $' . . We have already defined the norm . We shall also need the following dual bound: . (5) (recall that. #". . . . ! ). Obviously, $ $ % $ ' & $ (*) $ ( ' (, $ $ % $ ' & $ $ ' . ( ( ( 0 . . . . . . '. . . . . + '.

(75). . The set. . % of matrices. . . ,. . . The proof relies on the following Lemma. 3.1.5 Lemma Let. . . . '. . >. . 0 7 . . 7. . . . . . (7). . ( + . . is projectively finite. We leave it to the reader to check that this notion is independent of the choice of. . (6). . '$ . such that. . . . . . . . . -. .. and ..

(76) On Semigroups of Matrices in the (max,+) Algebra. (. Let. . . . . . . . . . ' %. +. . . . Indeed, after normalization, we may assume that >. ' . such that matrices ' and. 5. . Since there is at most + . . . . for some indices. . 2 4. . ( . . -0 0 0 0 0-0 . . . 6. . . . ( . .

(77). . . . . . . &. . belonging to the argmax in. . . &. . .

(78) . . . $. This implies that. & . . . .

(79). . $. . #

(80). .

(81)

(82). . .

(83). . .

(84). . . .

(85). . ( . . #. 6.

(86)

(87). 0. &. . . . , the Lemma is proven.. ' long enough, we have a factorization The primitivity assumption implies that8 for 8

(88)

(89) #

(90) !. and & ( is the “primitivity index” satisfying (4)). Then & . .

(91). . .

(92) . . . . . . #

(93). . . . .

(94). &. . . . . . with. . #

(95). (8). . . #

(96). . Moreover. )

(97). . ). . . . #

(98). . .

(99). 0. 0. It remains to apply Lemma 3.1.5 to conclude.. 3.2. 1. Preparation. We first recall or prove some lemmas of general interest. The first one is a well known combinatorial result due to Brown[5]. We say that a semigroup % is locally finite if any finitely generated subsemigroup of % is finite.. .

(100). 3.2.1 Lemma (Brown) Let 4 % 6 be a morphism from a semigroup % to a locally finite semigroup .

(101) Then % is locally finite iff for all idempotent ' , is locally finite..

(102) . We now give some lemma specific to the dioid or of semigroups of matrices over dioids. 3.2.2 Definition Let a proper partition #. . & ' %. 32. '. . . 4. '. . case. We first define a notion of reducibility. . . We say that % is reducible if there exists. be a dioid and % a subsemigroup of such that. 0-0 0 . . .

(103). . . &. .. 0. . It is easily Let be a morphism 6 , and % checked that 6 6 exists a constant permutation matrix , two morphisms 4 4 6 . 6 6 , (with # 4 ), such that.

(104). . . . . ' . . . .

(105). .

(106). . . . . . .. .

(107).

(108) .

(109). . 0.

(110). . % is reducible iff there 6 and a map (9). ! Moreover, this is clearly equivalent to saying that the matrix is reducible (in the usual sense , of the Perron-Frobenius theory). The interest of irreducible semigroups arises from the following Lemma, which shows that, with respect to the Burnside problem, we may only consider irreducible semigroups. ". !. .

(111). Lemma 3.2.3 holds in a dioid (and not in an arbitrary semiring) because of the following property: “in a dioid, the set of all possible sums of the elements of a finite set is finite”.

(112) 6. . Ste´phane GAUBERT. . . . %

(113) . 3.2.3 Lemma Let

(114) and . .

(115). . . % is torsion (resp. finite) iff. be a reducible semigroup satisfying (9). Then are torsion (resp. finite).. It is clear that the condition is necessary. Conversely, an easy induction shows that.

(116)

(117) is a finite sum of elements of the finite set .

(118).

(119). . .

(120). ,. .

(121). hence sufficient.. #

(122). .

(123). .

(124). (10). . ,. .

(125). . . .

(126). . .

(127). We now prove some specific 3.2.4 Lemma Let . . '. $. 2. For all irreducible bloc . '. 3.2.5 Corollary Let.

(128). . and the condition is also 1. lemma.. ' . . The following assertions are equivalent:. is torsion. 1.. . $ . of , +.

(129). .. or .. ' . . The following assertions are equivalent:. is projectively torsion , such that for all irreducible bloc 2. There exists ' 1.. . $. . of , +. $

(130). 1 or . .. . is projectively torsion iff . Corollary 3.2.5 if obtained from Lemma 3.2.4 by noticing that ! , that is iff is torsion. for some . . Proof of Lemma 3.2.4. From Lemma 3.2.3, we may assume that is irreducible. If + is nilpotent, hence it is torsion. Otherwise, it follows from the cyclicity property (3) that

(131) + .. .

(132). Consider the following characterization of the +.

(133). . . $. . .

(134). .. , then is torsion iff 1. spectral radius (cf. Lemma 3.1.1):. . '. &. 0. $. (11). . We next generalize this property to semigroups of matrices.. . . 3.2.6 Proposition Let "

(135). . . 0 0-0 . . 6.

(136). . . . . ". Moreover, the.

(137). 32 . then . recall that !. . % . , and. . . .

(138). . .

(139)

(140). . . D ! ..

(141). . . . . . ".

(142) . . 1' . &. . . for some morphism . ( .

(143). +.

(144)

(145). . "

(146). &. 4. . 6. . ' . Let. 0. , we have. .

(147)

(148).

(149). . is attained in both summations .. Proof of Proposition 3.2.6. Since. . 6. . Then +. . 0 0-0 . &. . . . . " .

(150) . . $. . . =" . $. '. . &. +. (12).

(151) On Semigroups of Matrices in the (max,+) Algebra. by (11). Similarly,. . .

(152). . . .

(153)

(154). &. +. 7. . " . . . . . ". .

(155) $. +. ". . . ". 0 0-0 . &. &. 2. by Lemma 3.1.2. We now prove the converse inequalities. Let . .

(156)

(157). . +. ". . 0 0-0. (13) . '. 2 $. . '.

(158). +. . . 0 &. . # 000 . such that.

(159). .

(160) "

(161). , each entry of corresponds to some entry of one of the . Precisely, for each . "

(162). ' # 0 -0 0 , take & ' # 0 0-0 such that (with the convention # # ). Let

(163)

(164) 0 0-0 . Then. Since . ". . . !. . /. $. . . . . . &. . /. &. '. &. +. The boolean semiring,. . .

(165)

(166). . . . . . .

(167). . . .

(168). . &. . . &. . .

(169). . . . 000. &. &. .

(170). '. ' &. "

(171). $ +. which shows that the converses inequalities in (13) and (12) hold.. /.. . , which can be seen as a subsemiring of will play an important role in the sequel.. . . . . (with . 1. .. . ),. .. 3.2.7 Lemma Let ' . be an idempotent matrix. Then, the irreducible blocs of are either or equal to some matrix , where denotes the / / -matrix whose entries are all equal to .. $ Proof Let. $. . $. $. . be an . irreducible / / -bloc. Since is idempotent, we have reduced to a single element. Otherwise, . $. $ $. . . If +. $.

(172). .. , then. $. is 1. The following Lemma is reminiscent of the theory of (classical) spectral radii of Hadamard products of nonnegative matrices (see Elsner, Johnson, Dias da Silva [12], Theorem 7). 3.2.8 Lemma Let. where. . ". '. . - be irreducible. Then "

(173) . . +. . . ". . . (14). .. ranges the set of diagonal matrix with non diagonal entries. ".

(174). ". . .

(175). . . ". . . . . . + for all . Assume Proof of Lemma 3.2.8. We get from 3.1.1,2 that + "

(176) " . Set. be an eigenvector, + and let by homogeneity that , associated i.e. " .2 # 0 0-0

(177) . Then, ". . Hence, the inf in (14) is equal to ! ! + #"

(178) . 1. . . . 4 Finiteness Results for 4.1. . . .

(179) -Semigroups. It Is Decidable if a Finitely Generated (max,+) Semigroup of Matrices Is Torsion. . We now prove Theorem 2.0.2, which is an extension of Simon’s decidability result [34]. Let 6 ". 6 0-0 0 and set . !. . 4.1.1 Lemma For % to be torsion, it is necessary that +. "

(180). .. %.

(181) 8. Ste´phane GAUBERT. Proof From Proposition 3.2.6, we get ".

(182). +. 1'. for some. . . . If % is torsion, then. .

(183). . .

(184)

(185) . +. &. is torsion, hence, +. . .

(186)

(187). . 1. This allows considering only the following case. 4.1.2 (Canonical form) We shall assume that . 32 . . ". . . . . . ". 0.

(188). . Indeed," provided , condition (15) becomes satisfied if we replace each +. as in Lemma 3.2.8. . . . . Thus, we are reduced to a semigroup of matrices of the elements ). We next introduce some morphisms from Let . 4. We naturally extend morphism. to. . . that we also extend to. 6. . . . E6. .

(189). . . . . . . 4. . .

(190). . 4. . $.

(191). if if .. . .. .. . if if .. 6. .. . . . 0. . .

(192). . .. . if . if if. #. . . It should be noted that in the case of the dioid Simon’s proof [34]. is not a morphism, but we have. . ..

(193). .. , the map analogous to. $ $ $ $ 0 .

(194)

(195)

(196)

(197) . 0 . .

(198). . Hence, is a morphism from following laws and : . . . 9#; %. '.

(199).

(200)

(201). .

(202) . . is the key of.

(203) . .

(204). 9#; %. .

(205). . (16) /.. . #. . equipped with the two.

(206). .. . '. . to the three elements dioid. . 4.1.3 Proposition Let ' following assertions are equivalent.. . .

(207). We claim that:. . 5. . .

(208). to. , with. . . . .

(209). 76. . We naturally extend. . $ . in a similar way. The product morphism is given by. Finally, we introduce the following map. . . $. . ( is the subdioid of comprising and to simpler structures.. .

(210). . . . . (componentwise). In the case on nonpositive reals, there is another useful 4. . . .

(211). by $. . (15).

(212). . . . Let denote the injection. ' ("! # $'%)$*. . . .

(213). $ ..

(214). ' +"! # "!$'%)# $, $%&$. It should be noted that is not a morphism. In particular, stating that is torsion (in stating that is torsion in the dioid , which is always the case since. . 6. . .

(215). . . The. ) has nothing to do with is finite..

(216) On Semigroups of Matrices in the (max,+) Algebra. . 9. is torsion

(217) 2. is torsion. 1.. .. . . 3. for each non strongly connected component of , there exists a circuit of composed only of arcs of weight , .. . 4. each non strongly connected component of Proof of the Proposition. (4). +.

(218). . contains at least a circuit of. .

(219). .. (3) is obvious.. (1) 2 (3): This follows from Lemma 3.2.4 and the fact that since all the entries of with weight has all its entries equal to .. . are . , a circuit. (3) 2 (2). This is clear from Lemma 3.2.4, since the irreducible blocs and the entries equal to are exactly

(220) the same for and . 1 This yields the following extension to. . of Simon’s algorithm [34].. . 4.1.4 Algorithm Compute the finite semigroup satisfies property 4.1.3,3.. .

(221). %. . . There is an equivalent version which uses the maps 4.1.5 Algorithm Compute the finite semigroup. . '. and for each .

(222) % , check that the matrix . and : .

(223). % and check property 4.1.3,4 for each ' . The Theorem is proved.. . When the semigroup. . . . . . % '. .

(224). . . ,. . . . . . . . .

(225). .

(226)

(227). $. . . # . #. . . +. .

(228). . +. . admits a single strongly connected component:

(229) 1 , which is equivalent to being non nilpotent.. #. $. . and define. $ $

(230). . . . . . # . #. . . . . . # #. #. . . #. #. . . . . . . A path of length is a family of 1. The weight of the path @

(231)

(232) @. . . . . . 0. $ is torsion. In order to show that, we reduce % $ # # ". .

(233). %. is torsion iff there is no nilpotent matrix. . 4.1.7 Example Let us consider the matrices. % % 9 ; . . % is primitive, we obtain a particularly simple result:. Proof Obvious from 4.1.5 since for all

(234)

(235)

(236) there is a circuit in iff +. We claim that have. . 1. 4.1.6 Theorem A primitive semigroup

(237) in .. . .

(238) 2. . to the canonical form 4.1.2. We. . indices @

(239)

(240) @ . A circuit is a path such that @ D @ . An arc is a path of length is equal to & ' ' &.

(241).

(242) 10. Ste´phane GAUBERT. ". .

(243). +. and ". %. ". ). . . . . # and set $ $ # # . ". .2. . . . $ is torsion iff ) . . . Let. . ). . . . . . " #. . . .2. . $.

(244). . . ). . #.

(245). 0. . . %$ ). is torsion. According to theorem 4.1.6, we have to check that. nilpotent elements. But. . %$ . . . . )

(246). . . . .. .. )

(247) . $ .. )

(248). . . hence we may bound from below a product of / matrices and ) ) nilpotent, this shows that contains no nilpotent elements, hence. . $. . )

(249). %. . . )

(250). $ . Since by is torsion.. . .

(251). & 4.1.8 Remark We conclude this section with a brief indication of complexity. Because

(252)

(253) .

(254) .

(255) .

(256) three cases are possible: (i) &

(257) , then & , (ii) & and & , then &

(258)

(259) & , then & . (iii) & .

(260) . .

(261). . Hence, the cardinality of are essentially equivalent.. 4.2. .

(262). .

(263). .

(264). In other words, the finiteness of. . Let. . &. . 0-0 0. ' '. .

(265). & , only & #,

(266). . . .

(267). $ .. .. % is equivalent to the following property. . +. . . '. . &. #. . . $. . We have $. . . is not. is Finite. . . 5. .. & ')%. +. 2. 32 4 . & . 0. (17). Proof Condition (17) is clearly necessary. Conversely, assume that (17) holds. Let of % , and let

(268) )4 .

(269) . . )

(270). . be a finitely generated semigroup. % is finite iff the non entries of. ,. . . .

(271). We now prove Theorem 2.0.1. Since the condition (15) must be satisfied, we may assume that % . $. has no. % and % are equal and bounded above by , and the two algorithms. A Finitely Generated Torsion Subsemigroup of. 4.2.1 Proposition Let % the matrices of % are bounded.. ). . .

(272). . . . /. . . . #.

(273). . . . &. . . . 2. . . 0-0 0 . .

(274) '. '. . . $. . . . 0-0 0 6. be generators. 0. . (18) &. . Let 0-0 0 $ .

(275) We for some indices set denote the maximal number

(276) of indices

(277) $ such that in a factorization of type (18). Then . This implies 4# / &

(278)

(279) 8

(280) is bounded. Since the factors of (18) are either equal to or less that , which shows that

(281) than , this shows that can only take a finite number of values. The Proposition is proved. 1.

(282). .

(283). +. 2. . . . . 2. . 2. 2. . 4. 2. . 8. . .

(284). .

(285). . . is finite, it remains to check To show that a finitely generated torsion semigroup % ' that the .

(286) non entries of the matrices of % are bounded. Due to Brown’s Lemma, we may assume that % for some idempotent matrix . According to Lemma 3.2.7, we shall assume that has the following form. . . .. $. . .

(287) On Semigroups of Matrices in the (max,+) Algebra. . . . . 11. (the general case is obtained by an immediate induction). Thus, we have a decomposition of the form (9)

(288)

(289) and . Moreover, for all , We have necessarily with . $. . . . .

(290)

(291). (otherwise + there exists a finite real. . +. . . . .

(292). . 32. . .

(293). . .

(294). . . . . .

(295). '. # . . . '. . . . + . . . contradicts the fact that. such that. . . . (19). . .

(296). . . is torsion). The proof of 3.1.4 shows that .

(297). ( + 0. .

(298). .

(299). .

(300). . . Moreover, Formula (10) shows that It follows from (19) that is bounded below.

(301) This implies that is bounded below and concludes the proof of Theorem 2.0.1.. . 4.3. . Proof of Theorem 2.0.3. . The proof relies on two lemmas. The first one gives a symmetrized version of a classical polynomial identity 6 denote non commuting indeterminates and define (valid in rings). Let 0 0-0. % . . 6. %. 6. . ; . .

(302) . . . ;. 0 0-0.

(303) . 0 0-0.

(304). . 6.

(305) . 6. 99 . . . . . where the sums are taken over the even and odd permutations of # 0 0-0 . The well known Amitsur. Levitski theorem (see e.g. [6]) states that the polynomial identity % % holds in the matrix ring (where denotes a commutative ring). Such combinatorial identities extend to semirings:. . . . % holds in . . . . 4.3.1 Lemma Let be a commutative semiring. Then, the identity %. This can be easily deduced from the classical Amitsur-Levitski theorem by a technique of Reutenauer and Straubing ([33], proof of Lemma 1,2), see also [13], Chapter 1, Proposition 2.1.5 and [16], Proposition 2.2.1. The proof of the theorem simply consists in adapting the argument of Straubing [38] (which shows that a finitely generated torsion semigroup of matrices over a commutative ring is finite) that we reproduce 0-0 0) such that for all completely here. Recall that a word is -divided if it admits a factorization 5 permutation ,. . (8. .

(306) . . 0 0-0. .

(307). . #. . . denotes the lexicographic order). Then, Shirshov’s Lemma8 states that if , there exists an integer

(308)

(309) such that, such that 7 , either admits a factorization for all word ' , either contains a -divided factor (i.e. & with # & where is -divided).. . . . . . . is the order of the semigroup 0 0-0 is torsion), and . We claim that 0 (20)

(310) (with respect to the military order, i.e. “length first, lexicographic . If & , then for some & , hence . Set ( , # which is finite since the theorem assumes that for all , . . .

(311). . 1'. We show this assertion by induction on 8 second”). Let such that . . . . . . .

(312).

(313) . . . .

(314).

(315). . . .

(316) . . . . . . . .

(317) .

(318).

(319). . .

(320).

(321). . .

(322) . .

(323) 12. . Ste´phane GAUBERT. .

(324). . . . )

(325). with & , where . ). . 000 . . . &. . . , and by the induction hypothesis, we are done. Otherwise, we have is -divided. The symmetrized polynomial identity (4.3.1) implies that . . .

(326) . . . . . therefore. .

(327) . . 000.

(328)

(329) . . 9#9. .

(330) . . . . )

(331) . . ). where the are strictly less than . This concludes the proof of (20). Since assumption (1) implies that it is finite.. 5. . .

(332). is bounded above, the 1. The Burnside Problem for Projective Linear Semigroups. The proof of Theorem 3.1.4 suggests that the following quantity will play an important role. . 5.0.2 Definition The projective width of a set. . % . ' is by definition . . . . %.

(333). . . . for some morphism , we shall write If % Theorem 3.1.4, we can state:. . .

(334). . . . .

(335). & &. . .

(336). 0. (21). instead of. .

(337). % . Arguing as in the proof of. - is projectively finite iff its projective width. 5.0.3 Lemma A semigroup %

(338) !8 Thus, we have to show that. . . .

(339). %. is finite. The key point in the proof of the Theorem is the following.. be a reducible projectively torsion semigroup, and take 5.0.4 Lemma Let % Then the following assertions are equivalent: 1. 2.. . . .

(340). is finite .

(341). and. . . .

(342). . . is finite..

(343). are finite..

(344) . . . . as in (9).. Let us assume that the Lemma 5.0.4 is proved. Then, using Brown’s Lemma, with . and the

(345)

(346) canonical morphism, we may assume that for some boolean idempotent matrix ' having the form of Lemma 3.2.7. Therefore, by Lemma 3.1.5, the projective width of all the irreducible

(347).

(348) is finite, hence by 5.0.4, components of , which together with Lemma 5.0.3, gives Theorem 2.0.4.. . . . . . It remains to show Lemma 5.0.4. (1) 2 (2) being obvious, we only prove the converse implication. We have to bound the differents terms of the form & & which appear in & &' in (21).. First, we claim that . . . . . . . . . .

(349).

(350). . . . . $. . . .

(351). . . .

(352). 0. ) . (22). . .,. * . It might seem simpler to speak of “diameter” instead of “width”. We do not use the term “diameter” because there is a natural

(353) . metric on the projective space, namely the “Hilbert’s projective metric” [4] defined by D

(354) .

(355) . When and are two matrices with the same boolean image, the distance coincides with the projective width

(356)

(357) . In other words, coincides with the diameter associated with the metric , but only for a subclass of two-elements sets.. .

(358) On Semigroups of Matrices in the (max,+) Algebra. Indeed, we have. Moreover, since. . . .

(359). .

(360). .

(361). . . (. . . . . . . . . 13. ( (. . .

(362). . . is torsion, by Corollary 3.2.5, . . .

(363). . . . . . +. . . .

(364).

(365)

(366). . . .

(367). .

(368). . +. . . . . .

(369). 0. . .

(370)

(371) . .

(372). . .

(373). . which shows (22). We next give a second bound. Let. +. . . . . ,.

(374). . We claim that.

(375). . . . . . . (with. '. . '. ), hence.

(376). . . . . .

(377). . . . .

(378) . .

(379). . . . . #

(380). . . . ). It remains to show that.

(381) . .

(382).

(383) . .

(384) .

(385). . .

(386). . .

(387). . . . .

(388). . .

(389). .

(390). .

(391). . .

(392). . for some factorization. .

(393). . . . .

(394). . . . . . . (23). .

(395). . .

(396). . ). . .

(397).

(398). .

(399).

(400). (24). .

(401). . . ,. . . . . . . .

(402). . . where . #

(403). .

(404).

(405). . .

(406).

(407) ( + 0 +

(408) (. . . . . . . . . . . .

(409). .

(410). . #

(411). . A dual argument shows that. '. . . . . #

(412). . . . . . . 0

(413)

(414)

(415) (by (7)) + . . (

(416). . .

(417). . . . . ( + Indeed, from formula (10) together with (6) we have

(418) . .

(419). . is also bounded. We have, by (10),(6) and (7): .

(420). .

(421). . (. . . . . .

(422)

(423) . #

(424).

(425). .

(426). . #

(427).

(428). .

(429). . . ' ). We may assume a factorization of the form for some factorizations ( (if it is not the case, we factorize in a dual way instead of ). . 1/ If. . . , we have . .

(430).

(431). . . .

(432). Set. (. . . . . .

(433). . . . .

(434). . . .

(435). . . (

(436) ( +

(437) .

(438). . .

(439). . . . . )). ,. . . (

(440) ( (.

(441). . .

(442).

(443). .

(444). . . .

(445). . . . . .

(446). . . .

(447). . .

(448).

(449). . . . . . .

(450).

(451). . . . or.

(452) 14. Ste´phane GAUBERT. We get from (22):.

(453). .

(454) . 2/ When. . . . .

(455). . .

(456). . +. , we set. . . Let us define. + ,+ ,+ ). . )).

(457).

(458). . . . .

(459). .

(460). .

(461). . .

(462). . . .

(463). 0. + ,+ ,+ . Putting together the above bounds, we obtain + + + + + + + 0 ). in a way similar to . . . + . . . .

(464) (. .

(465).

(466). . and obtain by a similar and simpler argument

(467).

(468). . ,. . . . + . . )).

(469) . . .

(470). . )

(471). . . )). . .

(472). . )). .

(473). ))

(474). . . )

(475). . .

(476). .

(477).

(478). . . . The Lemma is proved.. 6. 1. Decidability of the Limitedness Problem. We now prove Theorem 2.0.5. 6.0.5 Lemma Let & be the rational series with trim linear representation . "

(479) If & is limited, then + or . Proof 1/ Assume that. .. "

(480). +. . . .

(481). and let. ". !. .

(482) .. ,. . It follows from . &. .

(483) . . . . ". . . . . +. .

(484). +. ".

(485). . . 6 and from Lemma 3.1.1,(4) that there exists some constant such that &' as +

(486) . 6 & . Since the representation is trim, this implies that takes arbitrary small (i.e. near ) . non values. Hence & is not limited. "

(487) " 2 Assume $ 2/ that . From 3.1.1,3, there exists and / such that . Moreover, we have + "

(488) )4. $ for some word . Since the linear representation is trim, there exists some / and two words such that 5

(489)

(490)

(491) .. . . Hence, for all for some. +. 5. . . .. 1# , . Since *. . . . 6. ". .

(492) 6. $. &'. . 6. . .

(493). $. +. . . $ . ".

(494). 0. 6. $. , & is not limited. 1. Due to Lemma 6.0.5, we may perform the same reduction as in the decision algorithm for the torsion property (see 4.1.2). Therefore, we may assume that. . (but. and. . . .

(495). . . .

(496). .. can have positive entries).. The conclusion is an immediate consequence of the following Lemma.. (25).

(497) On Semigroups of Matrices in the (max,+) Algebra. 15 .. 6.0.6 Lemma Assume that (25) holds, then & is limited iff the non values of That is, we require that ,. +. 5. . . 5.

(498). &'. . .. . 2.

(499). &'. + . . .

(500). &'. . are bounded below.. 0. (26). Proof of Lemma 6.0.6. This is a straightforward variant of the proof of Proposition 4.2.1. 1. It remains to show that the boundedness property (26) is decidable.. . 6.0.7 Lemma Under the assumption (25), the boundedness property (26) holds for & iff it holds for the series

(501) ) & given by the linear representation .. +. .. Proof Some routine calculus show that there exists two non constants. +. .

(502) . &'. . . ). &. +.

(503) . . ). &.

(504). . and. +. ). such that. 0 1. ). ). the limitedness problem for the series & is decidable (since the representation of & lives in

(505) Since

(506) . . , which is isomorphic to the tropical semiring, this follows from Hashiguchi’s theorem [21], see also [20, 35, 36, 24, 26]), Theorem 2.0.5 is proved.. 7 Related Remarks 7.1. Counter Example. We show that the theorem of finiteness of primitive semigroups of projective linear maps with rational entries does not extend to the irrational case. Let. . . . . $ # 0 # is a small parameter to be fixed soon and . . #. . %$ . where is infinite. In 0-0 0 . We claim that 3 order to show that, we recall that linear projective maps can be identified to homographic functions –as in the conventional algebra–. More precisely, we set for ' ,.

(507) . We have. . %. 9#;.

(508). .

(509). . where the homographic map associated with is.

(510). . Similarly, we have.

(511). % 9# ;. . . . . . . 2.

(512). . #.

(513).

(514). %. 9#;. . . Clearly, it is enough that the semigroup to ,show We assume that and note that. .

(515). . . . . . 3. .

(516)

(517). . $.

(518). . . 0. 0. . .

(519)

(520). 0. (equipped with the composition product) is infinite. # 0. .

(521). . .

(522) 16. Ste´phane GAUBERT. . . . . . . . Let assume that. take diameter of # (for instance, # . Then, the us further is greater than all ' , either # , hence, ' ,

(523) either # '

(524) for

(525). .

(526) Let us define a sequence by and or (we choose of '

(527)

(528) , then, the choice is arbitrary).

(529) ' so that , if both This “walk” in is visualized. ' . # . On the on Figure 1. Starting from , we may choose or.

(530)

(531) . Similarly, we have set

(532)

(533) . Then, because

(534)

(535) , the picture,

(536) # . Let /

(537) (resp. /

(538) ) denote the number of choices of only possible choice is. . . . . . . . . (resp.. .

(539). .

(540). . &. . . &. /. . $.

(541). . .

(542) . $ # 0 . . . . # ). We have

(543). $ are nondecreasing functions of . Since and and and # # are linearly independent 0-0/0 are over , all the distinct. Since the are images of by some

(544) . .

(545) . % $ , this shows that (hence ) is infinite. elements of . Moreover, . . . $. . 2. sequence

(546) ) up to step (e.g., for the walk of the picture,

(547) , . .

(548). . . . Figure 1: The. . $. . . . . .

(549) . /. . /. .

(550). /. . . /.

(551). . /. . . 8. . 7.2. $. /. . .

(552). /. . /. . /. $. /. . . $. An Upper Bound for the Lyapunov Exponent . 0-0 0 (with ' ) occurs word the We consider a random walk '

(553) $ . That is, the

(554)

(555)

(556)

(557) $ $ $ with 0-0 0 $ where the #. probability $ 6 6 are given nonnegative numbers such that $ be a morphism. The maximal Lyapunov exponent of is defined as the limit Let 4 6. . . . . . . . . . *. *. . $. . . $. .

(558) $. . '. .

(559). . . '. . &. 0& 0 &. $. (see [2],[15]). We just notice here that there is a subclass of matrices for which the Lyapunov exponent if immediately obtained due to the characterization of finite semigroups. 7.2.1 Theorem Let. 4. . . 6. ' such that . . ' . 4. . . . +. .

(560). .

(561)

(562). %. 9;. . +. with the associated “worst case matrix” ". . . . . . . 5. .. .

(563). and define the morphism:

(564)

(565). . .

(566). 0. ,. Then. Moreover, if. . .

(567). . " +.

(568). . +. . .

(569)

(570). #" +.

(571). ,. . . is torsion, then the equality holds in (27).. +. . .

(572)

(573). .

(574) 0. (27).

(575) On Semigroups of Matrices in the (max,+) Algebra. 17. It should be noted that we have by construction. . hence, we can decide if algorithms 4.1.5,4.1.4.. . . " .

(576).

(577). . . . . +. +.

(578)

(579). is torsion by checking the non trivial inequality . . +. . . .

(580)

(581). .

(582) . . +.

(583)

(584). . .

(585) . and by using the. $ of Example 4.1.7 can be obtained almost is torsion, we have # 0 . 7.2.2 Example The Lyapunov exponent of the semigroup without computation. Since the normalized semigroup. . " +.

(586).

(587).

(588).

(589). . 7.2.3 Example In the same vein, we can build many semigroups whose Lyapunov exponents are immediately obtained. Let. . . . . . #. . #. .

(590). . .

(591). . . . . where * stands for arbitrary finite real numbers (we allow different values) and stands for arbitrary

(592)

(593)

(594) finite real numbers . Let (Kronecker’s ). Then, ' ' . 4 # , #

(595) . , is non nilpotent for all . Hence the condition of Theorem 4.1.6 is satisfied, and since # and we have. . . . . . #. . . .

(596) .

(597). . . .

(598) . hence . . . .

(599).

(600). . +. . . &. . .

(601)

(602). . . . .

(603)

(604). . +. We have Setting. ". !. ,. . *. . . . . . . .

(605)

(606). . ".

(607). is finite, then. . +. . .

(608)

(609). &. . . +. . .

(610)

(611). . ,. . . #".

(612)

(613) 6.

(614). . ,. . .

(615). . +. . &.

(616)

(617). 6. . . . . ),. (28) .

(618). &.

(619)

(620). +.

(621). +. . . ,. (.

(622). 6. +. " .

(623) . .

(624)

(625). +. +. *. .

(626)

(627).

(628) , we obtain. Then, it follows from (29) that. Moreover, if in (30).. . +. .

(629)