Partial Conway and iteration semirings

P artial Con w a y and it eration semirings S.L. Blo om Dept. of Computer Science Stev ens Institute of T ec hnology Hob ok en, NJ, USA Z. ´ Esik Dept. of Computer Science Univ ersit y of Szeged Szeged, Hungary GRLMC, Ro vira i Virgili Univ ersit y T arragona, Spain W. Kuic h Inst. for Discrete Mathematics and Geometry T ec hnical Univ ersit y of Vienna, Vienna, Austria Abstract A Con wa y semiring is a semiring S equipp ed with a unary o p eration ∗ : S → S , alw ays called ’star’, satisfying the su m star and pro duct star identities. It is known that th ese identities imply a Kleene type theorem. Some computationally important semirings, such as N or N rat h h Σ ∗ i i of rational p ow er series of w ords on Σ with co efficients in N , cann ot ha ve a total star op eration satisfying the Conw ay identities. W e introduce here p artial Conway semirings , which are semirings S which h a ve a star op eration defined only on an ideal of S ; when the arguments are appropriate, the op eration satisfies the ab ov e identities. W e develop the genera l theory of partial Con wa y semirings and pro ve a Kleene theorem fo r this generalization. 1 In tro d uction It is well-known that there exists no finite ba se o f identit ie s for the r egular languages eq uipped with the op erations o f unio n +, pro duct · and (Kleene) star ∗ ; cf. [6, 18, 19]. The notion of Conw ay semirings in volv es t wo imp ortant identities for the star opera tion: the sum star and the pro duct star identities, ( a + b ) ∗ = a ∗ ( ba ∗ ) ∗ ( ab ) ∗ = 1 + a ( ba ) ∗ b. It has b een s hown that Kleene’s theorem for la nguages and a utomata, a s w e ll as its ge neralization to w eig ht e d automata, are consequences of these identities. Th us , it is p os sible to derive Klee ne’s theorem by pure ly equationa l r easoning from the axioms of Co nw ay semirings, cf. [6, 3, 13]. Impo rtant ex amples of Conw ay se mir ings are • the b o olea n semiring B ; • the semiring s B rat h h Σ ∗ i i of rationa l p ow er ser ies with c o efficients in B , which are isomorphic copies of the semir ings of regular langua g es, • the contin uous o r complete semirings [6, 8, 3]. 1 How ever, man y computationally impor ta nt s emirings do not hav e a totally defined star o p eration satisfying the Conw ay identities. Some examples of such semirings a re the semirings S rat h h Σ ∗ i i of ra tional pow er s eries o ver Σ with co efficients in the semiring S , where S is either the s e mir ing N o f natura l num b ers o r a nontrivial ring (if 1 ∗ = 1 ∗ · 1 + 1, then 0 = 1). The semiring N can be embedded into a Co nw ay semiring , namely the semir ing N ∞ obtained by adding a p oint ∞ . By means of this embedding, Kleene’s theore m for Conw ay semiring s b eco mes indirectly applicable to weigh ted finite automata ov er N . On the other hand, such an em b edding do es not exist for all semirings, so that Kleene’s theorem for Conw ay semiring s do es no t cover w eighted finite automata ov e r such semirings. In this paper , w e intro duce p artial Conway semirings as a generaliza tion of Conw ay semirings. In a partial Conw ay semiring S , the domain D ( S ) of the star op e ration is an ide al of the semiring; further, when restricted to this domain, the sum s tar and pro duct s tar iden tities hold. W e prove a Kleene theore m for par tial Conw ay semirings, and thus obtain a single unified r e sult which is directly applicable in a ll of the ab ov e situations. W e also o utline the g eneral theory of pa rtial Conw ay semirings which parallels with the theo ry of Conw ay semirings. This g eneral theo ry provides the background for the Kleene theorem ment io ned above. Moreov e r , we also introduce p artial iter ation semirings , which a r e pa rtial Con way semirings satisfying Conw ay’s group identities, cf. [6, 16]. W e define p artial iter ative semirings as star semirings in which certain linear eq uations hav e unique so lutions. W e prov e that pa r tial iterative semirings a re partial iteration semirings. As a n a pplica tion o f this r esult, we show tha t for any semiring S and set Σ, the p ow er series s e miring S h h Σ ∗ i i is a pa rtial iterativ e semiring and thus a partial iteration semiring. The results of this pap er ar e used in [4], where the semirings N rat h h Σ ∗ i i ar e characterized as the free pa rtial itera tion s emirings, a nd the s e mir ings N rat ∞ h h Σ ∗ i i as the free algebra s in a subv a riety of iteration semirings sa tisfying three additional simple identities. 2 Semirings A semiring [14] is an alg ebra S = ( S, + , · , 0 , 1) such that ( S, + , 0) is a co mmu ta tive monoid, ( S, · , 1 ) is a monoid, moreov er 0 is an absorbing elemen t with resp ect to m ultiplicatio n and pro duct distributes over sum: 0 · a = 0 a · 0 = 0 a ( b + c ) = ab + ac ( b + c ) a = ba + ca for all a, b , c ∈ S . The op era tion + is called sum or ad dition , and the op eratio n · is called pr o duct or multiplic ation . A semiring S is called idemp otent if a + a = a for all a ∈ S . A morphis m of s emirings pr eserves the sum and pro duct o per ations and the constants 0 and 1. Since semirings are de fined by ident ities, the class of all semiring s is a v ariety (see e.g., [15]) as is the cla ss of all idemp otent semirings. An imp ortant ex a mple of a semiring is the semiring N = ( N , + , · , 0 , 1) of na tural n umber s equipp ed with the usual sum and pro duct o p e rations, a nd an imp ortant exa mple of an ide mp o tent semiring is the b o olean semiring B whose underlying set is { 0 , 1 } and whose sum and product op er ations 2 are the o pe rations ∨ a nd ∧ , i.e., disjunction and conjunction. Actually N and B are r esp ectively the initial semiring and the initial idemp otent semiring. W e end this section by describing three cons tr uctions on semirings . F or more information on semirings, the reader is r e fer red to Golan’s b o ok [14]. 2.1 P olynomial semirings and p o wer series semirings Suppo se that S is a s e mir ing and Σ is a set. Let Σ ∗ denote the free mo no id of all words ov er Σ including the empty word ǫ . A formal p ower series , or just p ower series o ver S in the (noncommuting) letter s in Σ is a function s : Σ ∗ → S . It is a common pr actice to repres ent a power series s as a forma l sum P w ∈ Σ ∗ ( s, w ) w , where the c o efficient ( s, w ) is w s , the v alue o f s on the word w . The supp ort of a s e ries s is the set supp( s ) = { w : ( s, w ) 6 = 0 } . When supp( s ) is finite, s is called a p olynomial . W e let S h h Σ ∗ i i and S h Σ ∗ i resp ectively denote the c ollection of a ll power ser ies and po lynomials ov er S in the letters Σ. W e define the sum s + s ′ and pr o duct s · s ′ of tw o series s, s ′ ∈ S h h Σ ∗ i i as follows. F or all w ∈ Σ ∗ , ( s + s ′ , w ) = ( s, w ) + ( s ′ , w ) ( s · s ′ , w ) = X uu ′ = w ( s, u )( s ′ , u ′ ) . W e may identify any elemen t s ∈ S with the series , in fa ct p olyno mial that maps ǫ to s and all other elements of Σ ∗ to 0. In particular, 0 a nd 1 ma y b e view ed a s p oly no mials. It is w ell-known that equipp ed with the ab ov e op er a tions and consta nt s , S h h Σ ∗ i i is a semiring which contains S h Σ ∗ i as a subsemiring. The semiring S h Σ ∗ i can b e c ha r acterized b y a univ er sal prop erty . Consider the natura l embedding of Σ into S h Σ ∗ i such that each letter σ ∈ Σ is mapp ed to the p olynomia l whos e supp ort is { σ } which maps σ to 1 . By this embedding, w e ma y v iew Σ as a s ubset of S h Σ ∗ i . Recall also that each s ∈ S is identified with a polynomia l. The following fact is well-kno wn. Theorem 2 .1 Given any semiring S ′ , any semiring morphism h S : S → S ′ and any function h : Σ → S ′ such that ( sh S )( ah ) = ( ah )( sh S ) (1) for al l a ∈ Σ and s ∈ S , t her e is a unique semiring morphism h ♯ : S h Σ ∗ i → S ′ which extends b oth h S and h . The condition (1) mea ns that for an y s ∈ S and letter a ∈ Σ, sh S c ommutes with ah . In particular, since N is initial, and s ince when S = N the conditio n (1) holds automatica lly , w e obtain that any map Σ → S ′ int o a semiring S ′ extends to a unique semiring morphism N h Σ ∗ i → S ′ , i.e., the p olynomial semiring N h Σ ∗ i is freely genera ted by Σ in the v ariety of semirings. In the s ame wa y , B h Σ ∗ i is freely generated by Σ in the v ariety of idempo tent s emirings. Note that a se ries in B h h Σ ∗ i i may b e identified with its supp ort. Thus a series in B h h Σ ∗ i i corre s p o nds to a language ov er Σ and a p olynomia l in B h Σ ∗ i to a finite la nguage. The sum opera tion corr esp onds to set union and the pro duct opera tion to c oncatenation. The constan ts 0 and 1 are the empt y set and the singleton set { ǫ } . 3 2.2 Matrix semirings and matrix theories When S is a semiring, then for ea ch n ≥ 0 the set S n × n of all n × n matrices over S is a lso a semiring. The sum op eration is defined point wise and product is the usual ma trix pro duct. The constants are the matrix 0 nn all o f whos e entries are 0 (often denoted just 0), and the diago nal matrix E n whose diagona l en tries are all 1 . In addition to s quare matr ices, we will also ha ve opp ortunity to co nsider rectangular matrices of arbitrar y size . A nice fr amework that arises with rectang ula r ma trices is that of a matrix theory . Let S b e a semir ing . The matrix the ory over S [10 , 3] is the categ ory M at S whose ob jects a r e the na tural num b ers and whose morphisms n → p are the n × p matrices over S , i.e., the ele ments of the semiring S n × p . Comp ositio n is matrix pro duct with the matrices E n being the identit y morphisms. Equipp ed with the p oint wise sum op eration and the zero matrix 0 np all of who se ent r ies are 0, each hom-set S n × p of Mat S is a commutativ e monoid. Mor eov er, comp osition distributes ov e r finite sums. In the catego r y Mat S , ea ch ob ject n is bo th the catego rical n -fold pro duct of o b ject 1 with itself and the n -fold copro duct of ob ject 1 with itself. The canonical copro duct injections 1 → n are the 1 × n matrices e i , i = 1 , . . . , n , having a 1 in the i th p o s ition and 0 e ls ewhere. The canonical pro jection morphisms n → 1 are the tr ansp oses e T i of these matrices. In a ny matr ix theor y M at S , we a sso ciate a morphism ˆ ρ : m → n w ith any function ρ : { 1 . . . . , m } → { 1 , . . . , n } by defining the ( i, j )th entry of ˆ ρ to be 1 if iρ = j and 0 otherwise. W e will call ˆ ρ a functional matrix and w r ite just ρ instead of ˆ ρ . An inje ctive functional matrix is a functional matrix corresp onding to a n injective function, and a p ermutation matrix is a functional matrix cor resp onding to p ermutation. Let Mat S and M at S ′ be matrix theories. A morphism Mat S → Mat S ′ is a functor that preser ves ob jects a nd the ca nonical co pro duct injectio ns and pro jections. It follo ws that a ny mor phism Mat S → Mat S ′ preserves the a dditiv e structure, and determines and is determined by a semiring morphism S → S ′ . Thu s , the ca tegory of matrix theories is equiv alent to the categ ory of semir ings. By this eq uiv alence , we may iden tify each matrix theory mo rphism h : Mat S → Mat S ′ with its restrictio n to the 1 × 1 matrices which is a semiring morphism S → S ′ . The image of a ma trix ( A ij ) ij under h is then given b y ( A ij h ) ij . An isomorphism of ma trix theories is a ma trix theor y mo r phism which is bijectiv e o n hom-sets. F or the above facts and a more abstract tre a tment of matrix theor ies the r eader is r eferred to [10]. 2.3 Dualit y The dual of a semir ing S = ( S, + , · , 0 , 1) is the semir ing S d = ( S, + , ◦ , 0 , 1) which has the sa me sum ope r ation and constants as S and who se pro duct op eration is defined b y a ◦ b = b · a , in the reverse order. Note that ( S d ) d = S , for all s e mir ings S . A dual morphism h : S 1 → S 2 betw e e n semirings S 1 → S 2 is a mo r phism S 1 → S d 2 , or equiv alently , a mo rphism S d 1 → S 2 . A dual isomorphi sm is a bijective dual mo rphism. No te that for any semir ing S , the identit y function ov er S is a dual isomorphism S → S d . Suppo se that M at S 1 and M at S 2 are matr ix theories. A dual matrix the ory morphism h : Mat S 1 → Mat S 2 maps morphis ms m → n to mor phisms n → m , i.e., matrices in S m × n 1 to matrices in S n × m 2 , suc h that ( AB ) h = ( B h )( Ah ) whenever A and B are matrices of appropriate size, moreover E n h = E n for each n . It is req uired that ca nonical injections are mapp ed to canonical pro jections and vice versa, so that e i h = e T i and e T i h = e i , for all 1 × n matrices e i . It follows that a dual matrix theory mor phism pr eserves the zero matrice s a nd the a dditiv e structure. A dua l isomor phism of matrix theories is bijective on eac h hom-s e t. The dual isomor phism S → S d can be lifted to matrix theo r ies. 4 Proposition 2.2 F or any semiring S , the matrix the ory Mat S d is dual ly isomorphic t o Mat S , a dual isomorphism Mat S → Mat S d maps e ach A ∈ S n × p to its tr ansp ose A T ∈ S p × n . In p articular, denoting c omp osition in Mat S d by ◦ we have: ( A + B ) T = A T + B T 0 T n = 0 n ( AB ) T = B T ◦ A T E d n = E n Pro of. It is clear that for eac h n , p the as signment A 7→ A T defines a bijection from the set of morphisms n → p in Mat S to the set of morphis ms p → n in Mat S d . Moreover, the additive structure and the iden tities E n are pres erved, and c o pro duct injections are mapp ed to copr o duct pro jections and vice versa. Thu s , it r emains to prove that ( A · B ) T = B T ◦ A T holds for all A ∈ S n × p and B ∈ S p × q in Mat S , where compo sition (i.e., matr ix pro duct) in Mat S is denoted · and matrix pro duct in Mat S d is denoted ◦ . But for all appropr iate i , j , the ( i, j )th entry of ( A · B ) T is ( A · B ) T ij = ( A · B ) j i = X k A j k · B ki = X k B ki ◦ A j k = X k B T ik ◦ A T kj = ( B T ◦ A T ) ij . ✷ Remark 2.3 When A or B is a 0-1 matrix, or more generally , when ea ch entry of A commut es with any en try of B , then we ha ve ( AB ) T = B T A T . 3 P artial Con w a y semirings The definition o f Con wa y semirings in volv es t wo imp orta nt ident ities of regular languages. Con- wa y semirings a pp ea r implicitly in Conw ay [6] and were first defined explicitly in [2, 3 ]. See also [17]. On the other hand, the applicability of Con way semirings is limited due to the fact that the sta r o p e r ation is total, whereas man y imp ortant semir ings o nly ha ve a partially defined star op eratio n. Moreover, it is not true that all such semirings can b e e mbedded int o a Conw ay semiring with a totally defined s tar op eration. Definition 3 .1 A p artial ∗ -semiring is a semiring S e quipp e d with a p artial ly define d star op er- ation ∗ : S → S whose domain is an idea l of S . A ∗ -semiring is a p artial ∗ -semiring S su ch t hat ∗ is define d on the whole semiring S . A morphism S → S ′ of (p artial) ∗ -semirings is a semiring morphism h : S → S ′ such that for al l s ∈ S , if s ∗ is define d then so is ( sh ) ∗ and s ∗ h = ( sh ) ∗ . Thu s , in a partial ∗ -semiring S , 0 ∗ is defined, and if a ∗ and b ∗ are defined then so is ( a + b ) ∗ , finally , if a ∗ or b ∗ is defined, then so is ( ab ) ∗ . When S is a partial ∗ -semiring, we let D ( S ) denote the domain of definition of the s tar op eratio n. Definition 3 .2 A partial Conw ay se miring is a p art ial ∗ -semiring S satisfying the fol lowing two axioms: 1. Sum star identit y : ( a + b ) ∗ = a ∗ ( ba ∗ ) ∗ (2) for al l a, b ∈ D ( S ) . 5 2. Pro duct star identit y : ( ab ) ∗ = 1 + a ( ba ) ∗ b, (3) for al l a, b ∈ S such that a ∈ D ( S ) or b ∈ D ( S ) . A Con way semiring is a p artial Conway semiring S which is a ∗ -semiring (i.e., D ( S ) = S ). A morphisms of ( p artial) Conwa y semirings is a (p artial) ∗ -semiring morphism. Note that in any partial Conw ay semiring S , aa ∗ + 1 = a ∗ (4) a ∗ a + 1 = a ∗ (5) 0 ∗ = 1 (6) for all a ∈ D ( S ). Moreover, if a ∈ D ( S ) or b ∈ D ( S ), then ( ab ) ∗ a = a ( ba ) ∗ . (7) It follows that also aa ∗ = a ∗ a (8) ( a + b ) ∗ = ( a ∗ b ) ∗ a ∗ (9) for all a, b ∈ D ( S ). When a ∈ D ( S ) we will denote aa ∗ = a ∗ a by a + and call + the plus o p er ation. Conw ay semirings give rise to Conw ay matr ix theories [3 ]. In the same wa y , partia l Conw ay semirings give rise to partial Conw ay ma tr ix theories defined below. W e say that a colle c tio n J of matric e s in Mat S is a matrix ide al if for any int eg ers m, n , it co ntains the zero matrix 0 mn and is clo sed under sum, moreov er, it is clo sed under multiplication with an y matrix: if A : m → n in J then for any B : p → m and C : n → q it holds that B A, AC ∈ J . It is easy to show that if I is an ideal of S , then the collection of all ma trices J = M ( I ), all of who se entries are in I is a matrix ideal of M at S , and that an y matrix ideal is of this sor t. Thus, any matrix idea l of Mat S is uniquely determined by an ideal of S . Definition 3 .3 Supp ose t hat S is a semiring and c onsider t he matrix the ory Mat S . We say that Mat S is a par tial Conw ay matr ix theor y if it is e quipp e d with a star op er ation A 7→ A ∗ , define d on the squar e matric es A : n → n , n ≥ 0 whose domain is t he c ol le ction of al l squ ar e matric es in a matrix ide al M ( I ) , mor e over, the matrix versions of t he su m and pr o duct star identities hold: ( A + B ) ∗ = A ∗ ( B A ∗ ) ∗ (10) for al l A, B ∈ M ( I ) , A, B : n → n , and ( AB ) ∗ = E n + A ( B A ) ∗ B , (11) for al l A, B ∈ M ( I ) , A : n → m , B : m → n . When Mat S is a p artial matrix the ory su ch t hat star is define d on al l s quar e matric es, then we c al l Mat S a Conw ay matrix theor y . A morphism of (p artial) m atrix the ories is a matrix the ory morphism which pr eserves st ar. Note the following special cas e s of (11): A ∗ = AA ∗ + E n (12) A ∗ = A ∗ A + E n (13) 0 ∗ nn = E n (14) 6 where A : n → n in M ( I ), n ≥ 0. Also, AA ∗ = A ∗ A for all A : n → n in M ( I ). Below we will denote AA ∗ by A + . If Mat S is a (partial) Con way matrix theory , then by identifying a 1 × 1 matrix ( a ) in M at S with the elemen t a , the semiring S b ecomes a (partial) C o nw ay se mir ing. Conv er sely , an y (partial) Conw ay semiring determines a (partial) Conw ay matrix theory , as we s how b elow. Definition 3 .4 Supp ose that S is a p artial Conway semiring with D ( S ) = I . We define a p artial star op er ation on the s emirings S k × k , k ≥ 0 , whose domain of definition is I k × k , the ide al of those k × k matric es al l of whose entries ar e in I . W hen k = 0 , S k × k is trivial as is the definition of star. When k = 1 , we u s e the star op er ation on S . As s uming t hat k > 1 we write k = n + 1 . F or a matrix  a b c d  in I k × k , define  a b c d  ∗ =  α β γ δ  (15) wher e a ∈ S n × n , b ∈ S n × 1 , c ∈ S 1 × n and d ∈ S 1 × 1 , and wher e α = ( a + bd ∗ c ) ∗ β = αbd ∗ γ = δ ca ∗ δ = ( d + ca ∗ b ) ∗ . By the above definition, we have a ls o defined a star opera tion on those square matr ic es in M at S which b elong to M ( I ). It is known (cf. [3]) that when S is a C o nw ay semiring, then equipp e d with the ab ov e star op er a tion, Mat S is a Conw ay matrix theory . Mor e g enerally , but with the same pro of, we ha ve: Theorem 3 .5 Supp ose that S is a p artial Co n way semiring with D ( S ) = I . Then, e quipp e d with the ab ove s tar op er ation, Mat S is a p artial Conway matrix the ory wher e the star op er ation is define d on the squar e matric es in M ( I ) . Corollar y 3 .6 If S is a (p artial) Conway semiring, then so is t he semiring S n × n , for e ach n . Corollar y 3 .7 The c ate gory of (p artial) Conway semirings is e quivalent t o the c ate gory of (p artial) Conway matrix the ories. Also the following result is kno wn to hold for Conw ay ma tr ix theories. Theorem 3 .8 Supp ose that M at S is a p artial Conway matrix the ory wher e star is define d on the squar e matric es in M ( I ) . Then the fol lowing identities hold. 1. The matrix star iden tity ( 15) for al l p ossible de c omp ositions of a squar e matrix in M ( I ) into four blo cks su ch t hat a and d ar e squar e matric es, i.e., wher e a : n → n , b : n → m , c : m → n and d : m → m . 2. The p ermutation identit y (16) ( π Aπ T ) ∗ = π A ∗ π T , (16) for al l A : n → n in M ( I ) and any p ermutation matr ix π : n → n , wher e π T denotes the tr ansp ose of π . 7 The pro of is the same as for Co nw ay matrix theories, cf. [3]. F or later use we note the following. When M at S is a pa rtial Con way matrix theory with star op er ation defined on the square matrices in M ( I ), and if A =  a b c d  is a matrix with entries in M ( I ), partitioned as ab ov e, then A + =  ( a + bd ∗ c ) + ( a + bd ∗ c ) ∗ bd ∗ ( d + ca ∗ b ) ∗ ca ∗ ( d + ca ∗ b ) +  (17) A ∗ =  ( a + bd ∗ c ) ∗ a ∗ b ( d + ca ∗ b ) ∗ d ∗ c ( a + bd ∗ c ) ∗ ( d + ca ∗ b ) ∗  (18) 3.1 Dualit y Suppo se that S is a partial ∗ -semiring. Then we ma y equip S d with the sa me star op eration. Since D ( S ) is also an ideal of S d , we have that S d is a pa rtial ∗ -semiring. L e t S and S ′ be ∗ -semirings. W e say that a function h : S → S ′ is a dua l morphism of partial ∗ -semirings if it is dual semiring morphism mapping D ( S ) to D ( S ′ ) whic h preser ves star. A dual isomo rphism is a bijectiv e dual morphism. Proposition 3.9 When S is a p artial Conway semiring, so is S d . Mor e over, the identity fun c- tion S → S is a dual isomorphism S → S d . Pro of. This fo llows from the fact that (9) holds in all partia l Con wa y semirings . ✷ Since for partial Con wa y semirings S , the semiring S d is als o a partial Con wa y semir ing, Mat S d is a par tial Conw ay matr ix theory . A dual morphism Mat S → Mat ′ S betw e e n partial Co nw ay matrix theories als o pr eserves star. A dual isomorphism is a dual morphism whic h is bijectiv e on hom-sets. Proposition 3.10 Supp ose that Mat S is a p artial Conway matrix the ory. Then the function A 7→ A T , A : m → n in Mat S , is a dual isomorp hism Mat S → Mat S d of p artial Conway matrix the ories. Pro of. W e k now that the assig nment A 7→ A T defines a dual isomorphism of the underlying matrix theo ries. Let I = D ( S ) = D ( S d ). It is clear that if A is in M ( I ) then A T is also in M ( I ). T o c o mplete the pro o f, w e still hav e to show that ( A T ) ⊗ = ( A ∗ ) T for all squa re matrices in M ( I ), where ⊗ denotes the star op era tion in Mat S d . T o prov e this, let A : n → n in Mat S . When n = 0 or n = 1, o ur cla im is clear. W e pro c e ed by induction on n . Ass ume that n > 1. Then let us wr ite A =  a b c d  , where a and d are square matrices of size ( n − 1) × ( n − 1) 8 and 1 × 1, resp ectively . Then, using (15), (18) and the induction hypothesis, ( A T ) ⊗ =  a T c T b T d T  ⊗ =  ( a T + c T ◦ ( d T ) ⊗ ◦ b T ) ⊗ ( a T + c T ◦ ( d T ) ⊗ ◦ b T ) ⊗ ◦ c T ◦ ( d T ) ⊗ ( d T + b T ◦ ( a T ) ⊗ ◦ c T ) ⊗ ◦ b T ◦ ( a T ) ⊗ ( d T + b T ◦ ( a T ) ⊗ ◦ c T ) ⊗  =  (( a + bd ∗ c ) ∗ ) T ( d ∗ c ( a + bd ∗ c ) ∗ ) T ( a ∗ b ( d + ca ∗ b ) ∗ ) T (( d + ca ∗ b ) ∗ ) T  =  ( a + bd ∗ c ) ∗ a ∗ b ( d + ca ∗ b ) ∗ d ∗ c ( a + bd ∗ c ) ∗ ( d + ca ∗ b ) ∗  T =  a c b d  ∗  T = ( A ∗ ) T . ✷ 4 P artial iteration semirings Many imp or tant (partial) Conw ay semirings satisfy the group identities as so ciated with the finite groups, introduced by Conw ay [6]. Such ∗ -semirings are the contin uous ∗ -semirings, or more generally , the inductiv e ∗ -semirings of [12], the ∗ -semirings that aris e from co mplete semirings [3], or the (partia l) iterative semirings defined in the next section. When a (partia l) Co nw ay semiring satisfies the gro up iden tities, it will b e called a (par tial) iteration semiring. Definition 4 .1 We say that the gro up identit y as s o ciated with a finite group G of or der n ho lds in a p artial Conway semiring S if e 1 M ∗ G u n = ( a 1 + · · · + a n ) ∗ (19) holds, wher e a 1 , · · · , a n ar e arbitr ary elements in D ( S ) , and wher e M G is the n × n matrix who se ( i, j ) th entry is a i − 1 j , for al l 1 ≤ i, j ≤ n , and e 1 is t he 1 × n 0 - 1 matrix whose first ent ry is 1 and whose other entries ar e 0 , fin al ly u n is the n × 1 matrix al l of whose ent r ies ar e 1 . Ident ity (19) ass erts that the sum of the entries o f the first row o f M ∗ G is ( a 1 + · · · + a n ) ∗ . F or example, the gro up iden tity asso ciated with the group of or der 2 is  1 0   a 1 a 2 a 2 a 1  ∗  1 1  = ( a 1 + a 2 ) ∗ which by the matrix star identit y can b e written as ( a 1 + a 2 a ∗ 1 a 2 ) ∗ (1 + a 2 a ∗ 1 ) = ( a 1 + a 2 ) ∗ . (It is known tha t in Conw ay semirings, this iden tity is further equiv alent to ( a 2 ) ∗ (1 + a ) = a ∗ .) Definition 4 .2 We say that a Conway semiring S is an iter ation semir ing if it satisfies al l gr oup identities. We say that a p artial Conway semiring S is a pa rtial iteration semir ing if it satisfies al l gr oup identities (19) wher e a 1 , · · · , a n r ange over D ( S ) . A morphism of (p artial) iter ation s emirings is a (p artial) Conwa y semiring morphism. We say that a (p artial) Conway matrix the ory is a (partial) matrix iteration theor y if it satisfies al l gr oup identities. A morphi sm of (p artial) matrix iter ation the ories is a (p artial) Conway matrix t he ory morphism. 9 It is clear that a (partia l) Conwa y semir ing S is a (partial) itera tion semiring iff Mat S is a (partial) iteration semiring . Also, the categ ory of (partial) iteration semirings is equiv alent to the categor y o f (partial) matrix iteratio n theories. Proposition 4.3 Supp ose that t he p artial Conway semiring S satisfies the gr oup identity (19) for al l a 1 , · · · , a n ∈ D ( S ) . Then S also satisfies u T n M ∗ G e T 1 = ( a 1 + · · · + a n ) ∗ , (20) for al l a 1 , · · · , a n ∈ D ( S ) , wher e e 1 , M G and u n ar e defin e d as ab ove. Thus, if S is an iter ation semiring, then (20) holds for al l finite gr oups G . Pro of. F or each i ∈ { 1 , . . . , n } , let π i denote the p er mut a tion matr ix corresp o nding to the bijection { 1 , · · · , n } → { 1 , · · · , n } , j 7→ ij , where the pro duct i j is computed in the group G . An easy calcula tion shows tha t π M G π T = M G . Th us, by the pe r mutation identit y , also π M ∗ G π T = M ∗ G . Since this holds for a ll i , als o ( M ∗ G ) i 1 = ( M ∗ G ) 1 i − 1 for all i . Thus, the e n tr ies of the first c olumn of M ∗ G form a p er mutation o f the ent r ies of the first row of M ∗ G . W e conclude that if (19) holds, then so do es (20). ✷ Remark 4.4 In Co nw ay semir ings, the group identit y (19) is equiv alent to (20). The group iden tities seem to be extremely difficult to verify in practice. How ever, they ar e implied by the simpler functorial sta r conditions defined b e low. Definition 4 .5 Supp ose that S is a p artial Conway semiring so that M at S is a p artial Co n way matrix the ory. L et I = D ( S ) , and let C b e a class of matr ic es in Mat S . We say that M at S has a functorial star with r esp e ct t o C if for al l A : m → m and B : n → n in M ( I ) and for al l C : m → n in C , if AC = C B then A ∗ C = C B ∗ . Suppo se that C is a cla s s o f matrices in a Conw ay matrix theo ry Mat S . Then let B ( C ) denote the class of blo ck diagonal rectangular matrices whose diagona l blocks are in C . Lemma 4. 6 If a Conway matr ix the ory Mat S has a functorial star with r esp e ct to C , then it also has a fun ctorial star with r esp e ct to the class B ( C ) . Pro of. It suffices to prove the following. Let A : m → m and B : n → n and C : m → n in Mat S with AC = C B such that A ∗ and B ∗ are defined. Mor eov er, supp ose that C =  c 1 0 0 c 2  where c : m 1 → n 1 , d : m 2 → n 2 with m 1 + m 2 = m and n 1 + n 2 = n . If Mat S has a functoria l star with resp ect to { c, d } , then M at S has a functoria l star with resp ect to { C } . T o prove this, let us write A =  a 1 a 2 a 3 a 4  , B =  b 1 b 2 b 3 b 4  where a 1 : m 1 → m 1 , etc. Since AC = C B , we hav e a 1 c = cb 1 a 2 d = cb 2 a 3 c = db 3 a 4 d = db 4 Thu s , since Mat S has a functoria l star with resp ect to { c , d } , a ∗ 1 c = cb ∗ 1 and a ∗ 4 d = db ∗ 4 . 10 Using these equations, it fo llows that ( a 1 + a 2 a ∗ 3 a 4 ) c = c ( b 1 + b 2 b ∗ 3 b 4 ) ( a 4 + a 3 a ∗ 1 a 2 ) d = d ( b 4 + b 3 b ∗ 1 b 2 ) Thu s , using ag ain the fact that Mat S has a functorial star with respect to { c, d } , it follo ws that ( a 1 + a 2 a ∗ 3 a 4 ) ∗ c = c ( b 1 + b 2 b ∗ 3 b 4 ) ∗ ( a 1 + a 2 a ∗ 3 a 4 ) ∗ a 2 a ∗ 4 d = c ( b 1 + b 2 b ∗ 3 b 4 ) ∗ b 2 b ∗ 4 ( a 4 + a 3 a ∗ 1 a 2 ) ∗ d = d ( b 4 + b 3 b ∗ 1 b 2 ) ∗ ( a 4 + a 3 a ∗ 1 a 2 ) ∗ a 3 a ∗ 1 c = d ( b 4 + b 3 b ∗ 1 b 2 ) ∗ b 3 b ∗ 1 , so that A ∗ C = C B ∗ . ✷ Proposition 4.7 Supp ose that Mat S is a (p artial) Conway matrix the ory. 1. Mat S has a functorial star with r esp e ct to the class of all inje ctive functional matric es and their tr ansp oses. 2. If Mat S has a functorial st ar with re sp e ct to the class of functional matric es m → 1 , m ≥ 2 , then Mat S has a fun ctorial star with r esp e ct to the class of al l functional matric es. 3. If Mat S has a functorial star with re sp e ct to the class of tr ansp oses of functional matric es m → 1 , m ≥ 2 , then M at S has a functorial star with r esp e ct to t he class of t r ansp oses of al l functional matric es. 4. If Mat S has a functorial star with r esp e ct to the class of al l functional matric es m → 1 , m ≥ 2 , then M at S is a p artial matrix iter ation the ory. Pro of. The fact that when Mat S is a Con wa y matrix theor y , then Mat S has a functor ial dagg er with resp ect to the class of injectiv e functional matrices and their trans po ses is proved in [3]. The same pro of applies for partia l Con wa y matrix theories. The second and third claims follow fro m the preceding lemma. The last fact is proved a s follows. Let G b e a finite gro up of or de r n and consider the matrix M G defined above. Using the notation in (19), we have M G u n = u n a where a denotes the sum a 1 + · · · + a n . Thus, if Mat S has a functor ia l sta r with resp ect to all functional matrices m → 1, m ≥ 2, then M ∗ G u n = u n a ∗ and e 1 M ∗ G u n = a ∗ . Since also au T n = u T n M G , if Mat S has a functorial sta r with res pec t to all transp oses of functional ma trices m → 1, m ≥ 2, then a ∗ u T n = u T n M ∗ G and a ∗ = u T n M ∗ G e T 1 . But this implies that e 1 M ∗ G u n = a ∗ . (See Remark 4.4.) ✷ An impo rtant identit y tha t holds in all iteration semirings S and matr ix iteration theories Mat S is the c ommut ative identity , cf. [3], which is a genera liz a tion of the group iden tities. It allows us to deduce A ∗ ρ = ρB ∗ from Aρ = ρB un der c ertain c onditions , where A and B are squar e matrices and ρ is a functional matr ix . The commutativ e identit y also holds in partial matr ix iteration theories (with the o bvious restriction on the applicability of the star op era tion). Since the dual o f an iteratio n semiring is also an itera tion semiring, see b elow, the dual c ommu tative identity of [3] also holds in (par tial) matrix iteration theories. 4.1 Dualit y Proposition 4.8 Supp ose that S is a p artial Conway semiring. Then S is a p artial itera tion semiring iff S d is. Thus, Mat S is a p artial matrix iter ation the ory iff Mat S d is. 11 Pro of. Suppo se that Mat S is a par tial Conw ay matrix theory with star opera tion defined on the square matrices in the matrix ideal M ( I ). L et M G = M G ( a 1 , . . . , a n ) b e the matrix asso ciated with the finite g roup G o f or der n , s ee Definition 4.1, where a 1 , · · · , a n are in I . Note that M T G is just M G ( a 1 − 1 , · · · , a n − 1 ), the matrix obtained from M G by replacing each o ccurr ence of a i with a i − 1 . Since Mat S is a matrix iteratio n theor y , the gr oup identit y asso cia ted with G holds in Mat S . In pa r ticular, e 1 ( M T G ) ∗ u n = ( a 1 − 1 + · · · + a n − 1 ) ∗ = ( a 1 + · · · + a n ) ∗ holds. Thus, by Prop ositio n 4.3, e 1 ◦ M ⊗ G ◦ u n = ( u T n ( M ⊗ G ) T e T 1 ) T = ( u T n ( M T G ) ∗ e T 1 ) T = u T n ( M T G ) ∗ e T 1 = ( a 1 + · · · + a n ) ∗ . ✷ 5 P artial iterativ e semirings In this section we exhibit a class of partial iteration semir ings. Definition 5 .1 A par tial iterative s emiring is a p artial ∗ -semiring S s u ch t hat for every a ∈ D ( S ) and b ∈ S , a ∗ b is the unique solut ion of the e quation x = ax + b . A morphi sm of p artial iter ative semirings is a ∗ -semiring morphism. W e note that any semiring S with a distinguished ideal I such that for all a ∈ I a nd b ∈ S , the equation x = ax + b has a unique solution can b e turned into a partial iterative semir ing , where star is defined on I . Indeed, when a ∈ I , define a ∗ as the unique solution of the eq ua tion x = ax + 1. It follows that aa ∗ b + b = a ∗ b for all b , so that a ∗ b is the unique solution of x = ax + b . W e also note that when S, S ′ are pa rtial iterative semirings, then any s e miring morphism h : S → S ′ with D ( S ) h ⊆ D ( S ′ ) automatica lly preser ves star. Indeed, when a ∈ D ( S ), then a ∗ = aa ∗ + 1, th us a ∗ h = ( ah )( a ∗ h ) + 1, showing that a ∗ h is a solution of the e q uation x = ( ah ) x + 1 ov er S ′ . But s ince ah is in D ( S ′ ), the only s olution is ( ah ) ∗ . Thus, a ∗ h = ( ah ) ∗ . Proposition 5.2 Every p artial iter ative semiring is a p artial Conway semiring. Pro of. Supp ose that S is a partial iterative semiring and a, b ∈ D ( S ). Since a + b ∈ D ( S ) and ( a + b ) a ∗ ( ba ∗ ) ∗ + 1 = aa ∗ ( ba ∗ ) ∗ + ba ∗ ( ba ∗ ) ∗ + 1 = aa ∗ ( ba ∗ ) ∗ + ( ba ∗ ) ∗ = ( aa ∗ + 1)( ba ∗ ) ∗ = a ∗ ( ba ∗ ) ∗ , it follo ws b y uniqueness of solutions that ( a + b ) ∗ = a ∗ ( ba ∗ ) ∗ . Also, if a o r b is in D ( S ), then ab ∈ D ( S ) and ab ( a ( ba ) ∗ b + 1) + 1 = a ( ba ( ba ) ∗ + 1) b + 1 = a ( ba ) ∗ b + 1 , so that ( ab ) ∗ = a ( ba ) ∗ b + 1, by uniqueness. ✷ Corollar y 5 .3 If S is a p artial iter ative semiring, t hen Mat S , e quipp e d with the star op er ation define d on matric es in Se ction 3 , is a p artial Conway matrix the ory. 12 It is known, cf. [5, 3], that if a class of functions in several v aria bles ov er a set has certain clos ure prop erties, and if each fixed p oint equatio n with resp ect to a function in the class ha s a unique solution, then the same ho lds for finite s ystems o f fixed po int equations in volving functions from the clas s . Moreov er , such systems ca n b e solved by succes sive elimination of the unkno wns . A sp ecialization of this res ult is given b elow. Theorem 5 .4 Supp ose that S is a p artial iter ative semiring with D ( S ) = I . Then the fol lowing holds in the p artial Conway matrix t he ory Mat S . F or any A : n → n in M ( I ) and any B : n → p , A ∗ B is the un ique solution of t he m atrix e quation X = AX + B . Pro of. W e pro vide a pr o of fo r completeness. Let A a nd B b e matrices as ab ove. Since Mat S is a partial Conw ay matrix theory , AA ∗ B + B = ( AA ∗ + E n ) B = A ∗ B by (12). T o complete the pro of, we hav e to show that the solution is unique. This is clear when n = 0 or n = 1. W e pro ceed by induction on n . Assuming n > 1, write A in the form A =  a b c d  where a and d are squa r e matrices of size m × m and k × k , resp ectively , where m, k > 0, m + k = n . Then let X =  x y  B =  e f  where x, e are o f s ize m × p a nd y , f a re o f size k × p . Using he ab ov e decomp osition of the matrices, we can write the equation X = AX + B as x = ax + by + e (21) y = cx + dy + f . (22) By uniqueness, x = a ∗ ( by + e ) and y = d ∗ ( cx + f ). Substituting the expression for x in (21) and the expressio n for y in (22) gives x = ( a + bd ∗ c ) x + e + bd ∗ f y = ( d + ca ∗ b ) y + f + ca ∗ e. But since a + bd ∗ c and d + ca ∗ b a re in M ( I ), ea ch o f these equations has a unique s o lution. ✷ The next fact fo llows from Theorem 5.4 and a res ult from [9]. F o r completeness, w e provide a pro of. Theorem 5 .5 Supp ose that S is a p artial iter ative semiring with D ( S ) = I and A : n → n and B : n → p in Mat S such that A k ∈ M ( I ) for s ome k ≥ 1 . Then t he e quation X = AX + B has a unique s olut ion ( A k ) ∗ ( A k − 1 B + · · · + B ) . Pro of. Let f ( X ) = AX + B . Then f m ( X ) = A m X + A m − 1 B + · · · + B for all m ≥ 1. By assumption, the equa tion X = f k ( X ) has a unique solution X 0 = ( A k ) ∗ ( A k − 1 B + · · · + B ). Our aim is to show that X 0 is the unique solution of X = f ( X ). But f ( X 0 ) = f ( f k ( X 0 )) = f k ( f ( X 0 )), and since X 0 is the unique so lution o f the equation X = f k ( X ), we conclude that X 0 = f ( X 0 ). Also, if X = f ( X ), then X = f m ( X ) for a ll m , so that any solution of the eq uation X = f ( X ) is a solution of the equa tion X = f k ( X ). ✷ Remark 5.6 Supp ose that S is a pa rtial itera tive semiring with D ( S ) = I . Note that if a ∈ S and k ≥ 1 a re such that a k ∈ I , then a m ∈ I for all m ≥ k . Let J = { a ∈ S : ∃ k ≥ 1 a k ∈ I } , 13 so that I ⊆ J . By Theorem 5.5, the equa tion x = ax + b ha s a unique solution fo r each a ∈ I and b ∈ S , and this unique solution can be written as ( a k ) ∗ ( a k − 1 b + · · · + b ) whenever a k ∈ I . Now suppo se that S is c o mmut a tive. Then J is also an ideal of S . This follows by noting that 0 ∈ J , moreover, if a k ∈ I and b k ∈ I , then ( a + b ) 2 k ∈ I . Moreover, if a k ∈ I , then for a ny b ∈ S , ( ab ) k = a k b k ∈ I . Thus, if we define a ∗ for a ∈ J as the unique solution of the equation x = ax + 1, then S is a partial iterative semiring, wher e the domain o f definition of the star op eration is J . Moreover, this star op eratio n agrees with the the or iginal one on the ideal I . Our next aim is to show that partial iterative semirings are partial itera tio n semirings and thus the matrix theories of par tial iterative semir ing s are partial iteration matrix theor ies. Theorem 5 .7 Supp ose that S is a p artial iter ative semiring. Then the p artial Conway m atr ix the ory Mat S has a functorial st ar with r esp e ct t o al l matric es. Thus, if AC = C B for some matric es A : n → n , B : m → m and C : n → m , wher e A, B ∈ M ( I ) , then A ∗ C = C B ∗ . Pro of. W e hav e AC B ∗ + C = C B B ∗ + C = C B ∗ , showing that C B ∗ is a solution of the equatio n X = AX + C . But the unique s olution is A ∗ C . Thus A ∗ C = C B ∗ . ✷ Corollar y 5 .8 A ny p artial itera tive semiring is a p artial iter ation semiring. Pro of. Let S be a par tial iterative semiring. W e already know that S is a par tial Conw ay semiring (cf. Prop os ition 5.2) and thus Mat S is a partial Conw ay matrix theo ry . By Theorem 5.7, Mat S has a functorial s ta r with resp ect to all matrices. Thus, b y Prop ositio n 4.7, Mat S is a partia l matrix iteration theory . ✷ W e give an application of the ab ov e cor o llary . Le t S b e a semiring and Σ a set, and consider the p ower series semiring S h h Σ ∗ i i . F ollowing [1], we ca ll a se r ies s ∈ S h h Σ ∗ i i pr op er if ( s, ǫ ) = 0. Clearly , the prop er series form an ideal. It is known, cf. [1], that for any ser ies s, r , if s is pr op er, then the equation x = sx + r has a unique solution. Moreover, this unique solution is s ∗ r , where s ∗ is the unique s olution of the equation y = sy + 1. Corollar y 5 .9 F or any semiring S and set Σ , S h h Σ ∗ i i , e quipp e d with the ab ove star op er ation define d on the pr op er series, is a p artial iter ative semiring and thu s a p artial iter ation semiring. Remark 5.10 Co nsider the ab ove partial iter ative semiring S h h Σ ∗ i i with sta r oper a tion defined on the ideal I o f prop er series . Let J b e defined as in Remark 5.6. Then J is the co llection of all cycle fr e e series , cf. [7]. As shown in Remark 5.6, if S is commutativ e then J is also an ideal, and the s tar opera tion ca n b e extended to the idea l J so that S h h Σ ∗ i i beco mes a par tial iterative semir ing with star defined on J . By the above Cor ollary , this partial iterativ e semir ing is a partial iteration semir ing . Remark 5.11 Supp ose that S is partial iterative semiring with star op eration defined on D ( S ) = I , and supp ose that S 0 is a subse mir ing o f S which is equipp ed with a unary op era tion ⊗ . Moreov e r , suppo se that S is the direct sum o f S 0 and I , so that each s ∈ S has a unique representation as a s um x + a with x ∈ S 0 and a ∈ A . It is shown in [3, 2] that if S 0 , equipp ed with the op eratio n ⊗ , is a Con way se mir ing, then there is a unique wa y to turn S into a Conw ay semiring whose star op er ation extends ⊗ . This oper ation also extends the s tar o pe ration or iginally defined on I . Mor eov er, when S 0 is an iteration semiring, then S is also an iteration semiring . In particular , if S is a Conw ay o r an iter ation semiring, then so is S h h Σ ∗ i i . 14 W e end this section by defining iterative s emirings. Definition 5 .12 A n itera tive semiring is a p artial iter ative semiring S su ch that D ( S ) is the c ol le ction of al l element s s ∈ S which c annot b e written in the form 1 + s ′ . A morphism of iter ative semirings is a p artial ∗ -semiring morphism. It follows from our results that every iterative se miring is a partial iteration semiring. F or example, N h h Σ ∗ i i is an iterative semiring. Question: Is there a (partial) iterative semiring whose dual is not iterative? 6 Kleene theorem The classica l Kleene theorem equa tes languages reco gnizable b y finite automata with the regular languages , and its gener alization by Sch¨ utzen b er ger equates p ow er series re c ognizable by finite weigh ted automata with rational power series . I n this section we esta blish a Kleene theorem for partial Conw ay s emirings. T o this end, we define a general no tion of (finite) a uto maton in partia l Conw ay semirings . Definition 6 .1 Supp ose that S is a p artial Conwa y semiring, S 0 is a su bsemiring of S and Σ is a subset of D ( S ) . An automaton in S over ( S 0 , Σ) is a t riplet A = ( α, A, β ) c onsisting of an initial vector α ∈ S 1 × n 0 , a tr ansition matrix A ∈ S 0 Σ n × n , wher e S 0 Σ is the set of al l line ar c ombinations over Σ with c o efficients in S 0 , and a final v ec tor β ∈ S n × 1 0 . The int e ger n is c al le d the dimension of A . The be havior of A is | A | = αA ∗ β . (Since A ∈ D ( S ) n × n , A ∗ exists.) Definition 6 .2 We say that s ∈ S is recog nizable over ( S 0 , Σ) is s is the b ehavior of some automaton over ( S 0 , Σ) . We let R ec S ( S 0 , Σ) denote the set of al l elements of S which ar e r e c o gnizable over ( S 0 , Σ) . Next we define rational elements. Definition 6 .3 L et S , S 0 and Σ b e as ab ove. We s ay that s ∈ S is ratio na l over ( S 0 , Σ) if s = x + a for some x ∈ S 0 and some a ∈ S which is c ontaine d in the le ast set Rat ′ S ( S 0 , Σ) c ontaining Σ ∪ { 0 } and close d u nder the ra tional op era tions + , · , + and left and right multiplic ation with elements of S 0 . We let Rat S ( S 0 , Σ) denote the set of r ational element s over ( S 0 , Σ) . Note that Rat ′ S ( S 0 , Σ) ⊆ D ( S ) and that the element a in the ab ov e definition is in D ( S ). Proposition 6.4 Supp ose that S is a p artial Conway semiring, S 0 is a subsemiring of S and Σ is a s u bset of D ( S ) . Then R at S ( S 0 , Σ) c ontains S 0 and is close d under sum and pr o duct. Mor e over, it is close d under st ar iff it is close d u nder the plus op er ation. Pro of. Since 0 ∈ R at ′ S ( S 0 , Σ), it follows that S 0 ⊆ Rat S ( S 0 , Σ). Let r = x + a and s = y + b be in R at S ( S 0 , Σ), where x, y ∈ S 0 and a, b ∈ Rat ′ S ( S 0 , Σ). Then r + s = ( x + y ) + ( a + b ) and rs = xy + ( xb + ay + ab ), so that r + s and r s a re in Rat S ( S 0 , Σ). Since R at S ( S 0 , Σ) is close d under sum and pro duct and contains 1, it is close d under sta r iff it is clos e d under plus. ✷ The following fa ct is clear. 15 Proposition 6.5 Supp ose that S is a p artial Conway semiring, S 0 is a subsemiring of S and Σ is a subset of D ( S ) . Then R at S ( S 0 , Σ) is c ontaine d in the le ast su bsemiring of S c ontaining S 0 and Σ whic h is close d u nder star. W e give tw o s ufficient conditions under which Rat S ( S 0 , Σ) is clos ed under star. Proposition 6.6 L et S, S 0 and Σ b e as ab ove. Assume that either S 0 ⊆ D ( S ) and S 0 is close d under st ar, or the fol lowing c ondition holds: ∀ x ∈ S 0 ∀ a ∈ D ( S ) ( x + a ∈ D ( S ) ⇒ x = 0) . (23) Then R at S ( S 0 , Σ) is close d under star. Mor e over, in either c ase, Rat S ( S 0 , Σ) is t he le ast sub- semiring of S c ontaining S 0 and Σ which is close d under star. Pro of. W e know that Rat S ( S 0 , Σ) is clos ed under star iff it is clo s ed under plus. Assume firs t that S 0 ⊆ D ( S ) and S 0 is closed under star, so that S 0 is a Conw ay subsemiring of S . W e know that any s ∈ Rat S ( S 0 , Σ) can b e written a s a sum x + a , where x ∈ S 0 and a ∈ Rat ′ S ( S 0 , Σ) ⊆ D ( S ). Now S 0 ⊆ D ( S ) by as sumption, and since D ( S ) is an ideal containing both x and a , it follows that s = x + a ∈ D ( S ). Since S is a par tia l Conw ay semiring, s ∗ = ( x ∗ a ) ∗ x ∗ = x ∗ + ( x ∗ a ) + x ∗ . By assumption, x ∗ ∈ S 0 . Also, x ∗ a ∈ R at ′ S ( S 0 , Σ), since Rat ′ S ( S 0 , Σ) is closed under multiplication with elements o f S 0 . Thus, since Rat ′ S ( S 0 , Σ) is closed under plus and m ultiplication with elements o f S 0 , we hav e that ( x ∗ a ) + x ∗ ∈ Rat ′ S ( S 0 , Σ). Since s ∗ is the s um of an element of S 0 and an element o f Rat ′ S ( S 0 , Σ), it follows that that s ∗ is in Rat S ( S 0 , Σ). Note that when S 0 ⊆ D ( S ) and S 0 is closed under s ta r, then, by the ab ov e arg ument, Rat S ( S 0 , Σ) ⊆ D ( S ) is also clo sed under star, so that it is a Conwa y semiring. Next, assume that (23) holds. Let s = x + a ∈ Rat S ( S 0 , Σ), where x ∈ S 0 and a ∈ Rat ′ S ( S 0 , Σ). W e w ant to sho w that if s is in D ( S ), then s + is a ls o in Rat S ( S 0 , Σ). B ut by (23), s ∈ D ( S ) only if x = 0 . In that ca se, s + = a + ∈ R at ′ S ( S 0 , Σ) ⊆ Rat S ( S 0 , Σ). ✷ Remark 6.7 Note that the second condition in the ab ov e prop os ition holds whenever each s ∈ S has at mos t one repr esentation s = x + a with x ∈ S 0 and a ∈ D ( S ). This happ ens when S is the dir e ct sum of S 0 and D ( S ). In the pro of of our Kle e ne theorem, we will ma ke us e o f the following fact. Lemma 6. 8 Supp ose that e ach ent r y of the n × n matrix A is in Rat ′ S ( S 0 , Σ) . Then the s ame holds for the m atr ix A + . Pro of. W e prove this fact by induction on n . When n = 0 or n = 1, our claim is clea r. Assuming that n > 0 write A =  a b c d  , where a is ( n − 1) × ( n − 1), d is 1 × 1. Then A + is given b y the formula (17). W e only sho w that eac h entry of the s ubmatrix ( a + bc ∗ d ) + is in R at ′ S ( S 0 , Σ). But a + bc ∗ d = a + bd + bc + d . By the induction hypo thesis, e ach entry of c + is in Rat ′ S ( S 0 , Σ). Since Rat ′ S ( S 0 , Σ) is closed under sum a nd product, and since e ach ent r y of a, b or d is also in this s et, it follo ws that each entry o f a + bc ∗ d is in Rat ′ S ( S 0 , Σ). Thus, using the induction hypothesis again, it follows that each entry of ( a + bc ∗ d ) + is in Rat ′ S ( S 0 , Σ). ✷ Proposition 6.9 F or any S, S 0 and Σ as ab ove, Rec S ( S 0 , Σ) ⊆ Rat S ( S 0 , Σ) . 16 Pro of. Le t A = ( α, A, β ) b e an automaton over ( S 0 , Σ). Then | A | = αA ∗ β = αβ + αA + β . Clearly , αβ ∈ S 0 . By the previous lemma, A + ∈ R at ′ S ( S 0 , Σ). Since Rat ′ S ( S 0 , Σ) is closed under left and right multiplication with elements of S 0 and since Rat ′ S ( S 0 , Σ) is clo sed under sum, it follows that αA + β is in Rat ′ S ( S 0 , Σ). Thus, | A | is the sum of an element of S 0 and an element of Rat ′ S ( S 0 , Σ), showing that | A | is in Rat S ( S 0 , Σ). ✷ W e now pro ve the con verse of the previous prop osition. Proposition 6.10 F or any S, S 0 and Σ as ab ove, R at S ( S 0 , Σ) ⊆ Rec S ( S 0 , Σ) . Pro of. Supp ose that s ∈ Rat S ( S 0 , Σ). W e hav e to sho w that ther e is an automaton o ver ( S 0 , Σ) whose b ehavior is s . First w e pr ove this cla im for the elemen ts o f Rat ′ S ( S 0 , Σ). W e show: F or each s ∈ Rat ′ S ( S 0 , Σ) there exists an automato n over ( S 0 , Σ) whose b ehavior is s such that the pr o duct of the initial and the fi n al ve ctor of A is 0. Assume tha t s = 0. Then consider the automaton A 0 = (0 , 0 , 0) o f dimension 1. W e hav e that | A 0 | = 0 and it is clear that the pro duct of the initial and the final vector is 0 . Next let s = a for some letter a ∈ Σ. The n define the following auto ma ton A a of dimension 2: A a =   1 0  ,  0 a 0 0  ,  0 1  . W e hav e | A a | =  1 0   1 a 0 1   0 1  = a. In the induction s tep ther e are five cases to consider. Supp os e that s = s 1 + s 2 or s = s 1 s 2 such that there exist automa ta A i = ( α i , A i , β i ) over ( S 0 , Σ) with | A i | = s i satisfying α i β i = 0, i = 1 , 2. W e construct automata A 1 + A 2 , A 1 · A 2 defining s 1 + s 2 and s 1 s 2 , resp ectively . Let A 1 + A 2 =  ( α 1 , α 2 ) ,  A 1 0 0 A 2  ,  β 1 β 2  and A 1 · A 2 =  ( α 1 , 0) ,  A 1 β 1 α 2 A 2 0 A 2  ,  β 1 α 2 β 2 β 2  . Then | A 1 + A 2 | = ( α 1 , α 2 )  A ∗ 1 0 0 A ∗ 2   β 1 β 2  = α 1 A ∗ 1 β 1 + α 2 A ∗ 2 β 2 = | A 1 | + | A 2 | , and | A 1 · A 2 | = ( α 1 , 0)  A ∗ 1 A ∗ 1 β 1 α 2 A + 2 0 A ∗ 2   β 1 α 2 β 2 β 2  = α 1 A ∗ 1 β 1 α 2 β 2 + α 1 A ∗ 1 β 1 α 2 A + 2 β 2 = α 1 A ∗ 1 β 1 α 2 A ∗ 2 β 2 = | A 1 | · | A 2 | . Also, ( α 1 , α 2 )( β 1 , β 2 ) T = α 1 β 1 + α 2 β 2 = 0 17 and ( α 1 , 0)( β 1 α 2 β 2 , β 2 ) T = α 1 β 1 α 2 β 2 = 0 . (Of cour se, we could ha ve used the fact that α 2 β 2 = 0 earlier in the definition of A 1 · A 2 , but we wan ted to show that the construction works ev en if this do es not hold.) Next, we show that when s = r + for some r which is the b ehavior of an automaton A = ( α, A, β ) ov er ( S 0 , Σ), such that αβ = 0 , then s is the b ehavior of an automaton A + . Since | A | = αA ∗ β = αβ + αA + β = αA + β th us r = αA ∗ β = αA + β . Now let A + = ( α, A + β αA, β ) . By ( A + β αA ) ∗ = A ∗ ( β αA + ) ∗ , w e hav e | A + | = αA ∗ ( β αA + ) ∗ β = αA + β ( αA + β ) ∗ = ( αA + β ) + = | A | + = s. By assumption, we have tha t αβ = 0. Last, if s = | A | and x ∈ S 0 , where A = ( α, A, β ) is an automaton o ver ( S 0 , Σ) with αβ = 0 , then xs = | x A | and sx = | A x | wher e x A = ( xα, A, β ) and A x = ( α, A, β x ). Also ( xα ) β = α ( β x ) = 0. W e have thus shown that R at ′ S ( S 0 , Σ) ⊆ Rec S ( S 0 , Σ). Finally , if s ∈ Rat S ( S 0 , Σ), so that s = x + a for some x ∈ S 0 and a ∈ Rat ′ S ( S 0 , Σ), then there is an automa ton A = ( α, A, β ) over ( S 0 , Σ) whos e behavior is a . Then define B =   x α  ,  0 0 0 A  ,  1 β  . It holds that | B | =  x α   1 0 0 A ∗   1 β  = x + αA ∗ β = x + a = s. ✷ W e hav e prov ed: Theorem 6 .11 Supp ose that S is a p artial Conway semiring, S 0 is a subsemiring of S and Σ ⊆ D ( S ) . Then Rec S ( S 0 , Σ) = Rat S ( S 0 , Σ) . Corollar y 6 .12 Supp ose that S is a Conway semiring, S 0 is a Conway subsemiring of S and Σ ⊆ S . Then Rec S ( S 0 , Σ) = Rat S ( S 0 , Σ) is the le ast Conway subsemiring of S which c ontains S 0 ∪ Σ . Corollar y 6 .13 Supp ose that S is a p artial Conway semiring, S 0 is a subsemiring of S and Σ ⊆ D ( S ) . Su pp ose that whenever x + a ∈ D ( S ) for some x ∈ S 0 and a ∈ D ( S ) t hen x = 0 . Then Rec S ( S 0 , Σ) = Rat S ( S 0 , Σ) is the le ast p artial Conway su bsemiring of S which c ontains S 0 ∪ Σ . The cas e when the partial Conw ay semiring is a power s eries semir ing deserves spe c ia l a ttention. Let S b e a semiring and Σ a set, and consider the partial iter ation semiring S h h Σ ∗ i i . Recall that the star o pe r ation is defined on the proper power ser ies a nd that S can be iden tified with a sub- semiring of S h h Σ ∗ i i . W e denote R at S h h Σ ∗ i i ( S, Σ) by S rat h h Σ ∗ i i and Rec S h h Σ ∗ i i ( S, Σ) by S rec h h Σ ∗ i i . Since S h h Σ ∗ i i is the direct sum o f S and the ideal of pro p er p ow er series, (23) is sa tisfied. Thus, S rat h h Σ ∗ i i is close d under the star op eratio n, and thus S rat h h Σ ∗ i i is partial iteration semiring. 18 Corollar y 6 .14 Supp ose that S is a semiring and Σ a set. Then S rat h h Σ ∗ i i is the le ast p artial iter ation s ubsemiring of S h h Σ ∗ i i c ontaining S ∪ Σ . Mor e over, S rat h h Σ ∗ i i = S rec h h Σ ∗ i i . Recall from Remark 5.1 1 that when S is a Conwa y or iteration semiring , then s o is S h h Σ ∗ i i , for any s et Σ. Corollar y 6 .15 Supp ose t hat S is a Conway semiring. Then S rat h h Σ ∗ i i is the le ast Conway subsemiring of S h h Σ ∗ i i c ontaining S ∪ Σ . Mor e over, S rat h h Σ ∗ i i = S rec h h Σ ∗ i i . The following r esult is used in [4]. Theorem 6 .16 Supp ose that S is a semiring, Σ is a set, and S ′ is a p artial Conway semiring. Then a function h : S rat h h Σ ∗ i i → S ′ is a morphism of p artial Conway semirings iff the fol lowing hold. 1. The r estriction of h onto S is a semiring morphism. 2. Σ h ⊆ D ( S ′ ) . 3. h pr eserves line ar c ombinations in S h Σ i , i.e., ( s 1 a 1 + · · · + s n a n ) h = ( s 1 h )( a 1 h ) + · · · + ( s n h )( a n h ) for al l s i ∈ S , a i ∈ Σ , i = 1 , . . . , n , n ≥ 0 . 4. h pr eserves the b ehavior of automata: F or every automaton A = ( α, A, β ) in S rat h h Σ ∗ i i , | A | h = | A h | , wher e A h is t he aut omaton ( αh, Ah, β h ) over ( S h , Σ h ) in S ′ . Pro of. It is clear that the conditions ar e necessa ry . Supp o se now that h satisfies the ab ove conditions. Since the r estriction of h o nt o S is a semiring mor phism, h preser ves the constants 0 and 1. T o prov e that h preserves sum, consider rationa l series s 1 = | A 1 | and s 2 = | A 2 | in S rat h h Σ ∗ i i , where A i = ( α i , A i , β i ) a re automata ov er ( S 0 , Σ) for i = 1 , 2 . Let A = A 1 + A 2 be defined as in the pro of of Theore m 6.11. Since h maps Σ int o D ( S ′ ) and preserves linear combinations in S h Σ i , we hav e that A h = ( αh, Ah, β h ) is an a utomaton over ( S 0 h, Σ h ). Since h preserves b ehavior of automata, ( s 1 + s 2 ) h = | A | h = | A h | = | ( A 1 + A 2 ) h | = | A 1 h + A 2 h | = | A 1 h | + | A 2 h | = | A 1 | h + | A 2 | h = s 1 h + s 2 h. The fact tha t ( s 1 s 2 ) h = ( s 1 h )( s 2 h ) can be pr oved in the same way using the construction of the automaton A 1 · A 2 . Last, we prov e that h pr eserves + . F or this reaso n, let s b e a prop er rationa l series in S rat h h Σ ∗ i i . Le t A = ( α, A, β ) b e an a utomaton over ( S, Σ) whose behavior αA ∗ β = αβ + αA + β = αA + β is s . Consider the automaton A + defined in the pro o f of Theorem 6.11. Then, | A + | = | A | + and | ( A h ) + | = | A h | + . Thus, since h prese r ves b ehavior, s + h = | A + | h = | A + h | = | ( A h ) + | = | A h | + = ( sh ) + . 19 ✷ References [1] J. Berstel and Ch. Reutenauer, R ational Series and Their L anguages , Springer, 1988. V eris on of Octob er 19 , 2007, h ttp://www- igm.univ-mlv.fr/ ∼ be r stel/. [2] S.L. Blo om and Z. ´ Esik, Ma trix a nd matricia l iteration theorie s , Part I, J. Comput. Sys. Sci. , 46(1 9 93), 381–408. [3] S.L. Blo om and Z. ´ Esik, Iter ation The ories: The Equational L o gic of Iter ative Pr o c esses , EA TCS Monog raphs on Theoretical Computer Science, Spring e r –V erlag , 1 9 93. [4] S.L. Blo o m and Z. ´ Esik, Axiomatizing ra tional pow er series, to app ear. [5] S. Bloo m, S. Ginali and J.D. Rutlege, Scalar and vector iteration, J. Computer System Scienc es , 14(197 7), 251–256 . [6] J.C. Co nw ay . R e gular Algebr a and Finite Machines , Chapman and Hall, Lo ndon, 1971. [7] M. Droste a nd W. Kuich, Semirings and formal p ow er serie s , in: Handb o ok of Weighte d Automata , Springer, to app ear. [8] S. Eilenber g, Automata, L anguages, and Machines . vol. A, Academic Press, 1974 . [9] C.C. Elgot, Mona dic computation a nd itera tive alg ebraic theories, in: L o gic Col lo quium 1973, Studies in L o gic , J.C. Shepherdso n, editor , volume 80, Nor th Holland, Amsterda m, 1975, 175– 230. [10] C.C. Elgo t, Matricial theor ies, J. of Algebr a , 42(1976), 391–421 . [11] Z. ´ Esik, Group axio ms for iteratio n, Information and Computation , 148 (1999), 131–18 0. [12] Z. ´ Esik and W. Kuich, Inductiv e ∗ -semirings, The or et. Comput. Sci. , 32 4 (2004), 3–33. [13] Z. ´ Esik and W. Kuich, Eq uational a x ioms for a theory o f automa ta, in: F ormal L anguages and Applic ations , Studies in F uzz ine s s and Soft Computing 14 8 , Springer, 2004, 183– 196. [14] J.S. Gola n, The The ory of Semirings with Applic ations in Computer S cienc e , Longman Scient ific and T echnical, 1993. [15] G. Gr¨ atzer, Universal Algebr a , Springer, 1979 . [16] D. Krob, Co mplete systems of B- r ational identities, The or etic al Computer Scienc e , 89(199 1), 207–3 43. [17] D. Kr ob, Matrix v er sions o f ap erio dic K -ratio na l identities, The or etic al Informatics and Appl lic ations , 25(19 91), 423–44 4. [18] V.N. Redko, On the deter mining totality of relations o f an algebr a of regular events (in Russian), Ukr ainian Math. ˇ Z. , 16 (1964), 120–126 . [19] V.N. Re dko, O n alg ebra of comm utative ev ents (in Russian), Ukr ainian Math. ˇ Z. , 1 6(1964 ), 185–1 95. 20