Axiomatizing rational power series

Axiomatizing Rational P o w e r Series S. L. Blo om Dept. of Computer Science Stev ens Institute of T e chnology Hob ok en, NJ, USA Z. ´ Esik ∗ Dept. of Compute r Science Univ ersit y of Szeged Szeged, Hungary GRLMC Ro vira i Virgili Unive rsit y T arragona, Spain Abstract Iteration semirings are Con wa y semirings satisfying Conw a y’s group iden tities. W e sho w that the semirings N rat h h Σ ∗ i i of ratio nal p ow er series with coefficients in the semiri ng N of natural num b ers are the free partial iteration semirings. Moreo ver, we c haracterize th e semi- rings N rat ∞ h h Σ ∗ i i as th e free semirings in the v ariety of iteration semirings defined by three additional simple id entitie s, where N ∞ is the completion of N obtained by adding a p oint of infinity . W e also show that th is latter v ariet y coincides with the v ariety generated by the complete, o r contin uous semirings. As a consequ ence of these results, we obt ain that the semirings N rat ∞ h h Σ ∗ i i , equipp ed with the sum order, are free in th e class of symmetric in - ductive ∗ -semirings. This characteri zation corresp onds to Kozen’s axiomatization of regular languages. 1 In tro duction One of the most basic algebra ic structures studied in Computer Science are the semirings Reg (Σ ∗ ) of reg ular (or ratio nal) languages ov er an alphab et Σ equipp ed with the star o p eration. Salomaa [32] has axiomatized these semirings of reg ular languag es using a few simple identities and the unique fixe d p oint rule asserting that if the regular lang ua ge a doe s not co ntain the empty word then a ∗ b is the unique solution of the fixed point equatio n x = ax + b . Ther e are sev eral wa ys of ex pressing the empty word prop erty using a firs t- order lang uage. Pro bably , the simplest wa y is b y the inequalit y 1 + a 6 = a . Using this, the unique fixed point rule can be fo r mulated as the first-order axio m ∀ a ∀ b ∀ x ((1 + a 6 = a ∧ ax + b = x ) ⇒ x = a ∗ b ) . Salomaa’s result then amounts to the ass ertion that for an y Σ, Reg (Σ ∗ ) is freely generated in the class of ∗ -semirings satisfying a finite num b er of (simple) identit ies and the ab ov e axiom. W e hav e thus a fin ite first-orde r axiomatization of regular languag es. Because o f the e xtra co ndition on a , the unique fixed point rule is not a quasi-identit y . A finite axiomatizatio n using only qua si-identities has b een fir st obtained by Archangelsky and Gors hkov, cf. [2]. A seco nd, and p erha ps more serious concern is tha t several natural ∗ -semirings which ∗ Pa rtially supported by gran t no. MTM2007-63422 f rom the M inistry of Education and Science of Spain. 1 satisfy all identities of regular languag es are not mo dels of the unique fixed p oint rule. Examples of such semirings ar e semir ing s of binary relatio ns with the reflexive-transitive closure op eration as star, since for binary r elations, the equation x = ax + b usually has several solutions, even if 1 + a 6 = a (i.e., when a not r eflexive). On the other hand, a ∗ b is least amo ng all solutions, so that ∀ a ∀ b ∀ x ( x = ax + b ⇒ a ∗ b ≤ x ) where a ∗ b ≤ x may b e vie wed a s abbr eviation for a ∗ b + x = x . And indeed, the semirings of regular la ng uages can be c har acterized as the free alg ebras in a quasi-v ar iety of semirings with a s tar op eration axiomatized b y a finite set of simple identities and the ab ov e le ast fixe d p oint rule , or the le ast pr e-fi xe d p oint rule ∀ a ∀ b ∀ x ( ax + b ≤ x ⇒ a ∗ b ≤ x ) . This r esult is due to Krob [25]. In [22, 2 3], Kozen also required the dual of the leas t (pre- )fixed po int r ule ∀ a ∀ b ∀ x ( xa + b ≤ x ⇒ ba ∗ ≤ x ) , and gave a simpler pro of of c ompleteness o f this system. Several other finite axioma tizations are deriv able fro m Krob’s a nd K ozen’s systems, see [12, 1 3, 9]. But the la rgest class of algebr as in which the semirings of regular lang uages are free is of course a v ariety . This v ariety , the class of all semirings with a s ta r op er ation satisfying all identit ies true of regular languages, is the sa me a s the v ariet y gener a ted by a ll ∗ -semirings of binary r elations. The que s tion whether this v ar iety is finitely ba sed w as answered by Redko [30, 31] and Conw ay [15], who show ed that there is no finite (first-order or equational) axiomatizatio n. The question of finding infinite equational bases w as considered in [7, 25]. The system given in Krob [25] consists of the Con wa y semir ing identities, the identit y 1 ∗ = 1, and Conw ay’s group identities [15] asso ciated with the finite (simple) groups. Con way semirings were first defined formally in [6, 8]. Conw ay semiring s are semir ings equipp ed with a star op eration s a tisfying ( a + b ) ∗ = a ∗ ( ba ∗ ) ∗ and ( ab ) ∗ = a ( ba ) ∗ b + 1 . Con way semirings satisfying the infinite collection of group iden tities ar e called iter a tion semir ings, cf. [18]. The terminology is due to the fac t that iteration s emirings are exactly the semiring s whic h are iteration algebr a s, i.e., s atisfy the axioms of itera tion theories [8] which ca pture the equational prop erties o f the fixed p oint o p er ation. Thu s, Kr o b’s r esult characterizes the semirings o f regular languages as the free iteration semirings sa tisfying 1 ∗ = 1 (whic h implies that s um is idempo tent). Another pro of of this res ult using iteratio n theories can be obtained by combin ing the ax iomatization of r e gular languag es fro m [7] and the completeness (of cer tain gener alizations of ) the g roup identities for iteration theor ies, established in [18]. In this pap er, we drop the idempo tence of the sum op eratio n and consider the semirings of r ational power series N rat h h Σ ∗ i i and N rat ∞ h h Σ ∗ i i ov er the semiring N of natural n um be rs a nd its co mpletion N ∞ with a p oint of infinit y . The star o p e ration in N rat h h Σ ∗ i i is defined only on those prop er power series having 0 as the co efficient of the empty w ord (the empty w o rd prop erty), wher eas the s ta r op eratio n in N rat ∞ h h Σ ∗ i i is tota lly defined. W e prov e that N rat ∞ h h Σ ∗ i i is freely generated by Σ in the v ariety V of all iter ation semirings s a tisfying the iden tities 1 ∗ 1 ∗ = 1 ∗ , 1 ∗ a = a 1 ∗ and 1 ∗ (1 ∗ a ) ∗ = 1 ∗ a ∗ . This r esult is also of in terest because V co incides with the v ariety gene r ated by those ∗ -semirings that aris e fro m (countably) complete o r contin uous semirings by defining a ∗ as the sum P n ≥ 0 a n . Mo reov er, we pr ov e that N rat h h Σ ∗ i i is freely generated b y Σ in the class of all p artial iter ation semir ings. As a consequence of the equational ax io matizations, w e s how that N rat ∞ h h Σ ∗ i i , equipp ed with the the sum order , is freely genera ted b y Σ in the class of or dered ∗ -semirings sa tisfying the fixed p oint ident ity aa ∗ + 1 = a ∗ and the least pr e-fixed p oint rule. The pap er is or ganized as follows. In Section 2 we rev iew the notion of semiring s and p ow er series. Section 3 is devoted to (partial) Conw ay and iteration semirings . In Section 4 we pr ovide a formulation of the K le e ne-Sch¨ utzen b e r ger theo rem for (partial) Conw ay semirings from [8, 1 1]. In the characterization of the semiring s N rat h h Σ ∗ i i a s the free pa r tial iteration s e mir ings, in a ddition 2 to the Kleene- Sch¨ utzen ber ger theor em, our main to ol will b e the co mm utative iden tities. W e establish several technical results for the commutative identit y in Se c tio n 5. Section 6 is devoted to proving the freenes s result for the semirings N rat h h Σ ∗ i i men tioned a b ov e. Then, in Section 7 we pr ove that the semirings N rat ∞ h h Σ ∗ i i are fr e e in the v ariety if iteratio n semiring s satisfying three additional s imple identities. L ast, in Section 8 w e characterize the semirings N rat ∞ h h Σ ∗ i i as the free symmetric inductiv e ∗ -semirings, and as the free inductiv e ∗ -semirings satsifying an additional inequation. 2 Semirings A semiring [20] is an algebra S = ( S, + , · , 0 , 1) such that ( S, + , 0 ) is a commutativ e mo noid, where + is ca lled sum or addition, ( S, · , 1) is a monoid, where · is called pro duct or m ultiplication. Moreov er, 0 is an abso r bing element with resp ect to multiplication a nd pro duct dis tributes o ver sum: 0 · a = 0 a · 0 = 0 a ( b + c ) = ab + ac ( b + c ) a = ba + ca for all a, b, c ∈ S . A semiring S is called idemp otent if a + a = a for all a ∈ S . A mor phism of semir ings pres erves the s um and pr o duct op era tions and the constants 0 and 1 . Since semiring s are defined by identit ies, the class o f all semirings is a v ariety (see e.g., [21]) a s is the class of a ll idemp otent semir ings. An impo rtant example of a semiring is the semir ing N = ( N , + , · , 0 , 1 ) of natur al num b er s equippe d with the usua l sum and product o p erations. An imp or ta nt exa mple of an idempotent semiring is the bo ole an semiring B whose underlying set is { 0 , 1 } and whose sum and pro duct op er ations are the op erations ∨ and ∧ , i.e., disjunction and conjunction. Actually N and B ar e resp ectively the initial semiring and the initial idempo ten t semiring. The semir ing N ∞ is defined on the set N ∪ {∞} so that it contains N as a subsemiring and n + ∞ = ∞ + n = ∞ a nd m ∞ = ∞ m = ∞ for all n, m ∈ N ∪ { ∞} , m 6 = 0. W e descr ib e tw o constructions on semirings. F or more information on s e mir ings, the rea de r is referred to Golan’s b o ok [20]. The first co nstruction is that of matrix semirings . When S is a s emiring, then for each n ≥ 0 the set S n × n of all n × n matrices ov er S is a lso a s emiring. The sum oper ation is defined point wise and pro duct is the usua l matrix product. The co nstants ar e the matrix 0 n all of whose entries are 0 (often deno ted just 0 ), a nd the diago nal matrix E n whose diagonal entries are all 1 . In addition to s quare matr ic e s, we will a ls o consider more gener al rectang ular matrice s with the usual definition of sum and pro duct. (Rectangular matrices ov er S form a semiadditive category that ca n ca nonically b e as signed to S but w e will avoid using catego rical notions.) When ρ is a function { 1 , · · · , n } → { 1 , · · · , p } , for some n, p ≥ 0, there is a corre sp onding n × p matrix over each semiring: it is a 0-1 matrix with a 1 on the ( i, j )th p os ition exactly when iρ = j . Such matrices will b e ca lled functional . A p ermutation matrix is a functiona l matrix that corr e sp onds to a permutation. The seco nd construction is that of p ower ser ie s and poly nomial se mirings, cf. [5]. Supp ose that S is a semiring and Σ is a set. Let Σ ∗ denote the free monoid of all words ov er Σ inc luding the empty 3 word ǫ . A p ower series over S (in the noncommuting letters in Σ) is a function s : Σ ∗ → S . It is a co mmon practice to r e present a p ow er se r ies s as a formal s um P w ∈ Σ ∗ ( s, w ) w , wher e the c o efficient ( s, w ) is ws , the v alue of s on the word w . The s upp ort of a ser ie s 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 r esp ectively denote the collection of all p ow er se ries a nd po lynomials ov er S in the letters Σ. More gener ally , when S ′ ⊆ S , we let S ′ h h Σ ∗ i i denote the set o f all p ow er s e r ies in S h h Σ ∗ i i a ll of whose co efficients a re in S ′ . The set of poly nomials S ′ h Σ ∗ i is defined in the same way . W e denote by S ′ h Σ i (no star) the collection of thos e p olynomials in S ′ h Σ ∗ i which are linear co mbinations ov er Σ. W e define the sum s + s ′ and pro duct s · s ′ of tw o series s, s ′ ∈ S h h Σ ∗ i i a s 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 an y element s ∈ S with the ser ies, in fact p olyno mial, which maps ǫ to s a nd all other elements of Σ ∗ to 0. In particular , 0 and 1 may b e viewed a s p olynomia ls. It is well-known that e q uipped with the ab ove op eratio ns and consta nt s, S h h Σ ∗ i i is a s e miring which co ntains S h Σ ∗ i as a subsemiring . Note that B h h Σ ∗ i i is isomo rphic to the semiring of langua g es in Σ ∗ , where addition c o rresp onds to set union and multiplication to concatenation. An isomorphism maps each series in B h h Σ ∗ i i to its supp ort, and the inv er se of this isomor phism maps ea ch langua ge L ⊆ Σ ∗ to its char acteristic series s defined by ( s, w ) = 1 if w ∈ L and ( s, w ) = 0 , o therwise. The following fact is well-kno wn. Theorem 2.1 Gi ven any semirings S, 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 , ther e is a unique semiring morphism h ♯ : S h Σ ∗ i → S ′ which extends b oth h S and h . The condition (1) means that for any s ∈ S a nd letter a ∈ Σ, sh S c ommutes with ah . In particula r, since N is initial, and since when S = N the condition (1) holds automatically , we 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 olyno mial semiring N h Σ ∗ i is freely genera ted by Σ in the v ariety of semirings. In the same wa y , B h Σ ∗ i is freely g enerated by Σ in the v ariety of idemp otent semirings. 3 Con w a y and iteration semirings In this section, we rev iew the notions of (partial) Conw ay semiring and iteration s e mir ing. The notions and facts pre s ented here will b e used in the freeness results. The definition of Conw ay semir ings inv olves tw o imp orta nt identities o f regula r languag es. They app ear implicitly in Co nw ay [15] and were first defined ex plicitly in [6, 8]. Partial Conw ay semirings appea r in [11]. Reca ll that an ide al of a semiring S is a set I ⊆ S which con ta ins 0 and satisfies I + I ⊆ I and S I ∪ I S ⊆ I . This section is based on [8] and [11]. 4 Definition 3.1 A partial ∗ -semiring is a semiring e qu ipp e d with a p artial ly define d star op er- ation a 7→ a ∗ whose domain D ( S ) is an ide al. A partial Conw ay semiring is a ∗ -semiring S satisfying the sum star and pro duct star identities: 1. Sum star identit y : ( a + b ) ∗ = a ∗ ( ba ∗ ) ∗ (2) for al l a, b ∈ D ( S ) . 2. Pr o duct star iden tity : ( ab ) ∗ = 1 + a ( ba ) ∗ b, (3) for al l a, b ∈ S such that a ∈ D ( S ) or b ∈ D ( S ) . A ∗ -semiring is a p artial ∗ -semiring S with D ( S ) = S , i.e. , the star op er ation is total ly defin e d. A Conw ay semiring is a p artial Conway semiring which is a ∗ -semiring. Morphisms h : S → S ′ of (p artial) ∗ -semirings and (p artial) Conway semirings pr eserve the ide al and st ar op er ation: if a ∈ D ( S ) t hen ah ∈ D ( S ′ ) and a ∗ h = ( ah ) ∗ . Note that in any partial Con way semiring S , aa ∗ + 1 = a ∗ (4) a ∗ a + 1 = a ∗ (5) 0 ∗ = 1 (6) for all a ∈ D ( S ). Mor eov er, for all a, b ∈ S with a ∈ D ( S ) or b ∈ D ( S ), ( ab ) ∗ a = a ( ba ) ∗ . (7) It then follows that also ( a + b ) ∗ = ( a ∗ b ) ∗ a ∗ (8) for a ll a , b ∈ D ( S ), which ca n b e used instead o f (2) in the definition of partial Co nw ay s emirings. By (4) and (5), for any a, b in a partial Conw ay s e miring S , if a ∈ D ( S ) then a ∗ b is a solution of the equation x = ax + b and ba ∗ is a s olution of x = xa + b . In par ticular, a + = aa ∗ = a ∗ a is a solution of both x = a x + a and x = xa + a . An impo rtant feature of (partial) Conw ay se mir ings is that squar e matrices ov er Conw ay semi- rings also form (partial) Con wa y semiring s. Definition 3.2 Supp ose that S is a p artial Conway semiring. We turn the semirings S k × k , k ≥ 0 int o p artial ∗ -semirings. Note that D ( S ) k × k , the c ol le ction of al l k × k matric es al l of whose entries ar e in D ( S ) is an ide al of S k × k . The st ar op er ation wil l b e define d on this ide al. When k = 0 , S k × k is trivia l as is the definition of star. When k = 1 , we use the star op er ation on S . Assu ming that k > 1 we write k = n + 1 . F or a matrix  A B C D  define  A B C D  ∗ =  α β γ δ  (9) wher e A ∈ D ( S ) n × n , B ∈ D ( S ) n × 1 , C ∈ D ( S ) 1 × n , and D ∈ D ( S ) 1 × 1 , and wher e α = ( A + B D ∗ C ) ∗ β = αB D ∗ γ = δ C A ∗ δ = ( D + C A ∗ B ) ∗ . 5 Proposition 3.3 When S is a (p artial) Conway semiring, so is e ach S n × n . Mor e over, t he matrix star identit y (9) holds for al l mat ric es  A B C D  with A ∈ D ( S ) n × n , B ∈ D ( S ) n × m , C ∈ D ( S ) m × n , and D ∈ D ( S ) m × m , al l n, m ≥ 0 . In fact, ( AB ) ∗ = A ( B A ) ∗ B + E n holds for all rectangular matr ices A ∈ D ( S ) n × m and B ∈ D ( S ) m × n . F or later use we note that the following p ermut ation identity ho lds in a ll (partial) Co nw ay semi- rings. Proposition 3.4 When S is a p artial Conway semiring, A ∈ D ( S ) n × n and π is an n × n p ermutation matrix with tr ansp ose π T , then ( π Aπ T ) ∗ = π A ∗ π T . F ollowing Conw ay [15], we asso ciate a n ident ity in (partial) C o nw ay semirings with each finite group. Let G b e a finite gro up of or der n . Without loss of gener ality we may assume that the elements of G a re the integers 1 , · · · , n . Moreover, b eca use the p er m utation identit y holds in all (partial) Conw ay semir ings, without loss of generality w e may fix a sequencing of the elements and ass ume that 1 is the unit element of G . Definition 3.5 We say that the group identit y asso ciated with a finite g roup G of or der n holds in a p artial Conway semiring S if e 1 M ∗ G u n = ( a 1 + · · · + a n ) ∗ (10) 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 whose ( i, j ) th entry is a i − 1 j , for al l 1 ≤ i, j ≤ n , and e 1 is the 1 × n 0 - 1 matrix who se first entry is 1 and whose other entries ar e 0 , final ly u n is the n × 1 matrix al l of whose ent ries ar e 1 . Equation (10) ass erts that the sum of the entries of the first row of M ∗ G is ( a 1 + · · · + a n ) ∗ . F o r example, the group identit y asso ciated with the gr o up of order 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 a s ( a 1 + a 2 a ∗ 1 a 2 ) ∗ (1 + a 2 a ∗ 1 ) = ( a 1 + a 2 ) ∗ . (It is kno wn that in Conw ay semirings, this ident it y is further equiv alen t to ( a 2 ) ∗ (1 + a ) = a ∗ .) Definition 3.6 We say that a Conway semiring S is an itera tion semir ing if it satisfies al l gr oup identities. W e say that a p artial Conway semiring S is a partial iteration semiring if it satisfies al l gr oup identities (10) wher e a 1 , · · · , a n r ange over D ( S ) . A morphism of (p artial) iter ation semirings is a ( p art ial) Conway semiring morphism. W e end this section b y r ecalling fr o m [8 , 11] that power series semirings are (partial) iter a tion semirings. Suppos e that S is a semiring and Σ is a set, and cons ider the semiring S h h Σ ∗ i i . A series s ∈ S h h Σ ∗ i i is called pr op er [5] if ( s, ǫ ) = 0. It is clear that the prop er ser ies form an ideal of S h h Σ ∗ i i . It is well-kno wn (see e.g. [5 ]) that when s is prop er and r is any se r ies, ther e is a unique ser ie s that solves the fixed po int equation x = sx + r , and that this solution is s ∗ r , where s ∗ is the unique solution of y = sy + 1. 6 Proposition 3.7 F o r any semiring S , t he p ower series semiring S h h Σ ∗ i i , e quipp e d with the star op er ation define d on pr op er s eries, is a p artial iter ation semiring. When S is a ∗ -semiring, it is p ossible to turn star into a to tal op eratio n. Given a series s ∈ S h h Σ ∗ i i , it can be w r itten in a unique w ay as s = s 0 + r , wher e s 0 ∈ S and r is prop er. Since s 0 is in S and S has a star opera tion, s ∗ 0 is defined. W e define s ∗ = ( s ∗ 0 r ) ∗ s ∗ 0 , where ( s ∗ 0 r ) ∗ is the unique solution of the equation x = ( s ∗ 0 r ) x + 1 as before . The following fact is a sp ecia l case of a more general res ult prov ed in [6, 8]. Proposition 3.8 When S is an iter ation semiring, so is S h h Σ ∗ i i . 4 The Kleene-Sc h ¨ utzen b erger theorem Let S denote a semiring, let Σ denote a set, and co nsider the p ow er ser ie s semir ing S h h Σ ∗ i i which is a par tial itera tion semir ing (or an iteratio n semiring , if S is). As usual, we iden tify each letter in Σ and e a ch element of S with a series. W e call a ser ies s in S h h Σ ∗ i i r ational if s b elongs to the least partial iteration subsemiring of S h h Σ ∗ i i containing S ∪ Σ , i.e. when s is contained in the least subsemiring of S h h Σ ∗ i i co ntaining S ∪ Σ clos ed under the star op era tio n. W e let S rat h h Σ ∗ i i deno te the partial iter ation s emiring o f all rational pow e r series in S h h Σ ∗ i i . The Kleene- Sch¨ utzen b er ger theorem [5] equates ra tional p ow er ser ies with the p ow er ser ies reco gnizable b y (w eighted) automata. F o r later use, b elow we give a genera l definition of automata applica ble to all par tia l Conw ay semir ings, see [11] and [8]. Definition 4.1 L et S b e a p art ial Conway semiring and supp ose that S 0 is a subsemiring of S and Σ is a s ubset of D ( S ) . An auto maton in S over ( S 0 , Σ) is a triplet A = ( α, A, β ) , wher e for some inte ger n , α ∈ S 1 × n 0 , β ∈ S n × 1 0 , and A ∈ ( S 0 Σ) n × n , wher e S 0 Σ is the set of al l line ar c ombinations of t he element s of Σ with c o efficients in S 0 . The int e ger n is c al le d t he dimension of A . The b ehavior of A is | A | = αA ∗ β . Thu s, when the partial Conw ay s emiring is S h h Σ ∗ i i , where S is a semiring, S 0 is S and Σ is the collectio n o f p ow er series c o rresp onding to the letters in Σ, we obtain the usual notion of a (weight ed) a utomaton. W e let S rec h h Σ ∗ i i denote the collectio n of all p ower series which ar e behaviors o f such automata. The Kleene-Sch¨ utzebreg e r theor em is: Theorem 4.2 S rat h h Σ ∗ i i = S rec h h Σ ∗ i i . F or a pro of, see [11]. Below we will c a ll an a uto maton (ov e r ( S, Σ)) in S h h Σ ∗ i i also an automa ton in S rat h h Σ ∗ i i . When A = ( α, A, β ) is an automa ton in S rat h h Σ ∗ i i and h is a function S rat h h Σ ∗ i i → S ′ int o a pa rtial iteratio n semiring which is a s emiring morphism on S , maps Σ into D ( S ′ ) and preserves linear combinations in S h Σ i , then A h = (( αh ) , ( Ah ) , ( β h )) is an automato n in S ′ (ov er ( S h, Σ h )). F or later use w e also give the following re s ult from [11]. Theorem 4.3 Supp ose that S is a semiring and Σ is a set, so that S rat h h Σ ∗ i i is a p artial iter ation semiring. Supp ose that S ′ is a p artial iter ation semiring and h is a function S rat h h Σ ∗ i i → S ′ . Then h is a morphism of p artial iter ation semirings iff the r est riction of h onto S is a semiring morphism, h maps Σ to D ( S ′ ) and pr eserves line ar c ombinations in S h Σ i ; mor e over, h pr eserves the b ehavior of automata, so that | A | h = | A h | for al l automata A in S rat h h Σ ∗ i i . 7 5 The comm utativ e iden tit y In the pro of of our results, we will deduce the equa lit y A ∗ ρ = ρB ∗ from the equality Aρ = ρB , where A is an m × m , B is an n × n matrix over a pa rtial iter a tion semiring , and ρ is an m × n functional matrix . The commut ative identit y , defined below, is a ge neralization of the gr oup ident ities which holds in a ll (partial) itera tion semirings. The comm uta tive ident it y allows us to infer the implicatio n a bove, under ce r tain conditions. The commutativ e identit y was int ro duced for ∗ -semirings in [8 ] but its origins in iteration theories go back to [1 7]. See also [18]. This section is rather technical and all pro ofs ma y b e skipped at first reading. In or der to illustrate the commut ative identit y and its use, consider the following situatio n. Assume that A , B and ρ are as ab ov e, but for simplicity a ssume that ρ as a function is surjective and monotone, collapsing the first m 1 int egers to 1, the next m 2 int egers to 2 etc. Then write A as a blo ck matrix ( A ij ) ij , where each A ij is a m i × m j matrix for all i, j = 1 , · · · , n . The condition that Aρ = ρ B means that e ach row sum of any A ij is b ij , the ( i, j )th entry of matrix B . Similarly , A ∗ ρ = ρB ∗ means that A ∗ can b e written as a matrix of blo cks of size m i × m j , i, j = 1 , · · · , n , and for eac h i and j , ea ch row sum of the ( i, j )th blo ck is equal to the ( i, j )th ent ry of B ∗ . Now assume that the following str o nger condition holds for the matrices A and B : Ther e exist some r ow matric es c ij , i, j = 1 , · · · , n such that e ach b ij is the sum of the entries of c i,j and e ach entry of e ach A ij is a sum of c ert ain entries of c ij such that e ach entry of c ij app e ars exactly onc e as a summand in e ach r ow of A ij . Then the co mmu tative identit y implies A ∗ ρ = ρB ∗ . By adding 0’s to the row matrices c ij we can make all of them size 1 × k , for some k , or alterna tively , as w e do b elow, we can make eac h c ij size 1 × k i , so that the size of c ij only dep ends on i . Thus the row matrice s c ij can b e arrang e d in the form of a blo ck matrix C = ( c ij ) ij as b elow. Before formally defining the commutativ e identit y , we introduce s o me notation. Let S be any semiring and co nsider ma trices A ∈ S m × n and B 1 , · · · , B m ∈ S n × p . W e let A | | ( B 1 , · · · , B m ) denote the matrix in S m × p whose rows are A 1 B 1 , · · · , A m B m , whe r e A 1 , · · · , A m are the r ows of A . Definition 5.1 Supp ose t hat S is a p artial ∗ -semiring. We say that the commutativ e identit y holds in S if for al l C ∈ D ( S ) n × k , m × n functional matrix ρ , k × m functional matric es ρ 1 , · · · , ρ m and k × n functional matric es τ 1 , · · · , τ n with ρ i ρ = τ iρ for al l i = 1 , · · · , m , (( ρC ) | | ( ρ 1 , · · · , ρ m )) ∗ ρ = ρ ( C | | ( τ 1 , · · · , τ n )) ∗ . Note that under the ass umptions we have Aρ = ρB for the matrices A = ( ρC ) | | ( ρ 1 , · · · , ρ m ) and B = C | | ( τ 1 , · · · , τ n ), and that the commutativ e identit y asserts that A ∗ ρ = ρB ∗ . The commutativ e identit y has a dua l w hich also holds in all (par tia l) iter ation s emirings, see [18, 1 1]. It can b e form ulated as follows. Definition 5.2 Supp ose t hat S is a p artial ∗ -semiring. We say that the dual commutativ e ident it y holds in S if for al l C ∈ D ( S ) k × n , m × n functional m atrix ρ , k × m functional matric es ρ 1 , · · · , ρ m and k × n functional matric es τ 1 , · · · , τ n with ρ i ρ = τ iρ for al l i = 1 , · · · , m , ρ T (( ρ T 1 , · · · , ρ T m ) | | ( C ρ T )) ∗ = (( τ T 1 , · · · , τ T n ) | | C ) ∗ ρ T . Here ( B 1 , · · · , B n ) | | A is the matrix who se columns are B 1 A 1 , · · · , B n A n , wher e A 1 , · · · , A n are the co lumns of A , A is m × n and B 1 , · · · , B n are p × m . 8 Definition 5.3 A semiring S is atomistic if for any a 1 , . . . , a m and b 1 , · · · , b n in S if a 1 + · · · + a m = b 1 + · · · + b n then ther e ex ist c 1 , · · · , c k in S and pa rtitions I 1 , · · · , I m and J 1 , · · · , J n of the set { 1 , · · · , k } such that a i = X p ∈ I i c p b j = X q ∈ J j c q for e ach i = 1 , · · · , m and j = 1 , · · · , n . Examples of atomistic semir ings ar e B , N and N ∞ . Proposition 5.4 Supp ose that S is atomistic and A ∈ S m × m , B ∈ S n × n ar e such that Aρ = ρB holds for some m × n functional matrix ρ . Then ther e is a matrix C ∈ S n × k , k × m functional matric es ρ 1 , · · · , ρ m and k × n functional matric es τ 1 , · · · , τ n with ρ i ρ = τ iρ for al l i = 1 , · · · , m such that A = ( ρC ) | | ( ρ 1 , · · · , ρ m ) B = C | | ( τ 1 , · · · , τ n ) . Pro of. It suffices to consider the case when ρ is surjective and monoto ne. Thus, the as sumption is that A is a blo ck ma tr ix ( A ij ) ij such that the sum of each row of each A ij is b ij , the ( i, j )th ent ry o f B . Since S is atomistic, for each ( i, j ) there is a row matrix C ij such that the sum o f its ent ries is b ij and each en try o f each row o f ea ch A ij can b e wr itten a s a sum of certa in e n tries o f C ij in such a wa y that each entry o f C ij app ears exa c tly once as a s ummand in each row of A ij . But this is clear since the semiring is atomistic. ✷ In a similar way , we hav e: Proposition 5.5 Supp ose that S is atomistic and A ∈ S m × m , B ∈ S n × n ar e mat ric es such that ρ T A = B ρ T holds for some m × n functional matrix ρ . Then t her e is a matrix C ∈ S k × n , k × m functional matric es ρ 1 , · · · , ρ m and k × n fun ctional matric es τ 1 , · · · , τ n with ρ i ρ = τ iρ for al l i = 1 , · · · , m such t hat A = ( ρ T 1 , · · · , ρ T m ) | | ( C ρ T ) B = ( τ T 1 , · · · , τ T n ) | | C. Proposition 5.6 Supp ose that S is atomistic. L et A ∈ S h Σ i m × m , B ∈ S h Σ i n × n b e matric es and let ρ b e a functional matrix of size m × n . If Aρ = ρB then ther e is a matrix C ∈ S h Σ i n × k , k × m functional m atr ic es ρ 1 , · · · , ρ m and k × n functional matric es τ 1 , · · · , τ n with ρ i ρ = τ iρ for al l i = 1 , · · · , m such that A = ( ρC ) | | ( ρ 1 , · · · , ρ m ) B = C | | ( τ 1 , · · · , τ n ) . Pro of. There exists a finite Σ 0 ⊆ Σ such that whenever A or B has a n entry which ha s a summand sσ where σ ∈ Σ and s is not 0, then σ ∈ Σ 0 . Now for ea ch σ ∈ Σ 0 , let A σ denote the m × m matr ix whose ( i, j )th entry for i, j = 1 , · · · , m is sσ wher e s is the co efficient o f σ in A ij , the ( i, j )th entry of A . If there is no such summand, let A σ ij = 0. Define the n × n matrices B σ , σ ∈ Σ 0 in the same wa y . W e then have A σ ρ = ρB σ , for each σ ∈ Σ 0 . Thu s, by 9 Prop ositio n 5.4, for each σ ∈ Σ 0 there is a matrix C σ ∈ S h{ σ } i m × k σ and functional matrices ρ σ i and τ σ j , i = 1 , · · · , m , j = 1 , · · · , n of appropriate size with ρ σ i ρ = τ σ iρ such that A σ = ( ρC σ ) | | ( ρ σ 1 , · · · , ρ σ m ) B σ = C σ | | ( τ σ 1 , · · · , τ σ n ) . Let Σ 0 = { σ 1 , · · · , σ p } , say . Define C =  C σ 1 · · · C σ p  and ρ i =    ρ σ 1 i . . . ρ σ p i    τ j =    τ σ 1 j . . . τ σ p j    for all i = 1 , · · · , m and j = 1 , · · · , n . Then for each i , ρ i ρ =    ρ σ 1 i . . . ρ σ p i    ρ =    ρ σ 1 i ρ . . . ρ σ p i ρ    =    τ σ 1 iρ . . . τ σ p iρ    = τ iρ . Also, ( ρC ) | | ( ρ 1 , · · · , ρ m ) =    C σ 1 1 ρ ρ σ 1 1 + · · · + C σ p 1 ρ ρ σ 1 1 . . . C σ 1 mρ ρ σ m 1 + · · · + C σ p mρ ρ σ 1 m    =    A σ 1 1 + · · · + A σ p 1 . . . A σ 1 m + · · · + A σ p m    = A, where A 1 , · · · , A m denote the rows of A . In a similar w ay , C | | ( τ 1 , · · · , τ n ) = B . ✷ Symmetrically , we hav e: Proposition 5.7 Supp ose that S is atomistic. L et A ∈ S h Σ i m × m , B ∈ S h Σ i n × n and let ρ b e a functional matrix of size m × n . If ρ T A = B ρ T then ther e is a matrix C ∈ S h Σ i k × n , k × m functional matric es ρ 1 , · · · , ρ m and k × n fun ctional matric es τ 1 , · · · , τ n with ρ i ρ = τ iρ for al l i = 1 , · · · , m such t hat A = ( ρ T 1 , · · · , ρ T m ) | | ( C ρ T ) B = ( τ T 1 , · · · , τ T n ) | | C. 10 6 F ree partial iteration semirings In this section, our aim is to show that for an y set Σ, N rat h h Σ ∗ i i is freely genera ted by Σ in the class of p artial iteratio n semiring s . F or this rea son, a ssume that S is a pa rtial iter ation semiring and h is a function Σ → D ( S ). W e can extend h to a semiring morphism N h Σ ∗ i → S . In particular, h is de fined on N and on N h Σ i , and in a p oint wis e ma nner, on matrices with en tries in N or N h Σ i . W e wan t to show that h can b e extended to a unique morphism h ♯ : N rat h h Σ ∗ i i → S of partial iteration semirings. F or this rea s on, we will consider automata A = ( α, A, β ) (in N rat h h Σ ∗ i i ) where α ∈ N 1 × n , β ∈ N n × 1 and A ∈ N h Σ i n × n for some n . Using the function h , w e define the image of A as the automaton A h in S : A h = ( αh, Ah, β h ). W e know from Theor em 4.3 that we are forced to define h ♯ by | A | h = | A h | , for all automata A . W e also know tha t if this function is well-defined, then it is a morphism N h h Σ ∗ i i → S of partial iteration s emirings (which clearly extends h ). So all w e ha ve to show is that h ♯ is well-defined. The pr o of of this fact relies o n a result proved in [4] that we reca ll now. Definition 6.1 L et A = ( α, A, β ) and B = ( γ , B , δ ) b e two aut omata (in N rat h h Σ ∗ i i ) of di men- sion m and n , r esp e ctively. We s ay t hat an m × n fun ct ional matrix ρ is a simu lation A → B if αρ = γ , ρδ = β and Aρ = ρB hold. Mor e over, we say t hat ρ is a dual sim ula tion A → B if ρ is a simulation A T → B T , wher e A T = ( β T , A T , α T ) and B T is define d in the same way. Note that ρ is a dual sim ulatio n A → B iff γ ρ T = α , B ρ T = ρ T A and ρ T β = δ hold. (More general s imulations w er e defined in [8]. The simulations defined ab ov e are the functional and dual functional s im ulations of [8]. In the pap er s [3, 4], the terms “cov er ing” a nd “ c o -cov e ring” are used for simulation and dual sim ula tio n. Moreover, only simulations and dual simulations corres p o nding to surjective functions are considered, since in the formulation o f Theorem 6.2 given in [4], the a utomata are “tr im”, i.e., without useless states .) Let ∼ denote the least equiv alence relation suc h that A ∼ B holds whenever there is a functional sim ula tion o r a dual functional sim ula tion A → B . Moreover, call t wo auto mata A and B e quivalent if | A | = | B | . The following result was proved in [4]: Theorem 6.2 T wo automata A and B in N rat h h Σ ∗ i i ar e e quivalent iff A ∼ B . So our tas k reduces to showing that for automata A and B in N rat h h Σ ∗ i i , if there is a functional or a dual functional sim ulation A → B , then | A h | = | B h | . Lemma 6. 3 Supp ose that A = ( α, A, β ) and B = ( γ , B , δ ) ar e automata in N rat h h Σ ∗ i i of dimen- sion m and n , r esp e ctively. Supp ose t hat ρ is an m × n fun ctional matrix which is a simulation A → B . Then | A h | = | B h | . Pro of. Since Aρ = ρB , it follows from Pr op osition 5.6 that ther e exists a matrix C ∈ N h Σ i n × k and k × m functional matrices ρ 1 , · · · , ρ m and k × n functional matrices τ 1 , · · · , τ n with ρ i ρ = τ iρ for all i such that A = ( ρC ) | | ( ρ 1 , · · · , ρ m ) a nd B = C | | ( τ 1 , · · · , τ n ). Thu s, also Ah = (( ρh )( C h )) | | ( ρ 1 h, · · · , ρ m h ) and B h = ( C h ) | | ( τ 1 h, · · · , τ n h ). Th us, by the commutativ e iden- 11 tit y , ( Ah ) ∗ ( ρh ) = ( ρh )( B h ) ∗ . Thus, | A h | = ( αh )( A h ) ∗ ( β h ) = ( αh )( Ah ) ∗ (( ρδ ) h ) = ( αh )( Ah ) ∗ ( ρh )( δ h ) = ( αh )( ρh )( B h ) ∗ ( δ h ) = (( αρ ) h )( B h ) ∗ ( δ h ) = ( γ h )( B h ) ∗ ( δ h ) = | B h | . ✷ Lemma 6. 4 Supp ose that A and B ar e finite automata as ab ove of dimension m and n , r esp e c- tively. S u pp ose t hat ρ is an m × n functional matrix which is a dual simulation A → B . Then | A h | = | B h | . Pro of. Since ρ is a simulation A T → B T , it follows as a bove that A T = ( ρC T ) | | ( ρ 1 , · · · , ρ m ) and B T = C T | | ( τ 1 , · · · , τ n ) for some C T , ρ 1 , · · · , ρ m and τ 1 , · · · , τ n with ρ i ρ = τ iρ . Thus, A = ( ρ T 1 , · · · , ρ T m ) | | ( C ρ T ) and B = ( τ T 1 , · · · , τ T n ) | | C . The pro of can b e completed as ab ove us ing the dual comm utative identit y . ✷ The main result of this sectio n is: Theorem 6.5 N rat h h Σ ∗ i i is fr e ely gener ate d by Σ in the class of p artial iter ation semirings. In detail, given any p artial iter ation semiring S and function h : Σ → D ( S ) , t her e is a unique p artial iter ation semiring morph ism h ♯ : N rat h h Σ ∗ i i → S extending h . Pro of. Given S and h , define h ♯ as follows. First, extend h to a semiring mor phism N h Σ ∗ i → S ′ . By Theorem 4.2, w e know that every ra tio nal ser ies in N rat h h Σ ∗ i i is the b ehavior of an automato n in N rat h h Σ ∗ i i . W e also know that for any ratio nal p ower series r ∈ N rat h h Σ ∗ i i reco gnized b y an automaton A , we are for ced to define rh ♯ = | A h | . By Theorem 6.2 , Lemma 6.3 and Lemma 6.4 , h ♯ is well-defined. It is clea r that h ♯ extends h . Moreov er, by Theor em 4.3, h ♯ is a morphism of partial iteratio n semirings. ✷ Remark 6.6 The pa pe r [11] als o defines p artial iter ative s emirings as partial ∗ -semirings S such that for ea ch a, b ∈ S , if a ∈ D ( S ), then a ∗ b is the unique solution o f the equation x = ax + b . It is shown that every par tial iter ative semiring is a par tial iter ation semiring, and that for any semiring S a nd set Σ, the p ower series semir ing S h h Σ ∗ i i is a partial iterative semiring . Th us, S rat h h Σ ∗ i i is also a partial iterative semiring. Since morphisms of partial itera tive s e mirings preserve star , it follows that N rat h h Σ ∗ i i is the free partial iterative semiring on Σ. This fact is related to a result proved in [29], where Morisaki a nd Sa k ai extended Sa lomaa’s axiomatization [32] of regular languag es to rational power series ov er fields (or more genera lly , princip al ide al domains ). Theorem 6.5 ca n be generalized. Co ns ider a p ower s e r ies s e miring S rat h h Σ ∗ i i where S is any semiring. W e can define simulations a nd dua l simulations and the relation ∼ for auto mata in S rat h h Σ ∗ i i over ( S, Σ) in the sa me wa y as ab ove. F or exa mple, when A = ( α, A, β ) and B = ( γ , B , δ ) a r e automa ta o ver ( S, Σ) of dimension m and n , then a sim ulatio n A → B is an m × n functional matrix ρ suc h that αρ = γ , Aρ = ρB and β = ρδ . If ρ is a simulation A → B , then αA k β = αA k ρδ = αρB k δ = γ B k δ 12 for a ll k , and th us | A | = | B | , i.e., A and B are equiv alent. In a simila r wa y , if ρ is a dua l simulation A → B , then | A | = | B | . Th us, if A ∼ B , then A a nd B ar e equiv alent. In the following g eneralizatio n of Theorem 6 .5 we w ill assume that also the c o nv erse prop erty is true, if A and B are equiv alent then A ∼ B . Theorem 6.7 L et S b e a semiring and Σ a set. Supp ose t hat if two aut omata A , B in S rat h h Σ ∗ i i over ( S, Σ) ar e e quivalent then A ∼ B hold s . Mor e over, supp ose that S is atomistic. Th en S rat h h Σ ∗ i i has the fol lowing universal pr op erty. Given any p artial iter ation semiring S ′ , semiring morphism h S : S → S ′ and function h : Σ → S ′ such that sh S c ommutes with ah for al l s ∈ S and a ∈ Σ , ther e is a unique p artial iter ation semiring m orphism S rat h h Σ ∗ i i → S ′ extending h S and h . The pro of is exa ctly the same. Theorem 6.7 is applica ble fo r exa mple to the bo olea n semiring B (see [8]), and the semiring s k defined in Sec tio n 7. How ever, for rings simpler characteriz ations exist, cf. [5]. Remark 6.8 Without the ass umption that S is atomis tic, we only have the following fact. Suppo se that S ′ is a partial Conw ay semiring satisfying the functorial star implic ations [8 ] Aρ = ρa ⇒ A ∗ ρ = ρa ∗ ρ T A = aρ T ⇒ ρ T A ∗ = a ∗ ρ T for all a ∈ D ( S ′ ) and A ∈ S ′ m × m whose ent ries are in D ( S ′ ), a nd for a ll m × 1 functional matrices, m ≥ 2. Then, a s sho wn in [8, 11], S ′ is a partial iter ation semir ing satisfying the functorial s tar implications Aρ = ρB ⇒ A ∗ ρ = ρB ∗ ρ T A = B ρ T ⇒ ρ T A ∗ = B ∗ ρ T for all A ∈ S ′ m × m , B ∈ S ′ n × n whose ent ries are in D ( S ′ ), and for all m × n functional ma trices ρ for any integers m, n . As ab ove, supp ose that if tw o automa ta A , B in S rat h h Σ ∗ i i over ( S, Σ) are equiv alent then A ∼ B . Then for any semiring morphism h S : S → S ′ and function h : Σ → S ′ such that Σ h ⊆ D ( S ′ ) and sh S commutes with ah , for all s ∈ S and a ∈ Σ, there is a unique partial iteratio n semiring mor phism S rat h h Σ ∗ i i → S ′ extending h S and h . 7 A c h aracterization W e ha ve seen that for any set Σ, N rat h h Σ ∗ i i is freely genera ted by Σ in the cla ss of p artial iteration semirings. In particular, N is initial in the class o f partial itera tion semiring s. This latter fact is also clear by noting that the star op eration is completely undefined in N a nd that N is initial in the cla ss o f semirings . The smallest iteration semiring which contains N as a subsemiring is N ∞ , the completion of N with a po int o f infinity denoted ∞ and star op era tion defined by 0 ∗ = 1 and n ∗ = ∞ for all n 6 = 0. In this section our aim is to show that the itera tion semirings N rat ∞ h h Σ ∗ i i are the free algebr a s in a subv ariety of iter ation semirings defined b y a few simple identities. By Prop ositio n 3 .8, N rat ∞ h h Σ ∗ i i is an itera tion semiring . The structure of the initial itera tion semiring was describ ed in [8]. Its elemen ts are 0 , 1 , 2 , · · · , 1 ∗ , (1 ∗ ) 2 , · · · , 1 ∗∗ , 13 ordered as indicated. Sum and pr o duct on the integers a re the standard op erations; the sum and pro duct on the r emaining elements are given by: x + y = max { x, y } , if x ≥ 1 ∗ or y ≥ 1 ∗ (1 ∗ ) n (1 ∗ ) p = (1 ∗ ) n + p x 1 ∗∗ = 1 ∗∗ x = 1 ∗∗ , if x 6 = 0 . Lastly , the star op er ation is defined by: x ∗ =    1 if x = 0 1 ∗ if x = 1 1 ∗∗ otherwise. Ident ifying 1 ∗ and 1 ∗∗ , the resulting co ngruence colla pses the ele men ts 1 ∗ , (1 ∗ ) 2 , · · · , 1 ∗∗ , so that the co r resp onding quotient ∗ -semiring is isomorphic to N ∞ . In this s ection we will characterize the iter ation semir ings N rat ∞ h h Σ ∗ i i as the free a lg ebras in the subv a riety V of iter ation semir ings sp ecified b y the fo llowing identities: 1 ∗ 1 ∗ = 1 ∗ (11) 1 ∗ a = a 1 ∗ (12) 1 ∗ (1 ∗ a ) ∗ = 1 ∗ a ∗ . (13) Proposition 7.1 The identity 1 ∗ = 1 ∗∗ holds in V . Pro of. Instantiating (13) with a = 1 a nd using (11 ) we have 1 ∗ 1 ∗∗ = 1 ∗ 1 ∗ = 1 ∗ . But by the ab ov e descr iption of the initial iter ation semir ing, 1 ∗ 1 ∗∗ = 1 ∗∗ in any iteration se miring. ✷ Since 2 ∗ = 3 ∗ = · · · = 1 ∗∗ in the initia l iter a tion semiring, it follows that n ∗ = 1 ∗ holds in V for any integer n ≥ 1 viewed as a term. Thu s, (11) ma y b e replaced b y the identit y 1 ∗ = 1 ∗∗ . Also, by (8), (1 + a ) ∗ = (1 ∗ a ) ∗ 1 ∗ , so that in view of (12 ), equation (13) is equiv alen t to (1 + a ) ∗ = 1 ∗ a ∗ . (14) More genera lly , w e hav e that ( n + a ) ∗ = 1 ∗ a ∗ (15) holds in V , for any n ∈ N ∞ , n 6 = 0 v iewed as a term. Also , in v iew of the other axioms, (13) is equiv alent to the simpler a ∗∗ = 1 ∗ a ∗ (16) since ( a + 1) ∗ = 1 ∗ ( a 1 ∗ ) ∗ = 1 ∗ + 1 ∗ ( a 1 ∗ ) + = 1 + 1 ∗ + 1 ∗ ( a 1 ∗ ) + = 1 + ( a + 1) ∗ = a ∗ a ∗∗ + 1 = a ∗∗ using only the Conw ay identities. Since N ∞ satisfies the identit ies (11), (12 ) and (13), and since 1 ∗ = 1 ∗∗ holds in V , we hav e: Corollar y 7.2 N ∞ is initial in V . 14 Also, for each set Σ, b oth N ∞ h h Σ ∗ i i and N rat ∞ h h Σ ∗ i i are in V . Consider the set of iter ation semiring terms , or just terms ov e r Σ defined b y t = 0 | 1 | a, a ∈ Σ | t + t | t · t | t ∗ . A ter m is called c onstant term or just c onstant if it cont ains no o ccur rence o f a ny letter in Σ. W e will s ay that tw o terms s, t ar e e quivalent if they are equiv alent mo dulo the defining identities of V , i.e., when the iden tity s = t holds in V . Each term t over Σ ev alua tes to a series | t | in N rat ∞ h h Σ ∗ i i as usual. Since N ∞ is initial in V , for any constant terms s, t we hav e | s | = | t | iff s = t holds in V . W e may thus identify ea ch constant term with an element of N ∞ . The class I of ide al terms is the least class of ter ms with the fo llowing prop erties . 1. 0 ∈ I and a ∈ I fo r all a ∈ Σ. 2. If s ∈ I and t ∈ I then s + t ∈ I . 3. If s ∈ I and t ∈ I o r t is a constant in N , then st and ts are in I . 4. If s ∈ I then s + is in I , where s + is a n abbrev ia tion for ss ∗ . Lemma 7. 3 When t is ide al, | t | is pr op er and | t | ∈ N h h Σ ∗ i i . The eas y pro of is omitted. It then follows that ea ch idea l term t a ls o ev aluates to a ser ies in the partial iteration semiring N rat h h Σ ∗ i i , and that this s e ries is the same a s the ev aluation of t in N rat ∞ h h Σ ∗ i i . Lemma 7. 4 F o r every term t ther e is an e quivalent t erm of the form t c + t 0 + 1 ∗ t ∞ , wher e t c is a c onstant in N , t 0 is an ide al term, and t ∞ is a term. Mor e over, if t c 6 = 0 then t ∞ is ide al. Pro of. This fact is implied by the following claim: F o r every term t ther e is an e quivalent term of the form t c + t 0 + 1 ∗ t ∞ , wher e t c is a c onstant in N ∞ , t 0 and t ∞ ar e ide al t erms. W e prove this fact by induction o n the structure of t . When t is 0 , 1 or a letter in Σ, our claim is cle a r. Supp ose that t = p + s . Then t is equiv alent to ( p c + s c ) + ( p 0 + s 0 ) + 1 ∗ ( p ∞ + s ∞ ). Assume now that t = ps . Then using (11) and (12), t is equiv alent to p c s c + ( p c s 0 + p 0 s c ) + 1 ∗ (( p c + p 0 ) s ∞ + p ∞ ( s c + s 0 ) + p ∞ s ∞ ). Finally , a ssume that t = s ∗ . If s c = 0, then t is equiv alent to 1 + s + 0 + 1 ∗ ( s 0 + s ∞ ) ∗ s ∞ s ∗ 0 as sho w n by the following computatio n using the sum star and pro duct star iden tities a nd (11), (12) and (1 3). ( s 0 + 1 ∗ s ∞ ) ∗ = s ∗ 0 (1 ∗ s ∞ s ∗ 0 ) ∗ = s ∗ 0 (1 + 1 ∗ ( s ∞ s ∗ 0 1 ∗ ) ∗ s ∞ s ∗ 0 ) = s ∗ 0 + s ∗ 0 1 ∗ (1 ∗ s ∞ s ∗ 0 ) ∗ s ∞ s ∗ 0 = s ∗ 0 + s ∗ 0 1 ∗ ( s ∞ s ∗ 0 ) ∗ s ∞ s ∗ 0 = s ∗ 0 + 1 ∗ s ∗ 0 ( s ∞ s ∗ 0 ) ∗ s ∞ s ∗ 0 = 1 + s + 0 + 1 ∗ ( s 0 + s ∞ ) ∗ s ∞ s ∗ 0 . If s c 6 = 0, then using (15) w e hav e that t is equiv alent to 1 ∗ ( s 0 + s ∞ ) ∗ = 1 ∗ + 1 ∗ ( s 0 + s ∞ ) + . In either cas e , s ∗ is of the req uir ed form. ✷ As an immediate co r ollary , we no te the following F atou prop erty: 15 Corollar y 7.5 If s ∈ N rat ∞ h h Σ ∗ i i and al l c o efficients of s ar e in N , then s ∈ N rat h h Σ ∗ i i . In our proo f that each itera tion semiring N rat ∞ h h Σ ∗ i i is freely ge nerated b y Σ in the v ariet y V w e will make use of the cor resp onding fact for the bo olean semir ing, proved in Krob [25]. Theorem 7.6 F or e ach Σ , B rat h h Σ ∗ i i is fr e ely gener ate d by Σ in t he variety of al l iter ation semirings satisfying 1 ∗ = 1 . Let W denote the v ariety of iteration s emirings sa tisfying 1 ∗ = 1. It is clear that W is a subv ariet y of V . W e introduce a co nstruction whic h assigns to every iteration semiring A in V an iteration semiring 1 ∗ A in W . Suppo se that A ∈ V . W e define 1 ∗ A = { 1 ∗ a : a ∈ A } . It is clear that 1 ∗ A contains 0 and is clos ed under sum and pro duct. Also, using (11 ), (12) and (13), (1 ∗ a ) + = 1 ∗ a (1 ∗ a ) ∗ = 1 ∗ a 1 ∗ a ∗ = 1 ∗ a + , showing that 1 ∗ A is closed under the “ plus op era tion” a 7→ a + . How ever, 1 ∗ A do es not necess arily contain 1 and is not necessar ily closed under star. Definition 7.7 F or e ach A ∈ V , we e quip 1 ∗ A with the fol lowing op er ations and c onstants. The sum + and pr o duct · op er ations and the c onstant 0 ar e inherite d fr om A , the c onstant 1 is 1 ∗ and the star op er ation ⊗ is define d by (1 ∗ a ) ⊗ = 1 ∗ (1 ∗ a ) ∗ . Note that ⊗ is well-defined, s inc e if 1 ∗ a = 1 ∗ b , for so me a, b ∈ A , then 1 ∗ (1 ∗ a ) ∗ = 1 ∗ (1 ∗ b ) ∗ . Also, by (13 ), (1 ∗ a ) ⊗ = 1 ∗ a ∗ . Using this, it follows that the plus op eration of 1 ∗ A determined by the star op eratio n ⊗ is the r estriction of the plus op er a tion of A . Indeed, for all a ∈ A , 1 ∗ a (1 ∗ a ) ⊗ = 1 ∗ a 1 ∗ a ∗ = 1 ∗ a + = (1 ∗ a ) + . Lemma 7. 8 F o r any A ∈ V , the assignment h : a 7→ 1 ∗ a , a ∈ A pr eserves al l op er ations and c onstants. Pro of. Clearly , w e hav e 0 h = 0 a nd 1 h = 1 . Also, ( a + b ) h = 1 ∗ ( a + b ) = 1 ∗ a + 1 ∗ b = ah + bh and ( ab ) h = 1 ∗ ab = 1 ∗ 1 ∗ ab = 1 ∗ a 1 ∗ b = ( ah )( bh ), for all a , b ∈ A . Finally , a ∗ h = 1 ∗ a ∗ = (1 ∗ a ) ⊗ = ( ah ) ⊗ . ✷ Corollar y 7.9 F or e ach A in V , 1 ∗ A is an iter ation semiring in W . Pro of. Since the morphism h is surjectiv e , we hav e 1 ∗ A ∈ V . Since also 1 ⊗ = 1 ∗ 1 ∗∗ = 1 ∗ 1 ∗ = 1 ∗ = 1 , it holds that 1 ∗ A ∈ W . ✷ F or the next co rollar y , no te that if t is a term ov er Σ a nd A is an iteration semir ing, then t induces a function A Σ → A a s usual. W e will denote this function by t A . Below we will write function co mpo sition in the diagr ammatic order . Corollar y 7.10 Supp ose that t is a term over Σ and A ∈ V . Then (1 ∗ t ) A c an b e factor e d as h ◦ (1 ∗ t ) 1 ∗ A , wher e h is the morphism A → 1 ∗ A of L emma 7.8 and h : A Σ → (1 ∗ A ) Σ , e 7→ e ◦ h . Pro of. Since h is a morphis m, h ◦ (1 ∗ t ) 1 ∗ A = (1 ∗ t ) A ◦ h = 1 ∗ (1 ∗ t ) A = (1 ∗ 1 ∗ ) t A = (1 ∗ t ) A . ✷ Corollar y 7.11 Supp ose that s, t ar e t erms over Σ and A ∈ V . Then 1 ∗ t = 1 ∗ s holds in A iff it holds in 1 ∗ A . 16 W e will use T he o rem 7.6 in the following w ay . Let Σ be a s e t and consider a term t over Σ. It is easy to see by induction that if 1 ∗ t ev aluates to a series r in B rat h h Σ ∗ i i , then in N rat ∞ h h Σ ∗ i i it ev aluates to the series whose no nzero co efficients are all ∞ and whos e supp ort is the same as that o f r . W e claim that if 1 ∗ t and 1 ∗ s ev aluate to the same series in N rat ∞ h h Σ ∗ i i , then the ident ity 1 ∗ t = 1 ∗ s holds in V . Let W denote the v ariety of iteration semir ing s satis fying 1 ∗ = 1. Since 1 ∗ t a nd 1 ∗ s ev aluate to the same series in N rat ∞ h h Σ ∗ i i , they ev aluate to the same series in B rat h h Σ ∗ i i . Thus, by Theorem 7 .6, 1 ∗ t = 1 ∗ s holds in W . Let A ∈ V . B y Co rollar y 7.9, 1 ∗ A ∈ W , so 1 ∗ t = 1 ∗ s ho lds in 1 ∗ A . By Corolla ry 7.11, this implies that 1 ∗ t = 1 ∗ s holds in A . Since A was an arbitr ary iteration s e miring in V , this mea ns that 1 ∗ t = 1 ∗ s holds in V . Lemma 7. 12 Supp ose that t, s ar e terms over Σ such t hat t he su pp ort of | t | is include d in the supp ort of | 1 ∗ s | . Then t + 1 ∗ s is e quivalent to 1 ∗ s . Pro of. By the ab ov e ar gument, 1 ∗ s = 1 ∗ ( t + s ) = 1 ∗ t + 1 ∗ s holds in V . Also, t + 1 ∗ t = (1 + 1 ∗ ) t = 1 ∗ t ho lds . T hus, t + 1 ∗ s = t + 1 ∗ t + 1 ∗ s = 1 ∗ t + 1 ∗ s = 1 ∗ s holds. ✷ W e now prove a s tr onger version of Lemma 7.4. Lemma 7. 13 F or every term t ther e is an e quivalent term of the form t c + t 0 + 1 ∗ t ∞ , wher e t c is a c onstant in N , t 0 is an ide al term, and t ∞ is a term. Mor e over, if t c 6 = 0 then t ∞ is ide al and | t 0 | and | 1 ∗ t ∞ | have disjoint supp orts. Pro of. W e know fro m Lemma 7.4 that t is e q uiv alent to a term of the form t c + t 0 + 1 ∗ t ∞ , where t c ∈ N , t 0 is an idea l term and if t c is no t 0 then t ∞ is a lso ideal. Now supp(1 ∗ t ∞ ) = supp( t ∞ ) is a regula r langua ge whic h we denote by R . Consider the ra tional series s 0 = | t 0 | and write it as the sum s 1 + s 2 , where ( s 1 , w ) = ( s 0 , w ) if w 6∈ R and ( s 1 , w ) = 0 otherwise , mo reov e r, ( s 2 , w ) = ( s 0 , w ) if w ∈ R and ( s 2 , w ) = 0 o therwise. It is known that s 1 and s 2 are rational (see [5]) and th us there exsist ideal terms t 1 and t 2 with | t 1 | = s 1 and | t 2 | = s 2 . Since | t 1 + t 2 | = s 1 + s 2 = s = | t 0 | , and since these terms a re ideal, by Theorem 6.5 w e hav e that t 0 = t 1 + t 2 holds in V . Since the supp ort of | t 2 | is included in the suppor t of | 1 ∗ t ∞ | , t 2 + 1 ∗ t ∞ = 1 ∗ t ∞ also ho lds in V . Summing up, t is eq uiv alent to t c + t 0 + 1 ∗ t ∞ which is in turn equiv alent to t c + t 1 + 1 ∗ t ∞ proving the claim. ✷ Theorem 7.14 F or e ach set Σ , N rat ∞ h h Σ ∗ i i is fr e ely gener ate d by Σ in V . Pro of. W e hav e alrea dy noted that N rat ∞ h h Σ ∗ i i is in V . By definition, Σ generates N rat ∞ h h Σ ∗ i i . But we still ha ve to sho w that if tw o ter ms ov er Σ ev alua te to the sa me series, then they are equiv alent. But a ny term t is equiv alent to some ter m of the form t c + t 0 + 1 ∗ t ∞ where t c is a constant in N and t 0 is ideal, and if t c > 0, then t ∞ is ideal. Now | t | = | t c | + | t 0 | + | 1 ∗ t ∞ | , whe r e | t c | ∈ N , | t 0 | ∈ N h h Σ ∗ i i and | 1 ∗ t ∞ | ∈ { 0 , ∞}h h Σ ∗ i i , i.e., each coefficient o f the series | 1 ∗ t ∞ | is 0 or ∞ . Mor eov er, | t 0 | and | 1 ∗ t ∞ | hav e disjoint suppo rts, and either | t c | = 0 or | 1 ∗ t ∞ | is proper . Thu s, if | t | = | s | , then | t c | = | s c | , | t 0 | = | s 0 | a nd | 1 ∗ t ∞ | = | 1 ∗ s ∞ | . By Co rollar y 7.2 we hav e that t c = s c holds in V . Since t 0 and s 0 ev aluate to the same serie s in N rat h h Σ ∗ i i , b y Theorem 6.5 we hav e that t 0 = s 0 holds in V . Finally , b y the ab ov e discussion, 1 ∗ t ∞ = 1 ∗ s ∞ holds in V , proving that t = s holds. ✷ 17 Corollar y 7.15 A series s ∈ N ∞ h h Σ ∗ i i is in N rat ∞ h h Σ ∗ i i iff s = s 0 + s ∞ wher e s 0 ∈ N rat h h Σ ∗ i i and al l nonzer o c o efficients of s ∞ ∈ N rat ∞ h h Σ ∗ i i ar e e qu al to ∞ . The series s 0 and s ∞ may b e chosen so that they have disjoint supp orts. Mor e over, a series s , al l of whose nonzer o c o efficients ar e e qual to ∞ , is ra t ional iff its supp ort is r e gular. Remark 7.1 6 The v ariety V is not finitely bas e d, since it has a non-finitely bas ed subv a riety W which has a finite re lative a x iomatization o ver V by the single identit y 1 ∗ = 1. See also [26]. Likewise, the v ariety of all itera tio n semirings is non-finitely based. Recall from [16] that a c omplete s emiring is a semiring S which is equipp ed with a summation op eration P i ∈ I s i for all index sets I satisfying P i ∈∅ = 0, P i ∈{ 1 , 2 } s i = s 1 + s 2 , mor eov er , pro duct distributes ov er a ll sums and summation is asso ciative: a ( X i ∈ I b i ) = X i ∈ I ab i ( X i ∈ I b i ) a = X i ∈ I b i a X j ∈ J X i ∈ I j a i = X i ∈∪ j ∈ J I j a i , where in the last equation the sets I j are pairwise disjoint. Count ably c omplete semirings are defined in the same with the a dditional cons traint that all sums are at mo st co untable. Clea rly , every complete semiring is countably c o mplete. An ω - c ontinuous semiring [8] is a semir ing S equipp ed with a par tial order such that S is an ω -co mplete partia l order ( ω -cp o) with bo ttom element 0 and the sum and pr o duct op erations are co ntin uous, i.e., they preserve the suprema of ω - chains. A c ontinu ous semiring is defined in the sa me way , it is a cpo with con tinuous o p er ations such that 0 is the bottom element. Each ω -co ntin uous semiring is a countably complete s emiring with X i ∈ I s i = sup { X i ∈ F s i : F ⊆ I finite } . Similarly , e ach contin uo us semiring is complete. The semiring N ∞ , equipp ed with the na tural order, is contin uous. It is well-known that equipp ed with the p o int wise or der, N ∞ h h Σ ∗ i i is also contin uous for each Σ. When S is co untably complete, we can define a star op er ation on S by a ∗ = P n ≥ 0 a n . Since ω -co ntin uous, contin uous and complete semirings ar e all countably co mplete, the same definition applies to these semirings . W e p oint out that the ∗ -semirings s o obtained are all in V . Indeed, it is known that when S is countably co mplete, then S is an iteratio n se mir ing (cf. [8]). W e hav e that 1 ∗ is a countable sum o f 1 with itself. Using distributivit y , it follo ws that 1 ∗ 1 ∗ = 1 ∗ . By distributivity , w e also hav e (12). Finally , 1 ∗ a ∗ and 1 ∗ (1 ∗ a ) ∗ are b oth equal to a countable sum P i ∈ I s i containing for each n a countable num b er o f summands s i equal to a n . By the ab ov e observ ations and the fact that the semirings N ∞ h h Σ ∗ i i are contin uo us a nd co ntain the semir ings N rat ∞ h h Σ ∗ i i , w e immediately have: Corollar y 7.17 Continuous, ω -c ontinuous , c omplete and c oun tably c omplete semirings, e quipp e d with the ab ove star op er ation, satisfy exactly the identities of the variety V . Remark 7.1 8 The set N ∞ carries another imp ortant se miring structure. Equipp ed with mini- m um as addition a nd addition as multiplication (and ∞ as the a dditiv e identit y element and 0 18 as the m ultiplica tive identit y), N ∞ is called the tr opic al semiring . It is known that the tr opical semiring has a non-finitely based eq uational theory , cf. [1]. K rob [27] has shown tha t the equality problem for rational p ow er series in tw o or more letters over the tropical semiring is undecidable. Rational p ower series in a single letter over the tropical se miring were treated in [14]. W e end this se c tio n by p ointing out how Theor em 7.6 can be der ived from Theor em 7.14. When k ≥ 1 is an integer, let k denote the quo tient of the iteration semiring N ∞ obtained b y collapsing k and ∞ and thus all ele ments of N ∞ at least k . When k = 1, k is just the B o olean semiring B with star op eration 0 ∗ = 1 ∗ = 1. Our result is : Theorem 7.19 F or e ach int e ger k ≥ 1 , k rat h h Σ ∗ i i is fr e ely gener ate d by Σ in t he variety of iter ation semiring satisfying t he identity 1 ∗ = k . Of course, in the statement of the Theorem, k a ls o denotes the term 1 + · · · + 1 ( k times). Since any iteration semir ing satisfying 1 ∗ = k s a tisfies (11), (12 ) and (13 ), Theorem 7.1 9 is immediate from Theor em 7.14 if we can show that the lea s t congr uence ∼ on N rat ∞ h h Σ ∗ i i whic h collapses k and 1 ∗ collapses any rationa l serie s in N rat ∞ h h Σ ∗ i i with a se r ies all of whose c o efficients are either less than k or eq ual to 1 ∗ (= ∞ ). By Cor ollary 7.15, it is sufficient to pr ov e this fo r rational series in N rat h h Σ ∗ i i . The rest of this section is devoted to proving this fact. Lemma 7. 20 Supp ose that s ∈ N rat h h Σ ∗ i i such that any n onzer o c o efficient of s is at le ast k . Then s ∼ 1 ∗ s . Pro of. Le t R = supp( s ) which is a reg ular la ng uage in Σ ∗ (cf. [5 ]), a nd let r denote the characteristic serie s o f R , so that for any word w , ( r, w ) = 1 if w ∈ R and ( r, w ) = 0 otherwise. It is kno wn that r is rational (this is true for any s emiring, cf. [5]) and th us k r is also ratio nal. Now it is known that t = s − k r is also rational, see Theo rem 1.8 in Chapter VI I of [5]. It is clear that k r ∼ 1 ∗ r = 1 ∗ s . Using this, w e hav e: s = k r + t ∼ 1 ∗ s + t = 1 ∗ s. ✷ Proposition 7.21 F or e ach inte ger k and e ach s ∈ N rat h h Σ ∗ i i ther e is a series r ∈ N rat h h Σ ∗ i i with s ∼ r such that al l c o efficients of r ar e either less than k or e qual to 1 ∗ . Pro of. In our argument, we will ma ke us e of the following known fact from [5]. Given any rational series s ∈ N rat h h Σ ∗ i i , s can b e wr itten as a sum of r ational series s 0 + · · · + s k such that each co efficient of any s i with i < k is 0 or i , and eac h co efficient of s k is 0 or ≥ k . By the previous lemma, s k ∼ 1 ∗ s k , and thus s k is congruent to the rational series s ′ k such that ( s ′ k , w ) = 1 ∗ if ( s k , w ) ≥ k and ( s ′ k , w ) = 0 other wise. W e conclude that s ∼ s 0 + · · · + s k − 1 + s ′ k which has the desired pr o p erty . ✷ 8 A second c haracterization In the previous section, we hav e characterized the semir ings N rat ∞ h h Σ ∗ i i as the free algebr as in a non-finitely based v ariet y V of ∗ -semirings. Since N ∞ has a natural o rder, N ∞ h h Σ ∗ i i may be 19 equipp e d with the p oint wis e or der. This order on N ∞ h h Σ ∗ i i is actually the same a s the sum or der: F or a ll series s, s ′ ∈ N ∞ h h Σ ∗ i i , s ≤ s ′ iff there is a series r with s + r = s ′ . More over, since N ∞ is a cont inu ous semir ing, c f. e.g. [1 9], so is N ∞ h h Σ ∗ i i . In particular, an y map x 7→ sx + r ov e r N ∞ h h Σ ∗ i i has the ser ies s ∗ r as its least pr e-fixe d p oint (since ss ∗ r + r ≤ s ∗ r and for all s ′ , if ss ′ + r ≤ s ′ then s ∗ r ≤ s ′ ). Moreov er, r s ∗ is the least pre-fixed point of the map x 7→ xs + r . The semiring N rat ∞ h h Σ ∗ i i , equipp ed with the p oint wise or der inherited fro m N ∞ h h Σ ∗ i i als o has these least pre-fixed p oint prop erties. How ever, in the main r e s ult of this section, w e will hav e to work with the sum order on N rat ∞ h h Σ ∗ i i which is not the sa me as the p oint wise or der. It is known that for r , s ∈ N rat h h Σ ∗ i i with r ≤ s in the point wise or der, the differ e nc e s − r may not be r ational (see [5]), so that there may not exist a r ational series r ′ with r + r ′ = s . Since N rat h h Σ ∗ i i = N rat ∞ h h Σ ∗ i i ∩ N h h Σ ∗ i i , the same holds for N rat ∞ h h Σ ∗ i i . But the ab ov e least pre-fixed po int pr op erty still holds in N rat ∞ h h Σ ∗ i i with the sum order , as will be shown be low. F or the rest of this pap er, by an or der e d semiring we shall mean a semiring S equipp ed with a partial order ≤ preserved by sum a nd pro duct: If a ≤ a ′ and b ≤ b ′ then a + b ≤ a ′ + b ′ and ab ≤ a ′ b ′ . F ollo wing [19], we call a ∗ -semiring an induct ive ∗ -semiring if it is an order ed semiring such that the following hold for all a, b, x ∈ S : aa ∗ + 1 ≤ a ∗ (17) ax + b ≤ x ⇒ a ∗ b ≤ x. (18) It then follo ws that for any a, b , a ∗ b is the lea st pre-fixed point of the map x 7→ ax + b , a nd is actually a fixed p o int . Moreover, it is known that the star op eratio n is also monotone in any inductive ∗ -semiring. A symmetric inductive ∗ -semiring S also s atisfies xa + b ≤ x ⇒ ba ∗ ≤ x (19) for all a, b, x ∈ S . In [23], Kozen defines a Kle ene algebr a as an idemp otent symmetric inductive ∗ -semiring. (Note tha t if an ordere d semiring S is idemp otent, then the partial order is the semilattice order : a ≤ b iff a + b = b .) A morphism of (symmetric) inductive ∗ -semirings is a ∗ -semiring mor phis m which preserves the o r der. The following result was proved in [19]: Theorem 8.1 E very inductive ∗ -semiring is an iter ation semiring satisfying 1 ∗ = 1 ∗∗ . W e c a ll a (symmetric) inductive ∗ -semiring sum or der e d , if its order r elation is given b y a ≤ b iff there is s ome c with a + c = b . Supp ose that S is an inductive ∗ -semiring. Since for any x ∈ S , 1 x + 0 = x , we ha ve that 0 = 1 ∗ 0 ≤ x . Thus, 0 is the lea st element o f S and since the order is preserved by addition, x ≤ x + y for all x, y ∈ S , so that the order on S is an extension o f the sum order . Proposition 8.2 Any inductive ∗ -semiring satisfies (11) and (13), i.e., the identities 1 ∗ 1 ∗ = 1 ∗ and 1 ∗ (1 ∗ a ) ∗ = 1 ∗ a ∗ . Mor e over, any inductive ∗ -semiring satisfies 1 ∗ a ≤ a 1 ∗ . Pro of. The identit y 1 ∗ = 1 ∗ 1 ∗ holds by Theor em 8 .1 a nd the description of the initial iteration semiring. Now for the inequality 1 ∗ a ≤ a 1 ∗ . W e hav e 1( a 1 ∗ ) + a = a (1 ∗ + 1) = a 1 ∗ . Thus, 1 ∗ a ≤ a 1 ∗ . Last for (13). On one hand, a ≤ 1 ∗ a , and thus a ∗ ≤ (1 ∗ a ) ∗ and 1 ∗ a ∗ ≤ 1 ∗ (1 ∗ a ) ∗ . On the other hand, 1(1 ∗ (1 ∗ a ) ∗ )+ a ∗ ≤ 1 ∗ (1 ∗ a ) ∗ + (1 ∗ a ) ∗ = (1 ∗ + 1)(1 ∗ a ) ∗ = 1 ∗ (1 ∗ a ) ∗ , and thus 1 ∗ a ∗ ≤ 1 ∗ (1 ∗ a ) ∗ . ✷ 20 Proposition 8.3 In any symmetric inductive ∗ -semiring S , 1 ∗ a = a 1 ∗ for al l a ∈ S . Pro of. W e have seen that 1 ∗ a ≤ a 1 ∗ . Since (1 ∗ a )1 + a = (1 ∗ + 1) a = 1 ∗ a , it holds that a 1 ∗ ≤ 1 ∗ a . ✷ Consider now N rat ∞ h h Σ ∗ i i equipp ed with the sum or der , denoted ≤ . W e claim that N rat ∞ h h Σ ∗ i i is a symmetric inductiv e ∗ -semiring. In [19], it is shown that if an order ed semiring equipped with a s tar oper a tion is ordere d by the sum order, then it is a symmetric inductiv e ∗ -semiring iff it satisfies aa ∗ + 1 = a ∗ and ax + b = x ⇒ a ∗ b ≤ x xa + b = x ⇒ ba ∗ ≤ x. In order to prov e these proper ties hold, we describ e a ll solutions o f a linear fixed po int equation ov er N ∞ h h Σ ∗ i i . Proposition 8.4 L et s, r b e series in N ∞ h h Σ ∗ i i . 1. If s is pr op er, then the e quation x = sx + r has s ∗ r as its unique solution. 2. If s = 1 + s 0 , wher e s 0 is pr op er, then the solutions of x = sx + r ar e t he series of t he form s ∗ r + 1 ∗ s + 0 t + t , wher e t is any series. 3. If s = k + s 0 , wher e s 0 is pr op er and k ∈ N ∞ , k 6 = 0 , 1 , then the solutions of x = sx + r ar e the series of the form s ∗ r + 1 ∗ s ∗ 0 t = s ∗ ( r + t ) , wher e t is any series. Pro of. F o r the firs t claim, see [5]. Assume that s = k + s 0 where s 0 is pro pe r and k ∈ N ∞ , k 6 = 0 . Consider the equa tion x = f ( x ), where f ( x ) = sx + r . Since N ∞ is contin uo us a nd x ≤ f ( x ), all solutions ca n b e obtained b y starting with a series t and forming the increas ing sequence f n ( t ), for n ≥ 0, and taking the supremum of this sequence. Since f n ( t ) = s n t + s n − 1 r + · · · + r , this gives s ∗ r + sup n ≥ 0 s n t . But for each n , s n t = ( k + s 0 ) n t , and using the expansion ( k + s 0 ) n = k n +  n 1  k n − 1 s 0 +  n 2  k n − 2 s 2 0 + · · · + s n 0 , we obtain that sup n ≥ 0 s n t =  1 ∗ s + 0 t + t if k = 1 1 ∗ s ∗ 0 t if k > 1 . ✷ Corollar y 8.5 L et s, r b e series in N rat ∞ h h Σ ∗ i i and c onsider the e quation x = sx + r with le ast solution s ∗ r . If z is any r ational solution, t hen ther e is a r ational series p ∈ N rat ∞ h h Σ ∗ i i with z = s ∗ r + p . Pro of. When s is prop er, s ∗ r is the only solution and our claim is clea r. Ass ume that s = 1 + s 0 , where s 0 is pro pe r (and rational). Then the least solution is s ∗ r = 1 ∗ s ∗ 0 r , and any other s olution z is of the form z = s ∗ r + 1 ∗ s + 0 t + t = 1 ∗ s ∗ 0 r + 1 ∗ s + 0 t + t . W e see tha t z = s ∗ r + z , so if z is rational, then it is the sum of the least solution with a r ational ser ies. The last ca se is when s = k + s 0 , where s 0 is prop er and k ∈ N ∞ , k > 1. Then the lea st solution is s ∗ r = 1 ∗ s ∗ 0 r , and any o ther solution z is o f the form z = 1 ∗ s ∗ 0 r + 1 ∗ s ∗ 0 t . W e a gain hav e z = s ∗ r + z , so tha t if z is rational, then it is the sum of the least solution with a r ational s eries. ✷ Symmetrically , we hav e: 21 Corollar y 8.6 L et s, r b e series in N rat ∞ h h Σ ∗ i i and c onsider the e quation x = xs + r with le ast solution r s ∗ . If z is any r ational s olut ion, t hen ther e is a r ational series p ∈ N rat ∞ h h Σ ∗ i i with z = rs ∗ + p . The main result of this sectio n is: Theorem 8.7 F or e ach Σ , N rat ∞ h h Σ ∗ i i (e quipp e d with t he su m or der) is fr e ely gener ate d by Σ in the class of al l inductive ∗ -semirings s atisfying a 1 ∗ ≤ 1 ∗ a . In detail, for any inductive ∗ -semiring S satisfying a 1 ∗ ≤ 1 ∗ a and for any fun ction h : Σ → S ther e is a unique inductive ∗ -semiring morphism h ♯ : N ∞ h h Σ ∗ i i → S extending h . Pro of. W e have alr eady pr ov ed that N rat ∞ h h Σ ∗ i i equipp ed with the sum order is an inductiv e ∗ -semiring satisfying a 1 ∗ ≤ 1 ∗ a . Suppose that S and h ar e g iven. Since S satisfies a 1 ∗ ≤ 1 ∗ a , S is an iteratio n semir ing in the v ariety V des crib ed in the previous section. By Theorem 7.14, h ex tends to a mor phism h ♯ of ∗ -semirings. T o see that h ♯ is preserves the or der, assume that s, s ′ ∈ N rat ∞ h h Σ ∗ i i with s ≤ s ′ . Then there exists a r ational serie s r in N rat ∞ h h Σ ∗ i i with s + r = s ′ . Since h ♯ preserves +, also sh + r h = s ′ h . But the o r der on S contains the sum order, so that sh ≤ s ′ h . The fact that h ♯ is unique follows fro m Theorem 7.14. ✷ Corollar y 8.8 F or e ach Σ , N rat ∞ h h Σ ∗ i i is fr e ely gener ate d by Σ in t he class of al l symmetric inductive ∗ -semirings. Corollar y 8.9 F or e ach Σ , N rat ∞ h h Σ ∗ i i is fr e ely gener ate d by Σ in t he class of al l su m or der e d (symmetric) inductive ∗ -semirings. Let k ≥ 1 and co nsider the ∗ -semiring k . Equipp ed with the natural o rder, k is a c o ntin uous semiring and thus k h h Σ ∗ i i is also a contin uo us semiring and a sy mmetric inductive ∗ -semiring. W e can write each ser ies s ∈ k h h Σ ∗ i i in a unique w ay a s the sum of s eries s 0 , s 1 , · · · , s k , where each nonzer o co efficient in an y s i is i . (Of cour se, s 0 is the 0 s eries.) Now by finiteness , it is known that s is ra tional iff each s i is rational iff the supp ort of each s i is regular. Using this, we do not hav e the problem encoun tered in connection with the o rdering of N rat ∞ h h Σ ∗ i i , the sum order and the p oint wise order ar e equiv alent on k rat h h Σ ∗ i i . Using Theo r em 7.19 w e ha ve: Theorem 8.10 F or e ach k ≥ 1 , k rat h h Σ ∗ i i is b oth the fr e e inductive ∗ -semiring and the fr e e symmetric inductive ∗ -semiring on the set Σ satisfying the identity k = k + 1 . Pro of. Let S b e an inductiv e ∗ -semiring satisfying k = k + 1. 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