Differential Meadows

Differen tial Meado ws Jan A. Bergstra ∗ Alban P onse Section Soft w are Engineering, Informatics Institute, Universit y of Amsterdam URL: www.scie nce.uva.n l/ ~ {janb, alban} Abstract A meado w is a zero totalised fi eld (0 − 1 = 0), and a cancellation meadow is a meadow without prop er zero divisors. In this paper we consider differenti al meado ws, i.e. , meado ws equipp ed with differentia tion op erators. W e giv e an equational axiomatiza tion of these op erators and thus obtain a fin ite basis for differential cancella tion meadow s. Using th e Zariski top ology w e prov e the existence of a differential cancellation meadow. 1 In tro duction A me adow is an algebra in the sig nature of fields with a n in verse op era to r that satisfies the equations of commutativ e rings with unit ( CRU ) tog ether with ( x − 1 ) − 1 = x ( R efl ) x · x · x − 1 = x ( RIL ) where the names of the e quations a bbreviate Refle ction and Restricte d In verse L aw , resp ec- tively . Mea dows were introduced in [2]. In [1] it was shown that the v ariety of mea dows satisfies precisely those equa tions whic h are v alid in all so-ca lled zero totalised fields ( ZTF s). A ZTF is a field equipp ed with an inv erse opera tor ( ) − 1 that has bee n made tota l by putting 0 − 1 = 0. Alternatively a nd following [4], we will qualify a z e ro totalise d field as a c anc el lation m e adow if it enjoys the following cancellation pro per t y: x 6 = 0 ∧ x · y = x · z = ⇒ y = z . (1) The mentioned re s ult from [1] ma y b e v iew ed as a completeness theorem: CRU + R efl + RIL completely axioma tises the equatio na l theor y E ( ZTF ) o f the class ZTF of z e ro totalise d fields. Another wa y of lo oking at this result is that it establishes that E ( ZTF ) has a finite ∗ Pa rtially supp orted by the Dutc h NWO Jacquard pro ject Symbiosis , pro ject num ber 638.003.611 1 basis. The pro o f of the finite basis theorem for E ( ZTF ) in [1] mak es use of the existence o f maximal ideals. Although co ncise a nd reada ble, that pro of is no n-element ary b ecause the existence of maxima l ideals requir es a non-elementary set theo retic principle, indep endent of ZF set theory . In [2] a finite basis theorem was established for E C ( ZTF ), the closed equations true in ZTF s. In [3 ] a pro of o f the finite basis result for E ( ZTF ) has b een given along the lines of the elementary proo f ab out E C ( ZTF ). The pro o f method is more g e neral than the pro of using maximal ideals beca use it generalizes to extended sig natures (see Theorem 1 b elow). In this pap er we apply this r esult to so-calle d differential meadows, i.e., meadows equipp ed with formal v a riables X 1 , ..., X n and differential op erator s ∂ ∂ X i . W e pr ovide a sho rt equational ax iomatization of the differential op erator s and thu s obtain a finite basis for different ial cancellation mea dows. This a ppea rs to b e an elega nt ax io ma- tization, e.g., ∂ ∂ X i (1 /x ) = − (1 / x 2 ) · ∂ ∂ X i ( x ) follows easily . Finally , we prove the existence of a differential cancellation mea dow, using the Zariski top olog y [7, 6] and a r epresentation result fro m [3]. The pap er is structured a s follows: in the next section we rec all cancellation meadows and the g eneric basis theorem, and introduce differential meadows. Then, in Sectio n 3 we prov e the ex istence o f a differen tial cance lla tion meadow. Some co nclusions are drawn in Section 4. 2 Cancellation and Differen tial Meado ws In this section w e fix some notation and e x plain cancella tion meadows and o ur gener ic basis theorem in detail. Then we introduce differential meadows. 2.1 Cancellation meado ws and a generic basis result A meadow is an a lg ebra in the s ignature o f fields that s atisfies the axioms in T a ble 1 . W e write Md for the set of a xioms in T a ble 1, thus (referr ing to the Introduction) Md = CRU + R efl + RIL . Let IL (Inv erse Law) stand for x 6 = 0 = ⇒ x · x − 1 = 1 , so IL states that there ar e no zero diviso rs. Note that IL and the cancellation prop erty (1) are equiv alent. A c anc el lation me adow is a meadow that also satisfies I L . F rom the axio ms in Md the following identities are deriv a ble: (0) − 1 = 0 , 0 · x = 0 , ( − x ) − 1 = − ( x − 1 ) , x · − y = − ( x · y ) , ( x · y ) − 1 = x − 1 · y − 1 , − ( − x ) = x. 2 ( x + y ) + z = x + ( y + z ) x + y = y + x x + 0 = x x + ( − x ) = 0 ( x · y ) · z = x · ( y · z ) x · y = y · x 1 · x = x x · ( y + z ) = x · y + x · z ( x − 1 ) − 1 = x x · ( x · x − 1 ) = x T able 1 : The s et Md of axioms for meadows W e write Σ m = (0 , 1 , + , · , − , − 1 ) for the sig nature o f (cancellation) mea dows. F urther - more, we often wr ite 1 /t o r 1 t for t − 1 , tu for t · u , t/u for t · (1 / u ), t − u for t + ( − u ), and freely use numerals and e x pone ntiation with cons tan t in teger exp onents. W e further use the notation 1 x for x x and 0 x for 1 − 1 x . Note that for all terms t , (1 t ) 2 = 1 t , 1 t · 0 t = 0 and (0 t ) 2 = 0 t . W e call an express io n 1 t a pseudo unit b ecause it is almost equiv alent to the unit 1, a nd for a similar rea son we say that 0 t is a pseudo z er o . The basis result fro m [3] admits gener alization if ps eudo units and ps eudo zer os pro pagate in the context rule for equa tional lo gic. W e reca ll the precise definition o f this form of propaga tion fr om that pap er. Definition 1. L et Σ b e an ex tension of Σ m = (0 , 1 , + , · , − , − 1 ) , the signatur e of me adows, and let E ⊇ Md b e a set of e qu ations over Σ . Then 1. (Σ , E ) has the pr op agation pr op erty for pseudo u nits if for e ach p air of Σ -t erms t, r and c ontex t C [ ] , E ⊢ 1 t · C [ r ] = 1 t · C [1 t · r ] . 2. (Σ , E ) has the pr op agation pr op erty for pseudo z er os if for e ach p air of Σ -terms t, r and c ontex t C [ ] , E ⊢ 0 t · C [ r ] = 0 t · C [0 t · r ] . W e now reca ll our ge ne r ic basis result fr om [3]: Theorem 1 (Gener ic basis theo rem fo r cancellatio n meadows) . If Σ ⊇ Σ m , E ⊇ Md is a set of e quations over Σ , and (Σ , E ) has the pseudo unit pr op agation pr op ert y and the pseudo zer o pr op agation pr op erty, then E is a b asis (a c omplete axiomatisation) of Mo d Σ ( E ∪ IL ) . 3 ∂ ∂ X i ( x + y ) = ∂ ∂ X i ( x ) + ∂ ∂ X i ( y ) (D1) ∂ ∂ X i ( x · y ) = ∂ ∂ X i ( x ) · y + x · ∂ ∂ X i ( y ) (D2) ∂ ∂ X i ( x · x − 1 ) = 0 (D3) ∂ ∂ X i ( X i ) = 1 (D4) ∂ ∂ X i ( X j ) = 0 if i 6 = j (D5) T able 2 : The s et of axioms DE 2.2 Differential Meado ws Given s o me n ≥ 1 we extend the sig nature Σ m of meadows with differentiation op erators and consta n ts X 1 , ..., X n to mo del functions to be differentiated: ∂ ∂ X i : M → M for i = 1 , ..., n and some meadow M . W e write Σ md for this extended signature. Equationa l axioms for ∂ ∂ X i are given in T able 2, wher e D4 and D5 define n 2 equational axioms. Obs e r ve that the Md ax ioms together with D3 imply ∂ ∂ X i (0) = 0. F urthermor e, using axiom D1 one easily pr ov es: ∂ ∂ X i ( − x ) = − ∂ ∂ X i ( x ). First we establish the exp ected co rollary of Theorem 1: Corollary 1 . The set of axioms Md ∪ DE (se e T ables 1 and 2) is a finite b asis ( a c omplete axiomatisation) of Mo d Σ md ( Md ∪ DE ∪ IL ) . Pr o of. The pseudo unit propag ation prop erty requires a chec k for ∂ ∂ X i ( ) o nly : ∂ ∂ X i (1 t · r ) = ∂ ∂ X i (1 t ) · r + 1 t · ∂ ∂ X i ( r ) = 1 t · ∂ ∂ X i ( r ) . (2) Multiplication with 1 t now yields the pro p erty . F rom (2) we get 0 t · ∂ ∂ X i ( r ) = ∂ ∂ X i ( r ) − 1 t · ∂ ∂ X i ( r ) (2) = ∂ ∂ X i ( r ) − ∂ ∂ X i (1 t · r ) = ∂ ∂ X i (0 t · r ) and multiplication with 0 t then yields the pseudo zero propag ation prope r t y . A differ ent ial me adow is a meadow eq uipp ed with fo r mal v ar iables X 1 , ..., X n and differ- ent iation op era tors ∂ ∂ X i ( ) that satisfie s the axioms in DE . 4 W e conclude his section with an eleg ant co nsequence of the fact that we a re working in the setting of meadows, namely the co nsequence that the differential of an inv erse follows from the DE axioms. Prop osition 1. Md ∪ DE ⊢ ∂ ∂ X i (1 /x ) = − (1 / x 2 ) · ∂ ∂ X i ( x ) . Pr o of. By ax ioms D3 a nd D2, 0 = ∂ ∂ X i ( x/x ) = ∂ ∂ X i ( x ) · 1 /x + x · ∂ ∂ X i (1 /x ), so 0 = 0 · (1 /x ) = ∂ ∂ X i ( x/x ) · (1 /x ) = ∂ ∂ X i ( x ) · 1 /x 2 + ( x/x ) · ∂ ∂ X i (1 /x ) (2) = 1 /x 2 · ∂ ∂ X i ( x ) + ∂ ∂ X i (( x/x ) · (1 /x )) RIL = 1 /x 2 · ∂ ∂ X i ( x ) + ∂ ∂ X i (1 /x ) , and hence ∂ ∂ X i (1 /x ) = − (1 / x 2 ) · ∂ ∂ X i ( x ) . 3 Existence of Differen tial Meado ws In this section w e show the existenc e of different ial mea dows with formal v a riables X 1 , ..., X n for a rbitrary finite n > 0. First we define a particular cancellation mea dow, and then we expand this meadow to a differential cancellation meadow b y adding formal differentiation. 3.1 The Zariski top ology congruence o ver C n 0 W e will use some terminoloy from alg ebraic g eometry , in pa rticular we will use the Za riski top ology [7, 6]. Op en (closed) sets in this top ology will b e indicated a s Z-op en (Z-closed). Recall that co mplements o f Z-close d sets are Z-op en and complements o f Z- ope n sets are Z-closed, finite unions of Z - closed sets a re Z-close d, and int ersections o f Z-closed se ts are Z-closed. Let C 0 denote the zero- to talized expansion of the complex n um ber s. W e will make use of the following facts: 1. The solutions of a set of poly nomial equa tio ns (with n o r less v ar iables) within C n 0 constitute a Z-closed subset o f C n 0 . Here ’p oly no mial’ has the conv e n tional meaning, not inv o lving div ision. T aking e q uations 1 = 0 and 0 = 0 resp ectively , it follows that bo th ∅ and C n 0 are Z-clo sed (and Z- op en as well). 2. Intersections of non-empty Z-op en sets ar e no n-empt y . In the following we consider terms t ( X ) = t ( X 1 , ..., X n ) 5 with t = t ( x ) a Σ m -term and we write T (Σ m ( X )) for the set of these terms. F o r V ⊆ C n 0 we define the equiv alence ≡ V C n 0 on T (Σ m ( X )) by t ( X ) ≡ V C n 0 r ( X ) if e ach a ssignment X 7→ V e v a luates b oth sides to equal v alues in C 0 . It follows immediately that for each V ⊆ C n 0 , T (Σ m ( X )) / ≡ V C n 0 is a meadow. In par ticular, if V = ∅ o ne obtains the trivial meadow (0 = 1) a s b oth 0 and 1 satisfy any universal quantification ov er an empty s e t. If V is a singleton this quotient is a cancellation meadow. In other ca ses the meadow may not satis fy the cancella tion prop erty . Indeed, suppo se that n = 1 and V = { 0 , 1 } and let t ( X ) = X . Now t (1) 6 = 0 . Thus t ( X ) 6 = 0 in T (Σ m ( X )) / ≡ V C 0 . If that is assumed to b e a cancellation meadow, how ever, o ne has 1 t ( X ) = 1, but 1 t (0) = 0, thus r efuting 1 t ( X ) = 1. W e now define the relation ≡ Z T C (Zariski T op ology Co ngruence ov er C n 0 ) by t ≡ Z T C r ⇐ ⇒ ∃ V ( V is Z- op e n, V 6 = ∅ a nd t ≡ V C n 0 r ) . The r elation ≡ Z T C is indeed a co ngruence for all meadow op erators: the equiv alence pr o p- erties follow easily ; for 0 ≡ Z T C 0 and 1 ≡ Z T C 1, take V = C n 0 , and if P ≡ Z T C P ′ and Q ≡ Z T C Q ′ , witnessed resp ectively b y V and V ′ , then P + P ′ ≡ Z T C Q + Q ′ and P · P ′ ≡ Z T C Q · Q ′ are witnessed b y V ∩ V ′ which is Z -op en and non-e mpt y because o f fact 2 ab ov e. Finally − P ≡ Z T C − P ′ and ( P ) − 1 ≡ Z T C ( P ′ ) − 1 are b oth witnessed by V . In [3] we defined the St andar d Me adow F orm (SMF) repr esent ation result for meadow terms. This res ult implies for T (Σ m ( X )) / ≡ Z T C that each term can be repres e nted by 0 or by p/q with p and q p olynomials no t equal to 0. W e notice that it is decidable whether or not a p olynomial equals the 0-po ly nomial by taking all corresp onding products of p ow ers of the X 1 , ..., X n together and then chec king that all co efficients v anish. A few mo re words on the SMF repr esentation r esult. SMFs ar e defined using levels: an SMF of level 0 is of the form p/q with p a nd q p olynomials, a nd an SMF of level k + 1 is of the form 0 p · P + 1 p · Q with P and Q b oth SMFs of level k . As a n example, let P b e the SMF of level 1 de fined by P = 0 1 − X 1 · 2 X 1 X 2 + 1 1 − X 1 · 1 + X 2 − 2 X 1 X 3 8 − X 1 X 2 3 . Now in T (Σ m ( X )) / ≡ Z T C , the po lynomial 1 − X 1 is on some Z- o pen non-empty set V not equal to 0 (see fact 1 ab ov e), thus 1 1 − X 1 ≡ V C n 0 1 and 0 1 − X 1 ≡ V C n 0 0, and hence P ≡ Z T C 1 + X 2 − 2 X 1 X 3 8 − X 1 X 2 3 . So, in T (Σ m ( X )) / ≡ Z T C , the SMF level-hierarch y collaps e s and terms can b e represented by either 0 o r by p/q with bo th p and q p olynomials not equal to 0. In the seco nd case 1 p/q = 1 and therefo r e it is a cancellatio n meadow. F urther more, equality is dec idable in this mo del. Indee d to chec k that 1 p = 1 (and 0 p = 0) for a p olynomial p it suffices to chec k that p is not 0 ov er the co mplex num b ers . Using the SMF representation all close d terms are either 0 or take the for m p/q with p and q no nz e ro p o lynomials. F or q and q ′ nonzero po lynomials we find that p /q ≡ Z T C p ′ /q ′ ⇐ ⇒ p · q ′ − p ′ · q = 0 which we hav e alrea dy found to b e dec ida ble. 6 3.2 Construct ing a differen tial c ancellation meado w In T (Σ m ( X )) / ≡ Z T C the differential op erato rs can b e defined as follows: ∂ ∂ X i (0) = 0 and, using the fa ct that differentials on p olyno mials ar e known, ∂ ∂ X i ( p q ) = ∂ ∂ X i ( p ) · q − p · ∂ ∂ X i ( q ) q 2 . Let V be the set of 0-p oints o f q a nd let U = ∼ V , the co mplement of V . Then p/q is different iable on U and the deriv ative coincides with the formal deriv ative used in the definition. This definition is representation independent: cons ide r p ′ /q ′ ≡ Z T C p/q with V ′ the 0-p oints of q ′ and U ′ = ∼ V ′ . Then there is so me non-empy and Z- ope n W such that p/q ≡ W C n 0 p ′ /q ′ . Now W ∩ U ∩ U ′ is non-empty and Z-op en, and o n this set, ∂ ∂ X i ( p q ) = ∂ ∂ X i ( p ′ q ′ ) . So, formal differentation ∂ /∂ X i preserves the co ng ruence prop erties. Fina lly , we check the soundness o f the DE a xioms: Axiom D1: Consider t + t ′ . In the ca se that one of t and t ′ equals 0, axio m D1 is obviously sound. In the remaining case, t = p/q a nd t ′ = p ′ /q ′ with a ll p olyno mials not equa l to 0 and t + t ′ = pq ′ + p ′ q qq ′ . Using ordinar y different iation on p olynomials we derive ∂ ∂ X i ( t + t ′ ) = ∂ ∂ X i ( pq ′ + p ′ q ) · q q ′ − ( pq ′ + p ′ q ) · ∂ ∂ X i ( q q ′ ) ( q q ′ ) 2 = ∂ ∂ X i ( p ) · q · ( q ′ ) 2 + ∂ ∂ X i ( p ′ ) · q 2 · q ′ − p · ∂ ∂ X i ( q ) · ( q ′ ) 2 − p ′ · ∂ ∂ X i ( q ′ ) · q 2 ( q q ′ ) 2 = ∂ ∂ X i ( p q ) · 1 ( q ′ ) 2 + ∂ ∂ X i ( p ′ q ′ ) · 1 q 2 = ∂ ∂ X i ( t ) + ∂ ∂ X i ( t ′ ) . Axiom D2: Similar. Axiom D3: Consider t , then either t = 0 or t/ t = 1, a nd in bo th case s ∂ ∂ X i ( t t ) = 0. Axioms schemes D4 a nd D5: W e derive ∂ ∂ X i ( X j ) = ∂ ∂ X i ( X j 1 ) = ( 0 if i 6 = j , 1 otherwise. Thu s, by adding formal differe ntiation to T (Σ m ( X )) we cons tructed a differential ca n- cellation mea dow. 7 4 Conclusions In this pap er we introduced differential meadows. W e provided a finite equational basis for differential cancellatio n mea dows a nd prov ed their ex is tence b y a co ns truction based o n the Zariski top olo gy . Different ial meadows gener a lize differen tial fields in the same way as meadows g e neralize fields. As sta ted in [1], exac tly the v on Neuma nn reg ula r rings a dmit expansio n to a meadow. The g eneral question, how e ver, which meadows can b e ex panded to differential meadows that satisfy the DE ax ioms is left op en. In [5] finite meadows hav e bee n characterized a s direct sums of finite fields . The e x istence of differe n tial meadows o v er a finite meadow is in particular left for further analysis. References [1] J.A. Berg stra, Y. Hirshfeld, a nd J.V. T uck er. Fields, meadows and abstract data types . In Arnon Avron, Nach um Ders howitz, a nd Alexander Rabinovich (eds.), Pil lars of Com- puter Scienc e (Essays De dic ate d to Boris (Bo az) T r akhtenbr ot on the Oc c asion of His 85th Birthday) , LNCS 4800, pa ges 166– 178. Springer-V er lag, 2008. [2] J.A. Bergstra and J.V. T uck er . The r ational n um ber s as an a bstract data t yp e. Journal of the AC M , 54(2), Article No. 7, 2 007. [3] J.A. Ber gstra and A. Ponse. A Generic B a sis Theorem for Ca nc e lla tion Meadows. Av ail- able at ar Xiv:0803.39 69 , March 2008. [4] J.A. B ergstra, A. Ponse, and M.B . v an der Z w aag. T uplix Calculus. Electro nic r e- po rt PRG0713, Progr amming Resear c h Gro up, Universit y of Amsterdam. Av a ila ble at arXiv:071 2.3423 , December 2 007. [5] I. Be thke and P .H. Ro den burg. Some pro per ties of finite meadows. Av ailable at arXiv:071 2.0917 , December 2 007. [6] R. Hartshor ne. Al gebr aic Ge ometry . Springer -V erlag, 1977 . [7] O. Za riski. The compactness o f the Riemann manifold of an abstr act field of alg ebraic functions. Bu l letin of t he Americ an Mathematic al So ciety , 50(10):6 83–69 1 , 1944. 8