Syntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland)

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1 Syntactical content of nite appoximations of patial algebas 1 Wikto Batol Inst. Matematyki, Uniw. Waszawski, Waszawa (Poland) batol@mimuw.edu.pl Xavie Caicedo Dep. Matematicas, Univ. de los Andes, Apatado Aeeo 4976, Bogota (Colombia) xcaicedo@zeus.uniandes.edu.co Fancesc Rossello Dep. Matematiques i Infomatica, Univ. Illes Baleas, Palma (Spain) dmifl0@ps.uib.es Abstact. In this pape we give a syntactical answe to the following question: What do we actually know about a patial algeba when we know its set of weak o elative subalgebas with cadinal smalle than a xed bound, if we do not have any infomation on how they ae linked to each othe within the algeba? 1 Intoduction A nite patial algeba isomophic to a weak o elative subalgeba (see Denition 1) of a patial algeba A may be undestood as a nite appoximation of A. Such nite appoximations ae ubiquitous in A (each nite subset of the caie of A suppots at least one weak subalgeba and exactly one elative subalgeba of A), and they completely detemine A, povided we know the way they glue (a patial algeba is the diect limit of its diected system of nite weak, o nite elative, subalgebas [2, Co ]). But, what do we actually know about a patial algeba when we know its nite appoximations (up to isomophisms), but without any infomation on how they ae linked to each othe within the algeba? Even moe, what do we know of a patial algeba when we know (up to isomophisms) all its weak o elative subalgebas with less than n points? In this pape we give a syntactical answe to this question. We dene the syntactical content of a type of nite appoximations as, oughly, the geatest set of fomulas of the fom \conjunction implies disjunction" that ae `captued' by these appoximations (see Denition 2 below), and we detemine it fo weak and elative subalgebas (with a xed bound on the cadinal of thei caie) of patial and total algebas. The poblem we addess hee is closely elated to the local equivalence poblem of patial algebas consideed in [1]; actually, this note is bon in pat fom the desie to bette undestand some of the esults obtained theein. Recall that, 1 This wok has been patially suppoted by the KBN gant 8-T11C-01011, and the DGCIyT gant PB C02-02

2 oughly speaking, two algebas ae locally equivalent when they have the same (up to isomophism) ssubstuctues of a xed type (e.g. nite weak o elative subalgebas). We want to point out that we do not conside nite closed subalgebas hee (elative subalgebas suppoted on nite sets closed unde the opeations), because they do not yield nice nite appoximations of algebas (thee ae patial algebas with no nite closed subalgeba), and, as one can see in [1], they do not yield nice esults in connection to the poblem addessed hee: anyway, we hope to epot on this (and othe cases) in a longe, full vesion of this note and [1]. To simplify things, we only deal hee with patial algebas ove nite (i.e., with nitely many opeation symbols) and homogeneous (i.e., one-soted) signatues, but the esults we obtain in this case ae easily genealized to moe geneal cases; cf. again [1]. 2 Peliminaies and notations Fo the convenience of the eade, in this section we ecall the basic denitions of the theoy of patial algebas used in this pape (except fo weak and elative subalgebas dened at the beginning of the next section); fo any notion not dened hee, as well as fo moe details about those dened, see [2]. In this section we also x some notation and conventions to be used thoughout the pape. We x fo the est of this pape a signatue = (; ), whee is a nite set of opeation symbols and :! IN is the aity mapping. We set = f' 2 j (') = ng fo evey n 2 IN. A patial -algeba (an algeba, fo shot) is a stuctue A = ; (' A ) '2 ), whee A is a set, called the caie of the algeba, and fo evey ' 2, ' A : A (')! A is a patial mapping with domain dom ' A A ('). We denote the class of all such algebas by Alg. Given an algeba denoted by a capital lette in boldface type, B, etc.), we always denote, unless othewise stated, its caie set by the same capital lette in slanted type, B, etc.). An algeba is nite when its caie is nite. The cadinal jaj of a nite algeba A is the cadinal of its caie. An algeba A is total when ' A is a total mapping, fo evey ' 2 ; we denote by TAlg the class of all total algebas. Two algebas A = ; (' A ) '2 ) and B = (B; (' B ) '2 ) ae isomophic when thee exists a bijection h : A! B (an isomophism) such that fo evey ' 2 and fo evey a; a 1 ; : : : ; a (') 2 A, ' A (a 1 ; : : : ; a (') ) = a iff ' B (h(a 1 ); : : : ; h(a (') )) = h(a). We x hencefoth a countably innite set of vaiables X = fx i j i 2 INg, disjoint fom. The set T (X ) of (-)tems with vaiables in X is dened as the least set T such that X [ (0) T and, if ' 2 and t 1 ; : : : ; t (') 2 T, then '(t 1 ; : : : t (') ) 2 T.

3 Given a tem t 2 T (X ) and an algeba A, we dene the (patial) tem function t A : A X! A (whee A X denotes the set of all valuations v : X! A) as follows: { If t = x i 2 X, then t A (v) = v(x i ) fo evey v : X! A. { If t = ' 2 (0), then t A (v) = ' A fo evey v : X! A if ' A is dened 2, and dom t A = ; othewise. { If t = '(t T 1 ; : : : t n ) fo some ' 2 and tems t 1 ; : : : ; t n, then v 2 dom t A n iff v 2 dom i=1 ta i and (t A 1 (v); : : : ; t A n (v)) 2 dom ' A, and if v 2 dom t A then t A (v) = ' A (t A 1 (v); : : : ; t A n (v)). Notice that the denedness and value of t A (v) only depend on the images unde v of the vaiables appeaing in t. Moeove, if ' 2, then the tem function associated to '(x 1 ; : : : ; x n ) is (essentially) the opeation ' A : A n! A. To simplify the notation, and unless othewise stated, when we wite in the sequel t A (v), we always assume that it is dened, i.e., that v 2 dom t A. An existence equation, an equation fo shot, is a pai (p; q) 2 T (X ) 2 tems, and will be witten p q in the sequel. Given an algeba A and a valuation v : X! A, the equation p q is satised in A w..t. v, in symbols ; v) j= p q, when v 2 dom p A \ dom q A and p A (v) = q A (v). So, fo instance, ; v) j= p p means simply that p A (v) is dened; theefoe, we denote the equation p p by 9Ip. Using equations as atoms, and the connectives :; _; ^; ); : : : (with thei usual logical meaning), we can build up fomulas and dene thei satisfaction in a patial algeba w..t. a given valuation; see [2, x7.1] fo details. In this pape we ae only inteested in a special type of such fomulas.?v A quasi-existence equation of type is a fomula of the fom i2i p i q i ) p q with I a nite set. A disjunctive quasi-existence equation, a _-equation fo shot, of type is a fomula of the fom ^ _ i2i p i q i ) j2j p 0 j q0 j with I and J nite sets; so, _-equations include, as special cases, quasi-existence equations and disjunctions of equations (taking jjj = 1 and I = ;, espectively). To simplify the notation, we usually omit the backets embacing the pemise and the conclusion in _-equations. We denote by L V the set of all _-equations W of some peviously xed type. Let now = i2i p i q i ) j2j p0 j q0 j be a _-equation. Then is satised in an algeba A w..t. a valuation v : X! A, in symbols ; v) j=, iff the following condition holds: If ; v) j= p i q i fo evey i 2 I, then ; v) j= p 0 j q0 j fo some j 2 J. 2 If ' 2 (0), we say that ' A is dened when ' A : A 0! A is total, and then we use the same symbol ' A to denote the image of this mapping. of

4 So, ; v) 6j= iff p A i (v) = qa i (v) fo evey i 2 I but, fo evey j 2 J, eithe v 62 dom p 0 A j, o v 62 dom q 0 A j, o p 0 A j (v) 6= q 0 A j (v). Now, an algeba A (globally) satises a _-equation, in symbols A j=, when ; v) j= fo evey v : X! A. It is clea that two isomophic algebas satisfy exactly the same _-equations (as we will see late, the convese implication is false, even fo total algebas). We say that an equation p q is a V consequence of a nite set of equations fp i q i g i2i when the quasi-equation i2i p i q i ) p q is a tautology (i.e., it is satised by all algebas); it is equivalent to say that p q is deduced fom fp i q i g i2i though Bumeiste's deduction ules fo existence equations [2, x6.4.8]. 3 Main esults We begin by ecalling the denitions of weak and elative subalgebas. Denition 1. Let A = ; (' A ) '2 ) and B = (B; (' B ) '2 ) be two algebas, with B A. i) B is a weak subalgeba of A when, fo evey ' 2, if b 2 dom ' B then b 2 dom ' A and ' B (b) = ' A (b). ii) B is a elative subalgeba of A when it is a weak subalgeba and, fo evey ' 2, if b 2 dom ' A \ B (') and ' A (b) 2 B then b 2 dom ' B. Notice that evey subset B of the caie of an algeba A suppots (in pinciple) many weak subalgebas of A, but only one such elative subalgeba, namely the geatest possible weak subalgeba of A suppoted on B. Given an algeba A, let S w ) and S ) be the classes of all algebas of cadinal at most n that ae isomophic to weak and elative subalgebas of A, espectively. Let also Sw) f and S f ) be the classes of all nite algebas that ae isomophic to weak and elative subalgebas of A, espectively. These ae the nite appoximations of A we conside in this pape. Denition 2. Let C be a class of algebas and let ~ S be an algebaic opeato coesponding to some type of nite subalgebas (fo instance, one of those dened above). A set ~ L of _-equations is the syntactical content of ~ S fo C when it is the geatest subset of L satisfying the following thee conditions: i) If V i2i p i q i ) W j2j p0 j q0 j belongs to ~ L then ~ L also contains evey _-equation of the fom V i2i p i q i ) W j2j 0 p0 j q0 j with J 0 J (we say ii) then that L ~ is well-fomed). Fo evey fomula 2 L ~ thee exists a non-empty nite set C S;C ~ () of nite algebas such that, fo evey algeba A 2 C, A 6j= iff thee exists some A 0 2 C ~ S;C () \ ~ S).

5 iii) Fo evey nite algeba A 0, thee exists a fomula S;C ~ 0) 2 L ~ such that, fo evey algeba A 2 C, A 0 2 ~ S) iff A 6j= ~ S;C 0). Thus, a well-fomed set ~ L of _-equations is the syntactical content of ~ S fo a class C when it is the geatest such set such that, fo evey A 2 C, the knowledge of ~ S) is equivalent to the knowledge of Indeed, notice that, fo evey A 2 C: ~L) = f 2 ~ L j A j= g: { To know whethe A satises a fomula 2 ~ L, one has only to check whethe some algeba in the nite set C ~ S;C () belongs to ~ S); { To know whethe a given nite algeba A 0 belongs to ~ S), one only has to check the non-satisfaction of ~ S;C 0) by A. In paticula, the following esult holds. Poposition 1. Let ~ L be the syntactical content of the opeato ~ S fo a class C of algebas. Then, given any two algebas A; B 2 C, ~ S) = ~ S(B) iff ~ L(B) = ~L). This gives a syntactical answe to the `local equivalence' poblem mentioned in the intoduction. Denition 3. Given a non-tautological _-equation, we shall call its complexity the geatest cadinal () of an algeba A such that A 6j= but A 0 j= fo evey stict weak subalgeba of A. We adopt the convention that tautological _-equations have complexity 0. Let L be the set of all _-equations of complexity at most n, and, fo evey ~L L, set ~ L = ~ L \ L. Notice that the complexity of a _-equation is always smalle o equal than the cadinal of the least initial segment 3 of T (X ) containing all tems appeaing in it. Poposition 2. Let L w denote the set of all _-equations of the fom ^ p i q i ) ( x j1 x j2 ) _ ( p 0 k q0 k) i2i (j 1;j 2)2J k2k V such that, fo evey k 2 K, 9Ip 0 k ; 9Iq0 k ae consequences of i2i p i q i. i) Fo evey n 2 IN, L w is the syntactical content of S w fo Alg. ii) L w is the syntactical content of Sw f fo Alg. 3 A subset Y T (X ) is an initial segment when, fo evey ' 2 and t 1; : : : t (') 2 T (X ), '(t 1; : : : t (') ) 2 Y implies t 1; : : : t (') 2 Y.

6 Poof. We will pove only (i), since (ii) follows immediately. To do that, L w being clealy well-fomed, we check points (ii) and (iii) in the denition of syntactical content, and then we show that L w is the geatest well-fomed set of _-equations satisfying point (ii) theein. a) Fo evey = V i2i p i q i ) ( W (j 1;j 2)2J x j1 x j2 ) _ ( W k2k p0 k q 0 k ) in L w, let C S () be a (minimal) set containing one, and only one, w ;Alg epesentative of evey isomophism class of algebas A 0 such that A 0 6j= and ja 0 j () n. This set is clealy nite (and it is empty iff is a tautology). We will show that it satises the popety equied in Denition 2. Let A be an algeba such that A 6j=, and let v : X! A be a valuation such that ; v) 6j=, i.e., such that p A i (v) = qa i (v) fo evey i 2 I but p0ka (v) 6= q 0 A k (v) fo evey k 2 K (notice that V all p 0 A k (v) and q 0 A k (v) ae dened because 9Ip 0 k and 9Iq0 k ae consequences of i2i p i q i ) and v(x j1 ) 6= v(x j2 ) fo evey (j 1 ; j 2 ) 2 J with j 1 6= j 2. Let V be the set of all vaiables appeaing (explicitly) in the tems of, and let A 0 be the least nite weak subalgeba of A containing v(v ) and such that p A0 i (v) and q A0 i (v) ae dened fo evey i 2 I. Then, fo any valuation v 0 : X! A 0 that coincides with v on V we have 0 ; v 0 ) 6j=, hence A 0 6j=. Since any stict weak subalgeba of A 0 satises, we have that ja 0 j () and thus it has an isomophic copy A 0 0 in C S (). This shows that if w ;Alg A 6j= then thee exists some A 0 0 in C S () \ S w ;Alg w ). Convesely, let A 0 be a nite algeba of cadinality less then () such that A 0 6j= and let A be an algeba containing A 0 as a weak subalgeba. Let v : X! A 0 be a valuation such that 0 ; v) 6j= : then p A0 i (v) = q A0 i (v) fo evey i 2 I but p 0 A 0 k (v) V 6= q 0 A 0 k (v) fo evey k 2 K (emembe that 9Ip 0 k and 9Iq 0 k ae consequences of i2i p i q i ) and v(x j1 ) 6= v(x j2 ) fo evey (j 1 ; j 2 ) 2 J with j 1 6= j 2. Taking as v : X! A the same valuation with taget set A, we also have p A i (v) = q A i (v) fo evey i 2 I (because they ae aleady dened, and equal, in A 0 ), p 0 ka 0 (v) 6= q 0 A 0 k (v) fo evey k 2 K (because they ae aleady dened, and dieent, in A 0 ), and v(x j1 ) 6= v(x j2 ) fo evey (j 1 ; j 2 ) 2 J with j 1 6= j 2, so ; v) 6j= and consequently A 6j=. b) Evey patial algeba has an empty weak subalgeba; thus, we can take as S (;) the equation x w ;Alg 1 x 1. Assume now that A 0 is a non-empty patial algeba of cadinality m 1, with caie A 0 = fa 1 ; : : : ; a m g. Let I 0 ) be the set of equations I 0 ) = f'(x i1 ; : : : ; x in ) x i0 j ' 2 ; n 0; i 0 ; : : : ; i n 2 f1; : : : ; mg; ' A0 (a i1 ; : : : ; a in ) = a i0 g and take as S w ;Alg 0 ) the _-equation ^ _ I 0 ) ) 1j 1<j 2m x j1 x j2

7 Notice that 0 ; v) 6j= S w ;Alg 0 ) fo any valuation v : X! A 0 such that v(x i ) = a i, i = 1; : : : ; m. Theefoe, if A 0 2 Sw), f then A 6j= S w ;Alg 0 ). Convesely, assume that A 6j= S w ;Alg 0 ), and let v : X! A be a valuation such that ; v) 6j= S w ;Alg 0 ). Taking a 0 i = v(x i ) fo evey i = 1; : : : ; m, we have that { if '(x i1 ; : : : ; x in ) x i0 2 I 0 ), i.e., if ' A0 (a i1 ; : : : ; a in ) = a i0, then ' A (a 0 i 1 ; : : : ; a 0 i n ) = a 0 i 0 ; { if 1 j 1 < j 2 m then a 0 j 1 6= a 0 j 2. Let A 0 0 = fa 0 1 ; : : : ; a0 mg, and conside the weak subalgeba A 0 0 = 0 0 ; ('A0 0 )'2 ) of A with dom ' A0 0 = f(a 0 i 1 ; : : : ; a 0 i n ) j (a i1 ; : : : ; a in ) 2 dom ' A0 g; ' 2 ; n 0: Then the mapping h : A 0! A 0 0 dened by h(a i ) = a 0 i, i = 1; : : : ; m, is an isomophism of A 0 onto A 0 0, and theefoe A 0 2 S w ). Notice that the complexity of S w ;Alg 0 ) is exactly ja 0 j = m. Theefoe, fo evey algeba A 0 of cadinality at most n we have constucted a _-equation ~ S w ;Alg 0 ) of complexity at most n such that, fo evey A 2 C, A 0 2 S w ) iff A 6j= S ~ w ;Alg 0 ), as we wanted. c) Assume that thee is some well-fomed set of equations L, ~ not contained in L w, and satisfying (ii) in Denition 2 w..t. the opeato S w and the class Alg. Then, L ~ will contain a fomula of the fom Vi2I p i q i ) p q, whee, say, 9Ip is not a consequence of the pemise. V Let A be a minimal algeba not satisfying i2i p i q i ) p q, and theefoe eithe. Then by (ii) A has a nite weak subalgeba A 0 that belongs to C S (); howeve, A and thus also A w ;Alg 0 has an extension satisfying, in contadiction with (ii). ut In the poofs of the next popositions, we shall only give the coesponding sets C ~ S;C () and _-equations ~ S;C 0); the poofs of the desied popeties ae simila to those in the pevious poposition, aleady pesented in detail. Poposition 3. a) L is the syntactical content of S w fo TAlg. b) L is the syntactical content of S f w fo TAlg. Poof. Given an abitay _-equation = V i2i p i q i ) W j2j p0 j q j in L, let C S () be a (minimal) set containing one, and only one, epesentative w ;TAlg of evey isomophism class of algebas A 0 of cadinality at most () that do not satisfy fo some valuation v : X! A 0. Moeove, given a nite algeba A 0, let S w ;TAlg 0 ) = S w ;Alg 0 ). ut Poposition 4. Let L denote the set of all _-equations of the fom ^ _ t j x ij ) _ ( p 0 k q0 k) i2i p i q i ) ( _ j2j k2k

8 whee evey t j is eithe a vaiable o a tem of the fom '(x i1 ; : : : ; x i(') ) fo some ' 2, and, fo evey k 2 K, 9Ip 0 k ; 9Iq0 k ae consequences of V i2i p i q i. a) Fo evey n 2 IN, L (esp. L ) is the syntactical content of S Alg (esp. TAlg ). b) L (esp. L) is the syntactical content of S f w fo Alg (esp. TAlg ). Poof. Fo evey 2 L C S, take C S w ;TAlg (). Moeove, fo evey nite algeba A 0 : fo () = C ;Alg S () and C w ;Alg S () = ;TAlg { S (;) = ;Alg S (;) = x ;TAlg 1 x 1 ; { If A 0 has caie A 0 = fa 1 ; : : : ; a m g, m 1, then let I 0 ) be the set of equations associated to A 0 as in the poof of Poposition 2, and let ut I c 0 ) = f'(x i1 ; : : : ; x in ) x i0 j ' 2 ; n 0; i 0 ; : : : ; i n 2 f1; : : : ; mg; (a i1 ; : : : ; a in ) 62 dom ' A o ' A (a i1 ; : : : ; a in ) 6= a i0 g Then take S ^ ;Alg 0 ) = S ;TAlg 0 ) to be the _-equation I 0 ) ) ( _ I c 0 ) 1j 1<j 2m x j1 x j2 ) The consequences of the pevious esults on the poblem of the local equivalence ae, by Poposition 1, the following. Coollay 1. Let A and B be any two algebas. i) Fo evey n 0, S w ) = S w (B) iff L w ) = L w (B), and Sw) f = Sw(B) f iff L w ) = L w (B). ii) Fo evey n 0, S ) = S (B) iff L ) = L (B), and S f ) = S f (B) iff L ) = L (B). iii) If A and B ae total algebas, then, fo evey n 0, S w ) = S w (B) iff S ) = S (B) iff L ) = L (B), and Sw) f = Sw(B) f iff S f ) = S f (B) iff L) = L(B). Refeences 1. W. Batol, X. Caicedo, F. Rossello, \Local equivalence of patial algebas." Pepint (1997). 2. P. Bumeiste, A Model Theoetic Oiented Appoach to Patial Algebas. Mathematical Reseach vol. 32, Akademie{Velag (1986).