A CLOSED ALGEBRA WITH A NON-BOREL CLONE AND AN IDEAL WITH A BOREL CLONE

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1 A CLOSED ALGEBRA WITH A NON-BOREL CLONE AND AN IDEAL WITH A BOREL CLONE MARTIN GOLDSTERN, MICHAEL PINSKER, AND SAHARON SHELAH Abstract. Algebras o the atural umbers ad their cloes of term operatios ca be classified accordig to their descriptive complexity. We give a example of a closed algebra which has oly uary operatios ad whose cloe of term operatios is ot Borel. Moreover, we provide a example of a coatom i the cloe lattice whose obvious defiitio via a ideal of subsets of atural umbers would suggest that it is complete coaalytic, but which turs out to be a rather simple Borel set. 1. Two problems about cloes o N 1.1. Descriptive set theory of algebras ad cloes o N. Let X be a set, ad deote for all 1 the set X X of all fuctios o X i variables by O (). The O := 1 O() is the set of all fiitary fuctios o X. A cloe is a subset C of O which cotais all projectios (i.e., all fuctios satisfyig a equatio of the form f(x 1,..., x ) = x k ) ad which is closed uder compositio, i.e., for all, m 1, all -ary f C, ad all m-ary g 1,..., g C, the m-ary fuctio f(g 1 (x 1,..., x m ),..., g (x 1,..., x m )) is also a elemet of C. I other words, C is required to be closed uder buildig of terms from its fuctios. The latter perspective shows that cloes arise aturally as sets of term fuctios of algebras with domai X; i fact, the cloes o X are precisely the sets of term fuctios of such algebras. Sice may properties of a algebra (e.g., subalgebras, cogrueces) deped oly o the cloe of the algebra, cloes are i that sese caoical represetatives of algebras, ad have bee studied itesively i the literature; for a moograph o cloes, see [Sze86]. While cloes arise i this way o base sets X of arbitrary (fiite or ifiite) cardiality, there is a additioal perspective o cloes from the viewpoit of descriptive set theory that ca oly be ejoyed o a coutably ifiite base set, as we will outlie i the followig. For otatioal ad coceptual coveiece, let us idetify X with the set of atural umbers N. The, for every fixed 1, we ca view O () = N N as a topological space whose topology is aturally give by equippig N with the discrete topology ad viewig N N as a product space. This space is homeomorphic to the Baire space (the metric space o N N i which two fuctios are closer the later they start to differ), ad a fuctio f O () is i the closure of a set F O () iff for every fiite subset of N there exists g F which agrees with f o this set. The set O the becomes the sum space of the spaces O (), i.e., the ope subsets of O are those sets F for which the -ary fragmet F () := F O () is ope i O () for every Date: February 3, Mathematics Subject Classificatio. Primary 08A40; secodary 54H15; 22A30; 03E15. Key words ad phrases. cloe; Borel set; complete aalytic; closed set; ideal; upper desity. Research of the first author supported by FWF project P N13. Research of the secod author supported by a APART-fellowship of the Austria Academy of Scieces. Research of the third author partially supported by NSF grat o: DMS , ad GIF (Germa-Israeli Foudatio for Scietific Research & Developmet) Grat o /2007. Publicatio 994 o Shelah s list. 1

2 2 M. GOLDSTERN, M. PINSKER, AND S. SHELAH 1. This space is itself homeomorphic to the Baire space, ad is i particular a Polish space, i.e., a separable topological space whose topology is geerated by a complete metric (cofer the textbook [Kec95]). The latter fact allows for the use of the otios of descriptive set theory o O. A subset of a Polish space is called aalytic iff it is the cotiuous image of a closed subset of the Baire space. For example, all Borel sets are aalytic. The coaalytic subsets of a Polish space are defied to be the complemets of aalytic sets. A coaalytic set Y i a Polish space S is called complete coaalytic iff for every Polish space S ad every coaalytic set Y therei, Y is the preimage of Y uder some cotiuous fuctio from S to S. I every ucoutable Polish space, i particular i O, there are aalytic sets which are ot coaalytic. A complete coaalytic set ca therefore ot be aalytic. A cetral theme of descriptive set theory is the ivestigatio of the complexity of subsets of Polish spaces. I this descriptive complexity hierarchy, the simplest sets are the closed ad the ope sets. Borel sets are still cosidered relatively simple, ad i fact, most sets of real umbers that appear i aalysis are Borel sets. Aalytic sets are more complicated tha Borel sets (similar to the differece betwee recursively eumerable ad recursive sets), ad coaalytic sets are cosidered to be slightly more complicated. As subsets of the Polish space O, sets of fiitary fuctios o N ca thus be classified accordig to this descriptive complexity. I particular, this applies to the fuctios of a algebra ad to cloes; we the call algebras ad cloes ope, closed, Borel, aalytic, etc. The cloes that arise most aturally are at the lowest level of the descriptive hierarchy: the closed cloes are precisely the polymorphism cloes of relatioal structures with domai N, i.e., the sets of fiitary fuctios preservig all relatios of some relatioal structure. Similarly to automorphism groups, polymorphism cloes cotai iformatio about their correspodig relatioal structure, ad are ivestigated i order to derive properties of this structure; we refer to [BCP10, BP11a, BP11b, BP] for applicatios of closed cloes i model theory ad theoretical computer sciece. Much further up i complexity, the first author of the preset paper proved that a certai importat cloe cotaiig O (1), called Pol(T 2 ), is complete coaalytic i order to show that this cloe could ot be the term cloe of ay algebra which has oly coutably may ouary fuctios [Gol]: for ay algebra which has all fuctios i O (1) ad coutably may o-uary fuctios is Borel, ad the cloe of term operatios of a Borel algebra is always aalytic. So far, o algebraic proof of the above result is kow, ad thus descriptive set theory is ot oly a way of classifyig cloes o N, but also a tool for provig theorems about such cloes. Fact 1. If a algebra is Borel or aalytic, the its cloe of term operatios is aalytic. The metioed exploitatio of this boud o the icrease i descriptive complexity whe passig from a algebra to its cloe of term operatios ispired the authors of the survey paper [GP08] to ask whether there exists a Borel algebra whose term cloe is ot Borel; the problem was stated as Problem N i the survey. Here, we will show that there exists a algebra which has cotais oly uary fuctios, is closed, ad has a term cloe which is ot Borel, providig a affirmative aswer to this problem.

3 A CLOSED ALGEBRA WITH A NON-BOREL CLONE A false coaalytic ideal cloe o N. A large class of cloes o N are ideal cloes (the class provides, i particular, 2 2ℵ0 coatoms i the lattice of cloes without use of the Axiom of Choice [BGHP09]). Let I be a ideal of subsets of N, that is, a dowward closed set of subsets which is closed uder fiite jois. The the set C I of all fiitary fuctios o N which sed powers of sets i I to sets i I is a cloe; from their defiitio which uiversally quatifies over all subsets of the atural umbers, cloes of this form ca be expected to be rather up i the descriptive hierarchy. Oe atural ideal o N is the followig. For each set A N, the upper desity d(a) is defied as d(a) := lim sup A [0, ). The family of sets of upper desity 0 forms a ideal I d. The correspodig cloe C Id was studied by the authors of [BGHP09], who at the time could ot determie whether or ot the cloe was precomplete, i.e., a coatom i the lattice of all cloes o N. Moreover, the secod author of that paper cojectured that C Id is, just like the cloe Pol(T 2 ) metioed above, complete coaalytic, i accordace with its obvious defiitio quatifyig over subsets of N; this would have explaied the difficulties whe tryig to decide whether or ot the cloe is a coatom. We disprove this cojecture by showig that C Id is i fact a Borel set of low complexity. Moreover, we show that C Id is ideed precomplete, solvig Problem 40 of [BGHP09] (also kow as Problem E i the survey paper [GP08]) Summary ad orgaizatio of the paper. We thus provide i this paper a example of a rather complex (o-borel) cloe which comes from a algebra that is rather simple (closed, ad moreover uary) (Sectio 2), ad a example of a rather simple (low Borel) cloe which seems to be complex i the sese that its obvious defiitio by meas of a otrivial ideal does ot suggest it is Borel, ad i the sese that determiig its precompleteess is a relatively hard task (Sectio 3). Our results solve Problems N ad E from [GP08]; the latter problem has bee stated as Problem 40 i [BGHP09]. 2. A closed algebra with a o-borel cloe Theorem 2. There exists a closed algebra o N whose cloe of term operatios is ot Borel. Moreover, this algebra ca be chose to cotai oly uary fuctios. Proof. Write N as a disjoit uio {0} T x T y A x A y, where T x, T y, A x ad A y are ifiite. Cosider the space Tx Ax of all fuctios from A x to T x, equipped with the metric that makes it the Baire space, ad cosider likewise Ty Ay. The there exists a closed subset B of the product space Tx Ax Ty Ay whose projectio oto the first coordiate is aalytic but ot Borel; see for example the textbook [Kec95]. Let F cotai the idetity fuctio id o N plus the set of all fuctios f i N N such that: f is the idetity o {0} T x T y, ad the pair (f Ax, f Ay ) is a elemet of B. The F is a trasformatio mooid sice f(f (x)) = f (x) for all f, f F such that f is ot the idetity fuctio. Moreover, it is clearly a closed subset of O (1) sice the set B is closed.

4 4 M. GOLDSTERN, M. PINSKER, AND S. SHELAH Now let h : N N defied by h() := {, {0} T x T y A x 0, A y. Set G to cotai all uary fuctios that ca be composed from elemets of the set {h} F. The G is the disjoit uio of {h}, F, ad the set G of all fuctios g i O (1) such that g is the idetity o {0} T x T y, ad g A x is i the projectio of B oto the first coordiate, ad g() = 0 for all A y. To see this, observe that h f = f for all f F, ad f h is the elemet g of G which agrees with f o A x. Sice the projectio of B oto the first coordiate is ot Borel, G is ot Borel. Hece G, as the disjoit uio of G with the closed set {h} F, is ot Borel either. Therefore, takig {h} F as the fuctios of the algebra proves the theorem. 3. The cloe preservig zero upper desity We ow ivestigate the cloe C Id of all fuctios which preserve the ideal of sets of upper desity 0. We first show that it is Borel, ad the that it is precomplete The complexity of C Id. I this part we prove of the followig theorem. Theorem 3. C Id is Borel. Defiitio 4. Let k 1 ad let f O (k). Each permutatio π of {1,..., k} iduces a fuctio f π O (k) by settig f π (x 1,..., x k ) := f(x π(1),..., x π(k) ). Moreover, for each 0 l < k, each tuple ā = (a 1,..., a l ) N l iduces a (k l)-ary fuctio fā by settig fā(y 1,..., y k l ) := f(a 1,..., a l, y 1,..., y k l ). We call each fuctio f π,ā a shadow of f. Whe l > 0, the we call f π,ā a proper shadow of f. The fuctios f π, which are just the fuctios f π,ā for a tuple ā of legth l = 0, ad i particular f itself, are called improper shadows of f. Observe that proper shadows of f have strictly smaller arity tha f. If f is uary, the it has o proper shadows. Defiitio 5. Let k 1. We call f O (k) miimal if the followig hold: f / C Id ; every proper shadow of f is i C Id. We will show that C Id is a Borel set as follows: a fuctio f is ot cotaied i C Id if ad oly if it has a miimal (proper or improper) shadow. Havig a miimal shadow will tur out to be equivalet to havig what is called a bad fuctio as a shadow. The set of bad fuctios has a ice defiitio which makes it a Borel set, ad havig a bad shadow is i tur easily show to be a Borel property, provig the theorem. Defiitio 6. Let k 1 ad f O (k). We say that f is bad iff the followig holds: There exists a ratioal umber ε > 0 such that for all i N there are, t i ad A [i, ) with the followig properties: A is sparse with respect to i: A [0, r) 1 2 i r for all r N; f[a k ] is dese i [0, t) with respect to ε: f[a k ] [0, t) ε t.

5 A CLOSED ALGEBRA WITH A NON-BOREL CLONE 5 Lemma 7. Let k 1, ad let f O (k) be bad. The f / C Id. Proof. Let ε as i the defiitio of badess, ad set 0 := t 0 := 0. By iductively applyig the defiitio of badess to i > max( j 1, t j 1 ), we ca fid tuples ( j, t j, A j ) for all j 1 such that 0 < 1 < 2 < ; 0 < t 1 < t 2 < ; A j [ j 1, j ) ad A j [0, r) r 2 j 1 for all r N; f[a k j ] [0, t j) ε t j. Now let A := j 1 A j. To see that A has upper desity 0, let ay ratioal umber δ > 0 be give; we will fid s N such that 1 m A [0, m) < δ for all m > s. To this ed, pick a atural umber v > 0 such that 1 2 < δ v 2. Now pick s N such that v 1 s < δ 2. The, for m > s, we have 1 m A [0, m) 1 m ( v 1 + A [ v 1, m) ) < δ A j [ j 1, m) m δ m v j v j m 2 δ j m m 2 j 1 δ v < δ. v j O the other had, f[a k ] has upper desity of at least ε, sice f[a k ] [0, t j ) ε t j for all j > 0. Hece, f / C Id. We will use the followig lemma i order to show that miimal fuctios are bad. Lemma 8. Let k 1, ad let f O (k) be miimal. Let B N be so that the fact f / C Id is witessed by B, i.e., B has upper desity 0, but f[b k ] has positive upper desity. The for each i 0 the set B\[0, i] also witesses that f / C Id ; i fact, the sets f[b k ] ad f[(b\[0, i]) k ] have equal upper desity. Proof. If k = 1, the f[b] f[b \[0, i]] f[b]\f[[0, i]], so the statemet follows immediately sice f[[0, i]] is fiite. Now assume k 2. Let S be the set of all proper shadows f π,ā of f for which the tuple ā cotais oly elemets of [0, i]; sice k 2, this set is o-empty. By the miimality of f, each f S is a elemet of C Id, ad so the set f [B k l f ] (for appropriate 1 l f < k) has upper desity 0. Sice S is fiite, the set D := f[b k ] \ f [B k l f ] f S still has the same positive upper desity as f[b k ]. It remais to check that f[(b\[0, i]) k ] D. Let d D. The d caot be writte as d = f(b 1,..., b k ) with all b i B ad at least oe b i i [0, i], as ay such d would be i f [B k ] for some proper shadow f S. O the other had, d ca be writte as d = f(b 1,..., b k ) with all b i B. Hece d f[(b \ [0, i]) k ]. Lemma 9. Let f O be miimal. The f is bad. Proof. Write k for the arity of f. Let B N be so that d(b) = 0 ad d(f[b k ]) > 0, ad let ε be a positive ratioal umber such that d(f[b k ]) > ε. Give i N we have to fid, t, A as i the defiitio of badess. Sice d(b) = 0, we ca pick m i so large that B [0, j) 1 j for all j m. Set 2 i D := B [m, ). The by Lemma 8, d(f[d k ]) = d(f[b k ]) > ε. Hece, we ca fid t i

6 6 M. GOLDSTERN, M. PINSKER, AND S. SHELAH such that f[d k ] [0, t) has size at least ε t. Now choose m such that f[d k ] [0, t) = f[(d [0, )) k ] [0, t). Fially, set A := D [0, ) = B [m, ). Lemma 10. Let f O. The followig are equivalet: (a) f / C Id. (b) There exists a shadow of f which is miimal. (c) There exists a shadow of f which is bad. (d) There exists a shadow of f which is ot i C Id. Proof of (a) (b). Let S be the set of shadows of f which are ot i C Id. Let g S have miimal arity. The g is miimal. Proof of (b) (c). Every miimal fuctio is bad, by Lemma 9. Proof of (c) (d). A bad fuctio caot be i C Id, by Lemma 7. Proof of (d) (a). Let f π,ā be a shadow of f which is ot i C Id, ad let B N be a set of upper desity 0 which is set to a set of positive upper desity uder f π,ā. Let B be the set obtaied by addig all etries of the tuple ā to B. The B still has upper desity 0, ad f seds B to a set of positive upper desity. Proof of Theorem 3. Clearly, the set B of bad fuctios is Borel sice its defiitio oly quatifies over atural ad ratioal umbers. We show that C (k) I d is Borel for all k 1. For each π i the set S({1,..., k}) of all permutatios of {1,..., k} ad each tuple ā = (a 1,..., a l ) N l, where 0 l < k, the mappig κ π,ā from O (k) to O which seds every f O (k) to f π,ā is cotiuous. By Lemma 10, O (k) \ C Id = π,ā[b] π S({1,...,k}) 0 l<k ā N l κ 1 Sice B is Borel, each of its cotiuous preimages κ 1 π,ā[b] is Borel, ad so is the coutable uio of these sets Precompleteess of C Id. We will ow show the followig. Theorem 11. C Id is precomplete, i.e., a coatom of the lattice of all cloes o N. The strategy is the followig: we will first show that every set B of positive upper desity ca be mapped by a uary fuctio from C Id oto a large set, that is, a set cotaiig ifiitely may itervals of the form [, 2]. We the show that for every large set C there is a set D of upper desity 0 such that C D ca be mapped oto all of N by a biary fuctio from C Id. These two facts together imply that every fuctio g / C Id, together with C Id, geerates (by buildig terms) a fuctio g mappig set of upper desity 0 oto N; ad it is well-kow that the oly cloe cotaiig C Id {g } is O. Hece, the oly cloe cotaiig C Id {g} is O as well, ad thus C Id is a coatom sice g was a arbitrary fuctio outside C Id. Lemma 12. T N has upper desity 0 if the limit of T [2k,2 k+1 ) 2 k, where k goes to ifiity, exists ad equals 0. Proof. Easy.

7 A CLOSED ALGEBRA WITH A NON-BOREL CLONE 7 Lemma 13. Let ε > 0 be a ratioal umber, ad f : N N satisfy f() ε for all 0. The for all A N we have d(f[a]) 1 ε d(a). I particular, f C Id. Proof. For each 1 we have 1 f[a] [0, ) 1 [ f A [0, ε )] 1 A [0, ε ) 1 = ε A [0, ε ) The ext lemma has bee show, for example, i [BGHP09] ad [CH01], but we iclude the proof for the reader s coveiece. I the followig, for F O we write F for the cloe of all term operatios over F, i.e., the smallest cloe cotaiig F. Lemma 14. Let g / C Id. The C Id {g} cotais a uary fuctio which is ot i C Id. Proof. Let g be k-ary, where k 1. Sice g / C Id, there is a ifiite set A N of upper desity 0 such that g[a k ] has positive upper desity. Let f 1,..., f k be fuctios from A ito A such that the fuctio (f 1 (),..., f k ()) is a bijectio from A oto A k. If we set f i () = 0 for all / A, the f i C Id for all 1 i k sice the rage of f i is cotaied i a set of upper desity 0. The uary fuctio h() := g(f 1 (),..., f k ()) ow maps A oto g[a k ]. Slightly modifyig the usual Ladau symbol, we will i the followig write O(x) for ay quatity i the iterval [0, x], for ay ratioal umber x 0. Lemma 15. Let B N have positive upper desity. The there is a uary fuctio f C Id ad a strictly icreasig sequece ( i ) i 1 of atural umbers such that f[b] i 1 [ i, 2 i ). Proof. Fix a positive atural umber e such that d(b) > 3 e. We first claim that there are ifiitely may N with B [, e). So let m N be give; we will fid m e with this property. Sice B has positive upper desity, we ca icrease m such that B [0, m) 3 em, ad also such that m > 2e 2. Now let := m e ; the m = e + O(e), ad e <. So we have B [, e) B [0, m) [0, ) [e, m) 3 e m O(e) 3 e 2 =. e Now we choose a ifiite strictly icreasig sequece ( i ) i 1 of atural umbers such that the itervals I i := [ i, e i ] are disjoit, ad all i have the above property. We ca the fid a fuctio f O (1) with the followig properties: f(x) x e for all x N (hece f C I d ), ad f[b I i ] [ i, 2 i ) for all i 1. The ext lemma is the crucial step of the proof of Theorem 11. Lemma 16. Let ( i ) i 1 be a strictly icreasig sequece of atural umbers, ad set C := i 1 [ i, 2 i ). The there exists a set D N of upper desity 0 ad a biary fuctio h C Id such that h[c D] = N. Proof. For otatioal simplicity we will aim for a fuctio h such that h[c D] N \ {1}. It is the easy to modify h to obtai h[c D] = N. Set 0 := 2; the for each atural umber k 1, the atural umber i(k) := max{i 0 : i 2 k } is well-defied. Clearly the sequece (i(k)) k 1 is weakly icreasig ad diverges to ifiity. ε

8 8 M. GOLDSTERN, M. PINSKER, AND S. SHELAH For all k 1, let d k := 2k. The d i(k) k = 2k + O(1), ad 1 d i(k) k 2 k 1. For all k 1, let D k be the iterval (2 k d k, 2 k ]. Note that these itervals are disjoit, ad that D k = d k. Let D := k 1 D k. To check that D has desity 0 we use Lemma 12: clearly 1 D [2 k, 2 k+1 ) = 1 d 2 k 2 k k+1 = 2 + O( 1 ) 0 for k. i(k+1) 2 k For all k 1, let R k := [ i(k), 2 i(k) ) D k. The cardiality of R k is i(k) d k = 2 k +O( i(k) ), so there exists a bijectio g k : S k [2 k, 2 k+1 ) betwee a subset S k of R k ad [2 k, 2 k+1 ). Eve though the sets [ i(k), 2 i(k) ) are ot ecessarily disjoit as the i(k) might ot be strictly icreasig, the sets D k ad therefore also the sets R k are disjoit. Hece we may defie a fuctio h O (2) by settig h(x, y) := g k (x, y) wheever (x, y) S k for some k 1, ad h(x, y) := 0 otherwise. The h maps C D oto {0} k 1 [2k, 2 k+1 ) = N \ {1}. It remais to check that h C Id. So let T N be of upper desity 0, ad let ε > 0 be a ratioal umber. Note that the set h[t T ] is the uio of the sets h[(t T ) R k ] [2 k, 2 k+1 ) {0}. For large eough k we have o the first coordiate: T [ i(k), 2 i(k) ) i(k) ε; o the secod coordiate: T D k d k = 2k + O(1). i(k) Hece the cardiality of (T T ) R k is bouded by ( ) 2 k i(k) ε + O(1) i(k) = 2 k ε (1 + O( i(k) 2 k )) 2k ε 2. So h[t T ] [2 k, 2 k+1 ) 2ε 2 k. As ε was arbitrary, we ca ow apply Lemma 12 to ifer that h[t T ] has upper desity 0. The ext lemma is agai kow from [BGHP09], but we give the short proof for the coveiece of the reader. Lemma 17. Let I be ay ideal o N, ad let C I the cloe of fuctios preservig I. If t O (k) is a fuctio for which t[z k ] = N for some Z I, the C I {t} = O. Proof. Let r = (r 1,..., r k ) : N Z k be a right iverse of t, i.e., t r is the idetity map o N. Ay fuctio h : N m N ca be writte as h = t (r h), where each of the fuctios r i h : N m N is i C I, as its rage is a subset of Z. Hece h C I {t}. Proof of Theorem 11. Let g / C Id. By Lemma 14 we may assume that g is uary. So there is a set A N of upper desity 0 such that B := g[a] has positive upper desity. Usig Lemma 15 we fid a set C of the form i 1 [ i, 2 i ), for a strictly icreasig sequece ( i ) i 1 of atural umbers, ad a fuctio f C Id such that f[b] C. From Lemma 16 we get a set D N of upper desity 0 ad a fuctio h C Id mappig C D oto N: A g B f C C D The assigmet (a, d) (f(g(a)), d) maps A D oto C D, so the biary fuctio t O (2) defied by t(x, y) = h(f(g(x)), y) h N

9 A CLOSED ALGEBRA WITH A NON-BOREL CLONE 9 maps A D oto N. Clearly t C Id {g}. Quotig Lemma 17 for Z := A D ow fiishes the proof. Refereces [BCP10] Mauel Bodirsky, Hubie Che, ad Michael Pisker. The reducts of equality up to primitive positive iterdefiability. Joural of Symbolic Logic, 75(4): , [BGHP09] Mathias Beiglböck, Marti Goldster, Lutz Heidorf, ad Michael Pisker. Cloes from ideals. Iteratioal Joural of Algebra ad Computatio, 19(3): , [BP] Mauel Bodirsky ad Michael Pisker. Topological Birkhoff. Trasactios of the AMS. To appear. Preprit arxiv.org/abs/ [BP11a] Mauel Bodirsky ad Michael Pisker. Reducts of Ramsey structures. AMS Cotemporary Mathematics, vol. 558 (Model Theoretic Methods i Fiite Combiatorics), pages , [BP11b] Mauel Bodirsky ad Michael Pisker. Schaefer s theorem for graphs. I Proceedigs of the Symposium o Theory of Computig (STOC), pages , Preprit of the log versio available at arxiv.org/abs/ [CH01] Gábor Czédli ad Lutz Heidorf. A class of cloes o coutable sets arisig from ideals. Studia Scietiarum Mathematicarum Hugarica, 37: , [Gol] Marti Goldster. Aalytic cloes. Preprit available at arxiv.org/abs/math/ [GP08] Marti Goldster ad Michael Pisker. A survey of cloes o ifiite sets. Algebra Uiversalis, 59: , [Kec95] Alexader Kechris. Classical descriptive set theory, volume 156 of Graduate Texts i Mathematics. Spriger, [Sze86] Áges Szedrei. Cloes i uiversal algebra. Sémiaire de Mathématiques Supérieures. Les Presses de l Uiversité de Motréal, Algebra, TU Wie, Wieder Hauptstraße 8-10/104, A-1040 Wie, Austria address: goldster@tuwie.ac.at URL: Équipe de Logique Mathématique, Uiversité Diderot Paris 7, UFR de Mathématiques, Paris Cedex 13, Frace address: marula@gmx.at URL: Istitute of Mathematics, The Hebrew Uiversity of Jerusalem, Jerusalem, Israel, ad Departmet of Mathematics, Rutgers Uiversity, New Bruswick, New Jersey address: shelah@math.huji.ac.il URL:

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