FINITE SUBSTRUCTURE LATTICES OF MODELS OF PEANO ARITHMETIC

proceedings of the american mathematical society Volume 117, Number 3, March 1993 FINITE SUBSTRUCTURE LATTICES OF MODELS OF PEANO ARITHMETIC JAMES H. SCHMERL (Communicated by Andreas R. Blass) Abstract. Some new finite lattices (for example, A/4, M7, and the hexagon lattice) are shown to be isomorphic to the lattice of elementary substructures of a model of Peano Arithmetic. The set of elementary substructures of a model JV of Peano Arithmetic forms a lattice Lt(yf), the substructure lattice of jv. It is unknown whether there are finite lattices that are not isomorphic to any substructure lattice. Indeed, it was conjectured in [8] that for any finite lattice L there is some jv 1= PA for which L = \X(JV). It was proved in [8] that if 2 < n < co, then there is JV such that Lt(^T) = n(«). (The partition lattices U(n) are defined below.) For 3 < n < co, the lattice Mn is the unique lattice that has n + 2 elements and n atoms. Thus, the lattice A/3, which is isomorphic to n(3), is isomorphic to a substructure lattice, although it is known from [2] or [4] that if yv is a model of True Arithmetic and 3 < n < co, then Lt(yT) ^ Mn. Other examples of finite lattices that are not substructure lattices of models of True Arithmetic are given in [8]. The lattices Mn for 4 < n < co and the hexagon lattice were identified in [8] as specific finite lattices that were not known to be isomorphic to substructure lattices. It is a consequence of the theorem in this note that M, whenever n = p" + 1 for some prime p, the hexagon lattice, and the lattice Mi are isomorphic to substructure lattices. It is still unknown whether or not Mx! is isomorphic to a substructure lattice. All previously known positive results about finite substructure lattices are contained in [8]. Prior to [8], Paris [4] proved that all finite distributive lattices occur as substructure lattices. (Also see [7].) Wilkie [9] showed that the pentagon lattice is a substructure lattice, and Paris [5] proved that M3 is a sublattice of an infinite substructure lattice. More generally, intermediate structure lattices will be considered. For # -< yv ^ PA, let Lt(yT/^) = {jf0 e la(yv): J? < JTq < jv}, regarded as a sublattice of Lt(yF). Notice that if Jf is a minimal model of PA and ^f -< jt, then Lt(^) = \l(j JJ?). Given a set A, let n(^) be the set of all partitions of A. If a, b A and n fl(a), then we write a «b (mod n) if {a, b} C C for some C n. Received by the editors February 11, 1991 and, in revised form, July 2, 1991. 1991 Mathematics Subject Classification. Primary 03H15, 03C62; Secondary 06B15. 833 1993 American Mathematical Society 0002-9939/93 $1.00 +$.25 per page

834 J. H. SCHMERL There are two extreme partitions 0^ and 1A in 11(A), where a & b (mod 1^) iff a = b, a «b (mod 0A) for every a, b A. We partially order n(^) so that if nx,n2 Yl then nx < n2 iff n2 is a refinement of 7Ti. In particular, 0^ < n < \A for each n U(A). With this partial order, n(^) becomes a complete lattice (which is the dual of the lattice usually considered). Thus, if nx, n2 U(A) then nx V n2 = {C0 : C0 = Ci n C2 ^ 0 for some Cx nx and C2 n2}. Let (A, y) be an algebra (in the sense of universal algebra); that is, each / e y is a function /: Ak -* A for some k < co. We let l\(a, Sr) be the sublattice of U(A) consisting of all partitions of A that correspond to congruences of (A, y). Thus, n e l\(a, Sr) iff whenever / e y is /V-ary and oq, ax,..., ak_x, bo, bx,..., bk_x A are such that a, «ft, (mod a) for each i < k, then /(a0, a.,...,a^_i)«/(fe0, &i,..., frt-i) (mod rc). The theorem can now be stated. Theorem. Let Jf ^ PA be countable and nonstandard, and let (A, y) be a finite algebra. Then Jf has a cofinal elementary extension JV such that U{Jr-/jr) = n(a, P). We need some definitions taken from [8]. Let L = (L, A, V) be a finite lattice. We denote the minimum and maximum elements of L by 0 and 1. A representation a of L in H(A) is an injection a: L -* U(A) such that a(0) = 0^, a(\) = \A and a(x\/y) = a(x) V a(y) for each x, y L. Notice that we do not require that a representation satisfy the identity a(x Ay) = a(x) Aa(y). Let a: L 11(A) be a representation. Then a is finite if A is finite, and a is nontrivial if \a(x)\ / 2 whenever x L. The representation a: L n(^) is a congruence representation if there is an algebra (A, Sr) such that a: L -> n(,4, J?~) is a lattice isomorphism. For n > 1 let a": L > 11(^4") be the representation such that for each c, d A" and x L, c «d (mod q"(x)) «for every i<n, ct & dj (mod a(x)). If we let S^ be the set of unary polynomial functions of (A, Sr), then Yl(A, S^x) = n(^4, S^). Therefore, when considering some l\(a, S^), we can assume that y consists only of unary functions. It is known [3] that every finite lattice has a congruence representation. However, it is an open question of some apparent difficulty whether every finite lattice has a finite congruence representation. The lattice Mxl is a specific lattice that has no known finite congruence representation. We present some examples of lattices that have finite congruence representations. Let L be a finite distributive lattice. Let A = {x L: x = 1 or x is a meetirreducible}, considered as a subposet of L. Let y be the set of all functions /: A» A such that f(a) > a for each a A. Then L and U(A, y) are isomorphic lattices. In fact, let a: L 11(A) be such that for each x L, a(x) is the partition whose only equivalence class that (possibly) is not a singleton is {a A : a> x}. Then a: L > Y1(A, y) is a lattice isomorphism.

MODELS OF PEANO ARITHMETIC 835 Let F be a finite field of order q, and let n = q + 1. Let A = F x F, and let / y iff there are a, b, c, d F such that f(x, y) = (ax + b, cy + d). Then U(A, y) 3 Mn. An observation due to Feit [ 1 ] implies that Mj has a finite congruence representation. Finally, let A = {0, 1, 2, 3, 4, 5, 6}. Then there is y such that n(,4, y) is isomorphic to the hexagon lattice. Specifically, n(^, y) = {(L,, 1^, {{0}, {1,2}, {3,4}, {5,6}}, {{0, 1,2}, {3,4}, {5,6}}, {{0}, {1,3}, {4,6}, {2,5}}, {{0,4,6}, {1,3}, {2,5}}}. We now proceed with the proof of the theorem. Lemma 1. Suppose L is a finite lattice, a: L > 11(^4) is a representation, and 2<n<co. Then (1) a" is a nontrivial representation of L; (2) // a is a finite representation then so is a" ; (3) if a is a congruence representation then so is a". Proof. (1) Clearly, if x L then \a"(x)\ = \a(x)\"?2. (2) If A is finite then so is A". (3) Let (A, y) be a unary algebra such that a: L -> U(A, y) is a lattice isomorphism. Without loss of generality assume that y has the identity function and all constant functions as members. Let % be the set of all functions g: A" A" for which there are fo, fi,..., fn-\ S? such that g((a0,ax,..., a _.)) = (/o(a0), f (a.),..., / _i(a _i)). We introduce the following notation to be used for the rest of this proof: for any x we denote by x the n-tuple (x, x,..., x). Now let?x be the set of all functions g: A" A" where for some i < n, g((an, ai,..., a _i)) = ~a[. Let ^ = b U &\. We claim that a": L -> n(^", ^) is a lattice isomorphism. First we show that a"(x) e n(^4",,f) for x L. Suppose g %, so that g((a0,ai,...,a _i)) = (/o(tfo), f\(a\),, fn-\(an-\)) Then c w a" (mod a"(x)) => for all / < n, c,- w a", (mod a(x)) => for all i < n, f(cj) w //(a1,) (mod a(x)) => </o(co),,/«-l(c -l)) «(/o(ao), > /n-i(rfn-i)) (moda"(x)) =*> g(c) «?(#") (mod a"(x)). Next suppose g e ^i, so that g((ao, ax,..., an-\)) = a7. Then c «cf (mod a"(x)) =$ Cj «a",- (mod a(x)) => cj «a",- (mod a"(x)) =*S(c)**(<0 (moda"(x)). Therefore, for all g 6 ^\ if c» a" (mod a"(x)) then g(c) w g(a") (mod a"(x)). Hence, a"(x) 6 U(A", "). Secondly, we show that if n U(A", 8?) then n = a"(x) for some x L. Let n U(A", &), and let /> e n(^) be such that a «& (mod p) etazs b (mod 7t).

836 J. H. SCHMERL To show that p f\(a, Sr), consider arbitrary a» b (mod p) => a «& (mod 7r) ^W)~W) =*/(a)«/(6) / e y. Then (modn) (mod//). Thus, // l\(a, y) so that p = a(x) for some x L. We claim that q"(x) = n. To see that a"(x) < /r, notice that c & d (mod 7t) => for each j < n, cf «a",- (mod 7r) => for each i < n, c, «a", (mod a(x)) => c k d (moda"(x)). To show that 7r < a"(x), consider c, d A" for which c «d (mod a"(x)). Let g0, Si,, gn-\ % be such that for i < n, gi((a0, a,,..., a _.)) = (60, 61,..., 6 -i}, where {dj if /' < /, a, if j = i, Cj if 7 > i for_all (a0, a.,..., a _.) /I". Then c = 0(co), gi(ci) * c?/(^/) (mod rc), i?/(^i) = c?!+i(c7+t), and g _i(a' _1) = a". Therefore, c «a" (mod rc). n Definition. Let a: L > FI(/4) and /?: L n(fl) be representations of L. Then a arrows ft (in symbols: a >/?) if whenever n 11(A) there is an injection 6: B -+ A such that (1) whenever x L and a, b B, then a «b (mod /?(*)) «- 0(a) «0(6) and (2) there is x e L such that whenever a, b B, then (mod a(x)) aztb (mod 0(x))& 6(a)* 6(b) (modn). We will refer to such a function 6 as a n-demonstrator for a /?. The next lemma is a consequence of the canonical partition theorem of Promel and Voigt [6]. This lemma was explicitly stated in Example 3.3 of [8] and was used there to prove the instance (see [8, Corollary 5.5]) of the theorem when L 9 11(A). Lemma 2. Let A be a finite set and let y: 11(A) > 11(A) be the identity representation. Then for each m > 1 there is n such that y" -» ym. The next lemma is the main ingredient in the proof of the theorem. Lemma 3. Suppose a: L -* Yl(A) is a finite congruence representation. there is n>\ such that a" a. Then Proof. Let a: L > H(A) be a finite congruence representation. Let (A,&~) be a unary algebra such that a: L -+ U(A, y) is a lattice isomorphism. Without loss of generality, we assume that the identity function is in y (so y ^ 0 ) and that y is closed under composition. Let m \Sr\ > 1. Letting y: 11(A) >

MODELS OF PEANO ARITHMETIC 837 n(^4) be the identity function, we can get n by Lemma 2 so that y" > ym. We claim that a" > a. In order to prove that a" -> a, consider some n Y1(A"). To obtain a ft-demonstrator for a" > a, let y = {^o, f\,, fm-1} > and let F: A > Am be the function for which F(a) = (fo(a), f(a),..., fm-\(a)) for each a A. Let 0j be a zr-demonstrator for 7" > ym, and then let 6 = 6X o F. We will show that 0 is a n-demonstrator for a" > a. Conditions (1) and (2) in the definition need to be proved. Proof of (1). Suppose c, d A" and x L. Notice that ak = yk o a for 1 < k < co. Then each of the following equivalences is clear. 6(c) x 6(d) (mod an(x))&6x(f(c))& 6x(F(d)) (mod y"(a(x))) &F(c)*F(d) (mod ym(a(x))) This proves (1) in the definition. &F(c)*F(d) (modam(x))» for each i < m, f(c) «f(d) «c «d (moda(x)). (mod a(x)) Proof of (2). Suppose that n U(A"), and let p U(A) be the partition such that a «b (mod p) & 6(a) «6(b) (mod n) whenever a, b A. In order to prove (2), we need to show that p X1(A, S* ). To do this, consider some i < m and arbitrary a, b A for which a «b (mod p). Then 6(a) w 6(b) (mod n). Hence, there is px Y1(A, SF) such that F(a) «F(b) (mod ym(px)), so that for each k < m, /fc(a) «/*( ) (mod px). Since y is closed under composition, for each j < m, f(f(a)) «fj(f(b)) (mod //,). It follows that F(y)(a)) «F(f(b)) (mod yw(//i)), so that 6x(F(f(a))) «6x(F(f(b))) (mod y"(px)); consequently, 0(^(a)) «&(/ (&)) (mod n), so that yj-(a) «^(/3) (mod //). This proves p U(A, SF). The following corollary generalizes Lemma 2. Corollary 4. Suppose a: L > n(^) is a finite congruence representation. Then for each m > 1 f/zeve w n such that a" am. Proof. This follows immediately from Lemmas 1 and 3 and the observation that (a")k and ank are isomorphic representations. We can now conclude the following corollary, which involves a technical definition from [8]. The reader is referred to [8] for a precise statement of this definition. Corollary 5. // L is a finite lattice that has a finite congruence representation, then for each n < co, L has an n-cpp representation. Proof. The «-CPP representation a of L is obtained as follows. Let a be a finite congruence representation of L. Let an = a2, which by Lemma 1 is a nontrivial finite congruence representation. Recursively obtain finite congruence representations an+x so that an+x an. This is possible by Lemmas 1 and 3. The representation an is an n-cpp representation of L. The theorem follows immediately from Corollary 5, Theorem 4.1 of [8], and Remark 5.1 of [8].

838 j. h. schmerl References 1. W. Feit, An interval in the subgroup lattice of a finite group which is isomorphic to M-,, Algebra Universalis 17 (1983), 220-221. 2. H. Gaifman, Models and types ofpeano's arithmetic, Ann. Math. Logic 9 (1976), 223-306. 3. G. Gratzer and E. T. Schmidt, Characterizations of congruence lattices of abstract algebras. Acta. Sci. Math. (Szeged) 24 (1963), 34-39. 4. J. Paris, On models of arithmetic, Conference in Mathematical Logic, London '70, Lecture Notes in Math., vol. 225, Springer-Verlag, Heidelberg and New York, 1972, pp. 252-280. 5. _, Models of arithmetic and the 1-3-1 lattice, Fund. Math. 95 (1977), 195-199. 6. H. J. Promel and B. Voigt, Canonical partition theorems for parameter sets, J. Combin. Theory (A) 35 (1983), 309-327. 7. J. H. Schmerl, Extending models of arithmetic, Ann. Math. Logic 14 (1978), 89-109. 8. _, Substructure lattices of models of Peano Arithmetic, Logic Colloquium '84, North- Holland, Amsterdam, 1986, pp. 225-243. 9. A. Wilkie, On models of arithmetic having non-modular substructure lattices, Fund. Math. 95(1977), 223-237. Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269 E-mail address: Schmerl@UCONNVM.UCONN.EDU