on spectra, and the negative solution of the decision problem for identities having a finite...

On Spectra, and the Negative Solution of the Decision Problem for Identities having a FiniteNontrivial ModelAuthor(s): Ralph MckenzieSource: The Journal of Symbolic Logic, Vol. 40, No. 2 (Jun., 1975), pp. 186-196Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271899 .

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THE JOURNAL OF SYMBOLIC LOGIC Volume 40, Number 2, June 1975

ON SPECTRA, AND THE NEGATIVE SOLUTION OF THE DECISION PROBLEM FOR IDENTITIES HAVING A

FINITE NONTRIVIAL MODEL'

RALPH MCKENZIE2

An algorithm has been described by S. Burris [3] which decides if a finite set of identities, whose function symbols are of rank at most 1, has a finite, nontrivial model. (By " nontrivial " it is meant that the universe of the model has at least two elements.) As a consequence of some results announced in the abstracts [2] and [8], it is clear that if the restriction on the ranks of function symbols is relaxed somewhat, then this finite model problem is no longer solvable by an algorithm, or at least not by a "recursive algorithm" as the term is used today.

In this paper we prove a sharp form of this negative result; showing, by the way, that Burris' result is in a sense the best possible result in the positive direction. Our main result is that in a first order language whose only function or relation symbol is a 2-place function symbol (9 (the language of groupoids), the set of identities (Vi)(u -r) that have no nontrivial model, is recursively inseparable from the set of identities (VD)(u -r) such that the sentence (VD)(Vx)(u i- A

-1(Qxx x) has a finite model. As a corollary, we have that each of the following problems, restricted to sentences defined in the language of groupoids, is algo- rithmically unsolvable: (1) to decide if an identity has a finite nontrivial model; (2) to decide if an identity has a nontrivial model; (3) to decide if a universal sentence has a finite model; (4) to decide if a universal sentence has a model. We note that the undecidability of (2) was proved earlier by McNulty [13, Theorem 3.6(i)], improving results obtained by Murskil [14] and by Perkins [17]. The other parts of the corollary seem to be new.

For proving these results, we rely on two theorems (an old one and a new one) about "spectra." By the spectrum of a first order sentence b, denoted Sp b, we mean the set of cardinalities of the finite models of b (0 being excluded). The spectrum of a set of sentences is defined in the same way. One knows from Bennett's abstract [2] that when the spectrum of a sentence is enlarged by adding to it 1 and all finite products of its members, the resulting set is the spectrum of a finite set of identities. As Bennett did not publish his proof, we shall prove it here, and show that one identity is sufficient. (T. Evans [5] independently proved a special case of Bennett's theorem, for universal Horn sentences. About the same time, G. Gratzer [7] proved that every set of positive integers including 1 and closed under multiplication is the spectrum of some infinite set of identities.)

Received February 6, 1974. 1 The main result of this paper was announced in [12]. 2 Research supported by National Science Foundation GP-35844X, and by an Alfred P.

Sloan Fellowship.

186 ? 1975, Association for Symbolic Logic



SPECTRA, THE DECISION PROBLEM 187

Our new result on spectra is the following: one can correlate effectively, with, each finite set of identities A, two positive integers m and n and a groupoid identity E, such that Sp E = {1} U {Un: u e Sp Z and u ? m}. [Moreover, in every nontrivial model of E the sentence (Vx)(-,x (Qxx) is true; and E has a nontrivial model iff Z does.]

Now it is well known (e.g., see [10]) that the sets of finitely satisfiable, and of refutable sentences of a first order language (with one or more binary relation symbols) are recursively inseparable sets. Our result announced in the first para- graph is an easy corollary of this, taking into account our version of Bennett's theorem and the theorem just stated.

A striking consequence of our work is that it shows the way, at least in principle, to write down a groupoid identity from whose finite nontrivial models all counter- examples to Fermat's last theorem can be recovered. (Physical limitations ensure that this identity will never be written down.)

?1. Preliminaries. The paper is on a basic level; the arguments are elementary. The reader should know a little first order logic, mainly the part called by Tarski "equational logic" [20]; a little universal algebra, mainly "congruences" and "subdirectly irreducible algebras"; and enough recursion theory to understand the concepts "recursive" and "recursively inseparable." If knowledge of these subjects is lacking, we recommend reference to, respectively, [1]; [6] and [9, Chapter 0]; [4] and [19].

We use Polish notation (sometimes with added parentheses for clarity) to denote the terms built up from function symbols and variables of a first order language. For example, (9vv0ov0,v),9 (&((vo, vo), vo) all denote the same term. The letters x, y, z as well as v0, v1, v2, ... are used to denote variables, with the understanding that distinct symbols denote distinct variables. In other contexts, x, y, z, ... will be used to denote not necessarily distinct elements of some set. An identity is a first order sentence of the form (V9a (u r), where a and X are terms. We write simply a -for the identity (VV)(a a-) where v = <vo, ,. , vn1> includes all variables appearing in a or in a.

An algebraic first order language is a first order language L whose nonlogical symbols are all function symbols. These function symbols are indexed by a set T, say {O,: t E T} is the set of function symbols, and there is a function <r(t): t E T> which defines the ranks of the symbols. Thus r(t) (= rat) is a nonnegative integer. A model for L is a universal algebra 2I = <A, (tt (t E T)>, in which A is a nonempty set, and for all t, Ctt is an r(t)-ary operation over A. (This set is called the universe of 21.)

We denote by w the set of nonnegative integers, and by 1 the set of positive integers. nA denotes the set of all sequences x = <x0,*, x., Xn-1> with xi E A (for all i < n). We concatenate the two sequences x = <x0, , xn-1> and y = <Yo . * y Yin->, and denote the result <xo, * , Xn1, Y05 . * Ym-> by x ^Y. The cardinality of a set A is written IA I.

The following results will be very useful. The first is a corollary of [11, Theorem 1.2].



188 RALPH MCKENZIE

THEOREM 1.1.3 Let E be a finite set of identities defined in a given first order language, and suppose that there is a term a = a(vo, vj, v2) defined in the language such that E implies the identities

(eO) u(vo, vo, V1) v1, u(vo, v1, vo) vo, u(vo, v1, v1) vO.

Then E is equivalent to a single identity (computable from E and a) defined in the same language.

PROOF. Let us write i(vo, v1, v2) for the term a(v2, a(vo, v1, v2), vo). Then E

implies the identities i(vo, v0, v1) i-(vo, v1, vo) i-(v1, v0, vo) vo, as required by [11, 1.2]. Moreover every identity yo yj is equivalent, in the presence of the identities (eO), to an identity a(yO, Y1, Vk) Vk where Vk does not occur in yo or yi.

(Substituting yj for Vk gives y a(yo, yy, lz yo by (eO).) Thus, E is equivalent to a set of identities of the form y x (x a variable). Hence the two assumptions of [11, 1.2] are verified.

REMARK 1.2. Let A be any nonempty set. The identities (eO) are satisfied by the operation D on A defined (for x, y, z E A) by

D(x,y,z)=z ifx=y, =x if x:Ay.

We shall call D the discriminating function over A. (These functions have been studied by universal algebraists. They play a basic role in the study of "primal algebras," as was shown in [18]. Part (2) of the next theorem is a corollary of primal algebra theory, but we prove it directly.)

THEOREM 1.3. Let L be an algebraic first order language, and a = a(vO, V1, v2) be a term of L with only the indicated variables. Let E be the set consisting of the identities (eO) from Theorem 1.1, as well as the following:

(el) a(vO, a(vO, v1, v2), V1) V1;

(e2) a(X, y, ...Vn 1) a(x, y, (***a(x, y, vo)] ... [a(x y, Vn- 1)])

(where n = rc9, one identity for each function symbol (9 of L).

The following are true: (1) A nontrivial model 2I for L is a subdirectly irreducible model of E iff at is the discriminating function over the universe of 21. (2) Every finite model of > is isomorphic to a direct product of subdirectly irreducible models of E. (3) For each universal sentence b defined in L, one can effectivelyfind an identity E

defined in L, such that b E-* is true in every subdirectly irreducible model of E. PROOF. Let us first notice that if a model 2i for L satisfies E then, for any

x, y E A,

O(x, y) = {<u, v> e 2A: ,a,(x, y, u) = a(x y, v)}

is the smallest congruence relation 0 of 2I with <x, y> E 0. In fact, it follows from satisfaction of (e2) that ((x, y) is a congruence. And (eO) yields <x, y> E ((x, y). And, also by (eO), if <u, v> E ((x, y) and if 0 is any congruence including <x, y>, then

u = a,(x, x, u)oa,(x, y, u) = Ua(x, y, v) oa,(x, x, v) = v;

3As I discovered after writing this paper, [16] contains this theorem and an admirably brief

and complete proof of it.




that is, <u, v> E 0. Notice also that 21 does satisfy E if 21 is any model for L in which at is the discriminating function.

To prove (1), let 21 k A, 21 is a nontrivial model for L. By the above, we only have to show that 21 is S.I. iff ao is the discriminator. Clearly, if at is the discriminator then, for x =A y E A, O(x, y) = 2A; hence 21 is not only S.I., but simple as well. Conversely, suppose that 21 is S.I. By definition, there are elements a, b E A (a =# b) such that <a, b> E O(x, y) whenever x =A y. Hence, for any x E A, if u(a, b, x) =A a then <a, b>e (0(a, u(a, b, x)), i.e.,

u(a, u(a, b, x), a) = g(a, g(a, b, x), b);

but the left side of this relation equals a, by (eO), and the right side equals b, by (el). This contradiction shows that g(a, b, x) = a for all x. That means 0(a, b) = 'A; hence O(x, y) = 2A whenever x, y are distinct in A; consequently g(x, y, z) = u(x, y, x) = x if x : y. Of course, (eO) implies o(x, y, z) = z if x = y. So we are done-at is the discriminating function.

To prove (2), let 21 be any finite model of E. By Birkhoff's subdirect representation theorem, there is a set F. consisting of homomorphisms of 21 onto S.I. models Q3f (f e F), which separates points. We choose F minimal, so that in particular F is finite, say F = {fo, * X n- } (If IAI = 1, then n = 0.) We may, and do, assume that 21 is equal to the isomorphic model, a subalgebra of 23 = Pi,< 3i (where 23i =

ft*(21)) so that, for a E A, we have a = <fo(a), ... * fn-.(a)>. By 1.3(1), already proved, git is the discriminating function over Bi. Of course,

at is computed coordinatewise from the gti. We claim that 21 = 23. To prove it, clearly it suffices to prove that if we fix any element e E A, then

for every i < n, b E Bi, the element x = j(i/b) satisfying

x, = ej for j =A i, and xi = b, belongs to A.

Indeed, the minimality of F implies that there are i, v E A with uj = vj for j =# i

and ui =A vi. We can suppose ui =# ei. Then w = ar(9, ,) e A and wj = ej (I : i)

while wi = ui :A ei. Moreover, there is 2 e A with zi = b (f, is onto). Then

a(2, ae , ), ) e) A is the desired element j(ilb). For (3), let b be a universal sentence. First, convert b into an equivalent sentence

in prenex normal form: (Vf)f where 0 has no quantifiers. Let x, y be distinct variables not occurring in v. In S.I. models of E (where o(v0, v1, v2) defines the discriminator) the following equivalences are implied by the formula x y (for any terms a, /:

ca /3?- u[X, u(a, /3, X), (, a, y)] y, -3 y yA a(3, y, X) /53,

cc y V 3y< (y, Ad 3)'/

a Y A /3 I Y<-*,r(aUY,/3) Y

Using these equivalences and making repeated replacements of subformulas, starting with ,, we can obviously produce an identity T y such that the following formula, universally quantified, is true in every S.I. model of E:

X y V [0bu-+ T y].



190 RALPH MCKENZIE

Since S.I. models have at least two elements, 5 then is equivalent to (WV, x, y)

(x y V - y) in S.I. models of A, which in turn is equivalent to (Vi, x, y) (u(y, x, -r) Ir) in these models. This is the desired identity.

?2. Bennett's theorem. If S c co 1, we write S* to denote the smallest set that includes S u {1} and is closed under multiplication. J. H. Bennett [2] announced a result and B. H. Neumann [15] proved another, which taken together are equivalent to the following theorem. The proof given here is new.

THEOREM 2. 1. Let 0 be a sentence of afirst order language L. There is an identity e = 80 (computable from b but belonging to a different language L') such that

(Sp o)* = Sp e and b has a nontrivial model if e80 does. PROOF. (We assume, as we may, that L has only a finite number of relation and

function symbols.) The first step is to put 5 into prenex normal form, (Qovo)... (Qkvk)O, where 0o is a formula without quantifiers (open formula), VO, V, Vk are variables, and each Qi is the quantifier "3" or ""V." Next, we expand the language by adding symbols for Skolem functions corresponding to the occurrences of "s " in the prenex normal form; we obtain a universal sentence 0Sk -('1z4 (with OL

open), called the Skolem normal form of S, whose models are just the models of

S expanded by adding "admissible" interpretations of the Skolem function symbols.

In the third step, we discard all relation symbols of L, replacing each relation symbol R that occurs in S by a new function symbol OR of the same rank. We adjoin three new function symbols: D (3-ary), and 0, 1 (constants). Call the new language L'. We convert the Skolem normal form of S into a universal sentence

(1) 02 = (V)02

of L' by replacing every atomic relational subformula, R(ao, * , an-1), of 01 by an atomic equality formula ORO ... an-l 1.

DEFINITION 2.2. A standard model for the language L' is a model 21 for L' satisfying:

(2.2.1) D' is the discriminating function over the universe A of 21;

(2.2.2) 21 satisfies 0 1;

(2.2.3) for each relation symbol R occurring in S, the sentence

(Vi)(ORVi 0 V ORi3 1) is satisfied by 21.

It is obvious that, for every cardinal K > 1, S has a model of power K iff 02 has a standard model of power K. Now 02 is universal, and the finitely many conditions specifying standard models are universal. Thus we can write down a universal sentence p whose models are the standard models satisfying S2. In Theorem 1.3, take L = L', u(vo, v1, v2) = Dvorvv2, and let Z be the (finite) set of identities defined there. Let E' be an identity equivalent to ,u in S.I. models of Z (by Theorem 1.3(3)), and choose E equivalent to Z U {E'} (by Theorem 1.1).

Now this e is the desired identity, satisfying Theorem 2.1. In fact, by Theorem 1.3(1) the standard models of 02 (which have the same cardinalities as the nontrivial models of S) are precisely the subdirectly irreducible models of e, and by Theorem




1.3(2), the finite models of e are precisely (isomorphic to) the finite direct products of finite standard models of 02. The conditions of Theorem 2.1 follow readily from this.

?3. Exponents of spectra. We formulate in this section a very easy theorem which may serve to put the others into sharper relief. If S ( c - 1 and n E w 1, we denote by S(n) the set of nth powers xn (x E S), and by S"ln) the set of x such that xn E S. For any finite set of identities A, we write r(Z) for the least upper bound of the ranks of function symbols appearing in E. By Eq(K) we mean the family of all spectra of finite sets of identities E satisfying r(Z) < K.

THEOREM 3.1. Let S U {1, m, n} c X 1. (1) IfS e Eq(m), then Suln e Eq(m n). (2) If S e Eq(m) and 1 > max(2, m/n), then S(n)T Eq(l). PROOF. To prove (1), we assume that S = Sp E where all function symbols of

E are of rank m. For each function symbol (t appearing in I we introduce n

(m . n)-ary symbols (9t (0 < i < n). We define a mapping from subterms of E to

n-tuples of terms constructed from the new symbols: if the variable vj appears in E, let ij = Kv?, , vn -1> (so that v0 = vK iff] = k and u = v, and each v0 is a variable); if T = (Qtuo" am-l occurs in E and ao, .., am- are already defined, put F =

<( .? / *1 > where O = a . e Finally, let

- {a9 T'a r T- and i < n}.

The reader can verify that 2 has a model with universe A iff E has a model with universe nA (for any set A).

To prove (2), we can assume that 1 > 2 1. n = m, S = Sp E, and every function

symbol of E is m-ary. We shall define z so that the models of X are essentially the nth powers n2i (21 k E) with additional structure to reflect the Cartesian decomposi- tion of n<2.

For the symbols of A, we take for each function symbol (9 of E an l-ary symbol ('; and three others, P (2-ary), PO (1-ary), and 0 (1-ary). A standard model of z

will be of the form

gZ = <ndA, ppO, 0,(i t (t E T)>,

derived from a model 2a = <A, (t (t E T)> of I by putting

P(95,Y) =

<XON Y1,.. I Yn-. > POM = <Xo, x0,0 *, x0>,

(i)( = <X1, ..., X-1, X0>,

(t1(g, * * *, Xl 1) = <(pt(g? ^ l

. . V .^ 1)-)>n

The construction of E so that the models of E are exactly those algebras isomorphic to some bY2t (21 k E) is straightforward; we leave it to the reader.

?4. Reduction to one equation with one binary function symbol. THEOREM 4.1. Assume that E is a finite set of identities involving at most m ? 1

function symbols, and that n ? max(2, 1 + (r E)/2). There is an identity e, whose only function symbol (9 is binary such that



192 RALPH MCKENZIE

(1) Spe ={1} u{kn: keSpE andk 2 m + 3}; (2) in every nontrivial model of e, the sentence (Vx)(-i 6Oxx x) is true; (3) e has nontrivial models iff E has them (and has nontrivial finite models iff 2 has

them). Moreover, there is an algorithm for finding e, given E. PROOF. (This proof ends with Lemma 4.3.) We begin by describing the sub-

directly irreducible models of e. The description is (necessarily, it seems) rather involved. We can assume that the functions in E are ?0, 0, 0m- 1 and that they are all of rank i = 2n - 2. Let 2I = <A, (i (i < m)> be any model of E with IAl ? m + 3. Choose a permutation ,T of A so that every orbit of 7T has at least m + 3 elements (i.e., for a E A and 1 < i < m + 3, 7ria #0 a).

We define an algebra 9[21, 7T] = Q3 as follows: Q3 = <B, (9'>, where B = nA, and for every x, 5, Y E B, (90Z, y) = z iff:

(bl) ifx = y, then 5 = <xl, , xn-1 TxO>;

(b2) if yo= 7Tx1 and x0 = 7T2 +iy1 (O < i < m), then Z = <You .. * Yn - 2()X1 i.. i, X1n- Yi, **, Yn-l)>;

(b3) if xO = x1 = *.* = xn-1 and yo = y1, then = <Xo,y, ,yn-2,7yn-1>;

(b4) if 5 = <x1, **, xn-1, 7xO>, then y = <x0, X0, T2xO>;

(b5) if x = <Y1, * *,Yn-l ,yo> then = <Yo, ,Yn-2, w'Yn- 1>;

(b6) if otherwise, then 7 = Y.

It takes little work to verify that the above clauses are consistent-that is, there is precisely one binary operation (9B satisfying them-but the task is an easy one. (It relies on the chosen property of 7T.)

We shall write B for the set of constant functions belonging to B. We proceed to show that there are terms constructed from (9 whose values, in every algebra Q3 = J[Zt, 7T], are, respectively: the discriminating function on B; the decomposi- tion functions which serve to canonically represent B as equivalent to nB (in particular, B is the range of one of these functions); a function which acts on B like 7T on A; functions (i which, restricted to B, act like (9 on A. The reader is invited to verify that the terms defined below do the job.

Let us put ?(vO) = (9(2)(v0v0)(3), and notice that

(E)'(-) = <X1, * * * Xn-l XO>

Po(VO) = 9(2)[9(2)((9V( )(3)](3) (PO(g) = <X0, XO>);

Pj(vo) = P0O( Q)((vo)) for 1 < j < n (Pi(x) = <xi, * * *, xi>); P (VO, * n - vn1) = Un- (... (U(P0v0)).. )

where Uj(v)= -((9(2)[(P(pJ))V](3)) (pOQ0? . . . , n

- ) = <X0

n - *, 1-l)

HI(vO) = 6(n)(vo) where E(v) = Cvv (HI(Kx, .. *, x>) = <px, ... *, ix>);

A(vo) = (n)(vo), where T(v) = -(0(2)V(3))

((<xA ( * *. x>) = <v-lx, * * *, vx>);




and for each i < m,

( vi(Vo,* Vn-1) = Pn-1 OP(a0 * * - O)Pu(oo .** -rn1)

in which uj+1 = v;, Tr+1 = Vn+i-1 for j < n - 2, and O =

Pn - 1(vovo), and uo = Po'(2 +')(vn-1) where F(v) = 'n- 1)((9VV)

(for k=K <Xk,* Xk> EB. O < k < , we have (eX . .. l-1) = K6(X)G4"*, (i(X)>).

We have constructed, above, all of the terms claimed to exist except for the one which defines the discriminating function on B. To do that, we first represent the discriminating function on B. Put

D(vo, V1, V2) = PO[Cv2P(vO, V1, V1, 1, vD].

Note that, for x, y, z e B, Db(, y, z) = z if x- = y and D(x, y, z) = x if x : y. Now we put

DO(x, y, u, v) = D(u, bAx, y, u), D(x, y, v)),

and note that, for x, y, a, v e B, DO(9, Y, u, v) is v if x = y, and is a if x :# y; and we put

DL(x, y, u, v) = Vn- 1(... V1(Do(Pox, Poy, u, v))..*

where Vj(z) = D(z, D(v, DO(Pix, Piy, u, v), u), u) [for 1 < i < n], and note that so long as a, v e B. we have that D1(, Y, a, v) is equal to v if x = y, and is equal to a otherwise; and, finally, we put

D(vo, v1, v2) = P(0o, .* *,n-0)

in which, for i < n, ui = D(vo, v1, Pi(VO), Pi(v2)). Clearly from the above, DF is the discriminating function on B.

Now we are virtually done. Since the discriminating function is uniformly represented by a term, we know from Theorem 1.1 that, for any finite set of identities D true in every algebra Y[2t, n-], there is a single identity e such that e implies D and e is true in every such algebra. We now define a very inclusive set of iden-

tities to accomplish our purpose. The terms occurring in these identities were defined above.

DEFINITION 4.2. By P(Z) we denote the set comprised of all identities listed in A-F below (in which the only function symbol is (9). By e8 we denote an identity equivalent to P(Z) (constructed with the help of Theorem 1.1 -see part C to verify the hypothesis of Theorem 1.1):

A. (i) Pix Pjix (for all i, j < n); (ii x -1 P (POX,

* * * , Pn - X);

(iii) PAP(xO * *, xn-1) Pixi (for all i < n). B. Identities whose meaning is that each C9j (j < m) has its range included in the

range of PO (i.e., P0C9j 6?j); plus identities whose meaning is that the functions ?70 * * * O ?- 1, restricted to the range of Po, satisfy the identities E. (These identities are easily obtained.)

C. The identities (eO), (el), (e2) from Theorem 1.3, taking D for or. (Here (e2) is just one identity Dvov1CxOx1x Dvov1CO[Dvov1xo][Dvov1x1].)

D. The identity Dx(C9xx)y x.



194 RALPH MCKENZIE

E. (i) HAPox AHPox Pox; (ii) DPoxH(1)Poxy Pox (for 1 < i < m + 3).

F. Six identities F(i)-F(vi) corresponding to clauses (bl)-(b6) in the definition of C9(Q = 9[2t, n]):

(i) (9xx P(Plx ** Pn,,x IX HPox);

(ii) oP (HI (2 + Op1y, p1X, * * , P," _Xp x)(HIpX, P, y, * , P- y) 11 P(POY, * * * P.P - 2Y, igi(PiX, * * , Pn - god Pl y, * , Pn - 1 0);

(iii), (iv) are obtained in an obvious way, following the above pattern; (V) ?7(2)X(3)

P (POX, . * * n2 Anl (v) 9~2~~3~ (P~x...,Pn - 2x, APn - 1)

(vi) we will not try to write this one down. We first convert (b6) into a universal sentence. In words, "for all x and y, if x =# y, and if either Poy =# HPIx or Pox =# fl(2+i)P1y for any i < m, and if . . ., then Cfxy = y." We then take for F(vi) an identity equivalent to this universal sentence (by Theorem 1.3(3)) in S.I. models of the identities C.

(This is the end of Definition 4.2.) LEMMA 4.3. (1) Let W{ be any model of E with IA I ? m + 3 and let v be as above.

Then Y[2t, A-] is a subdirectly irreducible model of en. (2) Let Q3 be any model of e8 and put 2(Q3) = <P'(B), Oil(j < m)> = 21. Then 2{

satisfies E; IBI = nIAI; and if 93 is nontrivial, then IA] ? m + 3 and (C9(x, x) =# x for all x E B. Moreover, if 93 is subdirectly irreducible, then 93 -[&(93), HF8 [ A].

(3) Every finite model of ez is isomorphic with a direct product of the algebras Y[2t, T] (2t, vTas in 4.3(1)).

PROOF. Most of 4.3(1) should be obvious by now. Notice that the choice of v

ensures that the identities D and E(ii) in (Z) are true in Y[2t, 4]. For 4.3(2), it is clear that 4.2A,B imply IBI = nIA I and 2{ k E. If 93 is nontrivial,

satisfaction of 4.2C,D,E imply that CQV(x, x) # x and JAI ? m + 3. If 93 is subdirectly irreducible, then satisfaction of 4.2C allows us to conclude from Theorem 1.3(1) that De is the discriminating function on B. Satisfaction of 4.2F then will yield that 93 [ O(3), He A].

Statement 4.3(3) is derived from 4.3(2) and 1.3(2). The above lemma clearly establishes our theorem. [One might wonder about

4.1(3). Observe that if E has nontrivial finite models, it has finite models of power exceeding m + 3 (direct powers of any nontrivial model).]

REMARK 4.4. It seems interesting to consider whether Theorem 4.1 can be strengthened slightly. Theorem 3.1(2) makes it seem plausible that n > max(2, (r E)/2) would be a strong enough assumption on n. I doubt it, but I think that n 2 2 and 2n ? 1 + r E is likely to be sufficient.

?5. Conclusions. The main conclusion to be extracted from this work is that problems which consist of discovering whether a given equation can be satisfied identically (for all values of the variables) in some nontrivial finite groupoid can be very difficult, and in fact, as a class, these problems admit no algorithmic solution. We can give precise form to both assertions.

The truth of Fermat's Last Theorem is a well-known conjecture which has defied all attempts at proof for three hundred years. Let us call the positive integer




x a Fermat number if there exist positive integers u, v, w, k (k > 2) such that X = w k = Uk + Vk. The conjecture is that there are no Fermat numbers.

COROLLARY 5.1. One can describe a procedure which will in a finite length of time mechanically produce a (very long) groupoid identity e, such that Sp e is the set of finite products of squares of Fermat numbers (including 1 as the empty product). This e has nontrivialfinite models if Fermat's theorem fails.

PROOF. The set of Fermat numbers is spectrum of a first order sentence i. 0 can be found so that after introducing Skolem functions and proceeding through the constructions of Theorem 2.1, r(eo) = 3 and e<, has 21 distinct function symbols. One can put E = {e,} in Theorem 4.1, and n = 2, thence producing e = en. Since there are no Fermat numbers < 24, this e is as desired.

COROLLARY 5.2. The sets A consisting of groupoid identities that imply (Vx, y)(x y), and B consisting of groupoid identities (V1)(u r) such that (VD, x)( r A -/m(9xx x) has afinite model, are recursively inseparable.

PROOF. Combining 2.1 and 4.1, one can obtain a recursive mappingffrom first order sentences to groupoid identities, so that f 1A is the set C of sentences that imply (Vx, y)(x . y), andf- 1B is the set D of all sentences that have a finite model with more than one element. Now Lavrov [10] proved the recursive inseparability of the sets of refutable and finitely satisfiable sentences. From this it follows easily that C and D are recursively inseparable. Hence A and B must also be.

REFERENCES

[1] J. L. BELL and A. B. SLOMSON, Models and ultraproducts: An introduction, North-Holland, Amsterdam, 1969.

[2] J. H. BENNETT, Equational spectra, this JOURNAL, vol. 30 (1965), p. 264. [3] S. BURRIS, Models in equational theories of unary algebras, Algebra Universalis, vol. 1

(1972), pp. 386-392. [4] Yu. L. ERSHOV, I. A. LAVROV, A. D. TAIMANOV and M. A. TAITSLIN, Elementary theories,

Russian Mathematical Surveys, vol. 20 (1965). [5] T. EVANS, The spectrum of a variety, Zeitschrift fur mathematische Logik und Grundlagen

der Mathematik, vol. 13 (1967), pp. 213-218. [6] G. GRXTZER, Universal algebra, Van Nostrand, Princeton, N.J., 1968. [7] , On the spectra of classes of algebras, Proceedings of the American Mathematical

Society, vol. 18 (1967), pp. 729-735. [8] G. GRXTZER and R. MCKENZIE, Equational spectra and reduction of identities, Notices of

the American Mathematical Society, vol. 14 (1967), p. 697. [9] L. HENKIN, J. D. MONK and A. TARSKI, Cylindric algebras: I, North-Holland, Amsterdam,

1971. [10] I. A. LAVROv, Effective inseparability of the sets of identically true formulae and finitely

refutable formulae for certain elementary theories, Algebra i Logika (2), vol. 1 (1963), pp. 5-19. [11] R. MCKENZIE, Equational bases for lattice theories, Mathematica Scandinavica, vol. 27

(1970), pp. 24-38. [12] , Simple undecidable problems about finite groupoids, Notices of the American

Mathematical Society, vol. 20 (1973), A-505. [13] G. MCNULTY, The decision problem for equational bases of algebras, Thesis, University

of California, Berkeley, 1971. [14] V. L. MURSKIi, Nondiscernible properties of finite systems of identity relations, Doklady

Academic Nauk SSSR, vol. 196 (1971), pp. 520-522; English transl., Soviet Mathematics. Doklady, vol. 12 (1971), pp. 183-186.



196 RALPH MCKENZIE

[15] B. H. NEUMANN, On a problem of G. Grdtzer, Publicationes Mathematicae (Debrecen). [16] R. PADMANABHAN and R. W. QUACKENBUSH, Equational theories of algebras with

distributive congruences, Proceedings of the American Mathematical Society, vol. 41 (1973), pp. 373-377.

[17] P. PERKINS, Unsolvable problems for equational theories, Notre Dame Journal Formal Logic, vol. 8 (1967), pp. 175-185.

[18] A. F. PIXLEY, The ternary discriminator function in universal algebra, Mathematische Annalen, vol. 191 (1971), pp. 167-180.

[19] H. ROGERS, JR., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.

[20] A. TARSKI, Equational logic and equational theories of algebras, Contributions to mathematical logic (K. Schitte, Editor), North-Holland, Amsterdam, 1968, pp. 275-288.

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