Recursive unary algebras and trees

Annals of Pure and Applied North-Holland

213

Logic 67 (1994) 2133268

Recursive unary algebras and trees Bakhadyr Khoussainov Mathematical Sciences Institute, Cornell University, Ithaca. NY 14853, USA Novosibirsk University, Novosibisrk. Russian Federation Tashkent Universify. Tashkent, Uzbekistan Communicated by A. Nerode Received 1 January 1993 Revised 15 June 1993

Abstract Khoussainov, 213-268.

B. Recursive

“nary

algebras

and trees, Annals

of Pure and Applied

Logic 67 (1994)

A unary algebra is an algebraic system d = (A,f,, ,L). wheref,, . ,fmare unary operations on A and n EW. In the paper we develop the theory ofeffective “nary algebras. We investigate well-known questions of constructive (recursive) model theory with respect to the class of unary algebras. In the paper we construct “nary algebras with a finite number of recursive isomorphism types. We give the notions of program, uniform, and algebraic dimensions of models, and then we investigate these notions on unary algebras. We find connections between algebraic and effective properties of r.e. representable unary algebras. We also deal with finitely generated r.e. (positive) unary algebras. We show the connections between trees and “nary algebras. Our interests also concern recursive automorphisms groups, r.e. subalgebra and congruence lattices of effective “nary algebras.

Contents 0. 1. 2. 3. 4. 5. 6. I. 8.

Introduction. Main definitions. Strongly homogeneous algebras. Trees and “nary algebras. Highly recursive “nary algebras, . . R.e. unary algebras and a specification problem Algorithmic dimensions of algebras . . . Algorithmic dimensions of trees. .. Subalgebras, automorphisms and homomorphisms

213 215 216 222 227 233 244 259 263

0. Introduction We develop the theory of recursive wary algebras and a fragment of the theory of recursive trees. We refer to unary algebras throughout this paper, as algebras. Correspondence 14853, U.S.A.

to: B. Khoussainov,

Elsevier Science B.V. rcnr ni ~Q_nn77ia?lFnnSn_x

Mathematical

Sciences

Institute,

Cornell

University,

Ithaca,

NY

214

B. Khoussainov

There are several results on algebras.

For instance,

the theory of a unary function

is

decidable [4, 12,381, the theory of two unary functions is not decidable [15], the second-order monadic theory of two successors is decidable [38]. Groups of automorphisms

of algebras

unary

have been

theories

have been investigated studied

[22,23,40].

[lo].

Algebras

The number

of models

with one unary

of

function

admitting elimination one unary operation

of quantifiers have been described [15]. Recursive algebras with have been investigated [33,34-J. There are characterizations of

o and o,-categorical

algebras

been investigated In recursive

[40,41].

Varieties

of algebras

and word problems

have

[21,25]. model

theory,

important

directions

are investigations

of recursive

representations, finding the interactions of effective, algebraic, and model-theoretic properties, and the study of reducibilities of recursive representations [l-lo, 41-441. There are a large number of results for recursive Boolean algebras, vector spaces, linear orderings, abelian groups, systems with unary predicates, lattices, graphs, and fields [l-lo, 16-181. In this paper, we investigate recursive unary algebras. Here is a brief summary. In Section 1 we give the main definitions for recursive model theory, algebras and trees. In Section 2 we define the class of strongly homogeneous algebras. We give criteria for strongly homogeneous algebras to be recursive representable and recursively categorical. In Section 3 we construct a partial functor between the category of algebras with a unary symbol and the category of trees. This functor preserves some effective properties. We investigate connections between recursive representations of trees and algebras. For instance, it is known that any r.e. extension of the theory of trees possesses a recursive model [6]. We construct an example of an r.e. unary theory with no recursive models. We define a class of algebras such that any r.e. extension of the theory of this class has a recursive representable algebra. In Section 4 we define the class of highly recursive algebras. We investigate recursive isomorphism types of these algebras. We give the effective-algebraic characterizations of recursively categorical and strong recursively stable highly recursive algebras. Section 5 is the central part of the paper. We investigate the structure of recursive algebras over r.e. partitions of o. The approach is to find structural properties of algebras over r.e. partitions. This approach turns out to be useful for obtaining several results, in particular for solutions of partial cases of a Bergstra-Tucker problem [27], and for finding counterexamples to Mal’sev’s theorem [20,27]. We also give an algebraic characterization of simple sets. We formulate a specification problem and the notion of a specified algebra. We construct examples of unspecified algebras. In Section 6 we construct algebras with a finite number of recursive representations. V.A. Uspensky and A.L. Semenov defined algebraic, uniform, and program reducibilities of recursive representations. These definitions allow us to introduce algebraic, program, and uniform preoderings in the class of recursive representations of a model [7]. We investigate these reducibilities on algebras.

Recursive

In Section 7 we characterize only.

We prove

that

these

algebraically stable. In the last section representations recursive a recursive

wary

recursively trees

we construct

algebra

in any

categorical

an

trees consisting stable,

algebra

with

if and

representation.

recursive

of finite blocks

only

countably

of automorphisms,

recursive

with no non-trivial

215

and tree.7

are recursively

and with a continuum

automorphism

algebras

if, they are

many

recursive

but with no non-trivial Finally,

homomorphisms

we construct

such that any two

elements of the algebra are separable by a continuum number of homomorphisms. The category of recursive models is equivalent to the category of constructive models. Therefore, the notions formulated in both categories are equivalent. We use these two categories

for convenience.

1. Main definitions the equality Let B = (F;‘, . . . , Flk; Py’, . . . , PTS) be a signature containing Let d be a system of signature g. Let v : o + A be a map onto A.

sign.

Definition 1. A pair (d, v) is a computable system, if there exist total recursive functions f,“‘, . . . ,fik, r.e. relations py’, . . , pp such that for all xi,. . , xn,, y,,.

. . ,YrnjEO, iE{l,. Fi”i(v(xJ,

. . ,k},jE{l,.

. . ,s):

. . . 94%,)) = v(f(x1,.

. . 2%I,))

and Pj”l64Y1)~~

. . >V(Y,,))

-

PimJ(YI,.

. 9Ym,).

Let (s!, v) be a computable system. We define an equivalence relation ye”on o by (x, y) E~,H v(x) = v(y). The recursive functions f;‘, . ,fknk are well-defined with respect to qY.Therefore, we may say that the algebraic system J$ is representable over qy. Definition 2. Let (JzZ, v) be a computable system. Then: 1. (d, v) is constructive, if the sets q,, py’, . , pp are recursive. 2. (&, v) is negative, if the complements 3. (sz’, v) is positive, if the set qy is r.e.

of the sets ye”, py’, . . , pFs are r.e.

A constructive system is recursive iff it is positive and negative simultaneously. There is another, equivalent, approach to the notion of constructive system. Namely, by a recursiue system we mean one with domain O.Iand uniformly recursive atomic relations. Let (&, v) be a constructive system. Then, there exists a recursive system 59, an isomorphism j?: & + g and a recursive function 4 such that flv = 4. Two recursive systems are recursively isomorphic if there exists an isomorphism between these systems which is itself recursive. The system & has a recursive (positive, negative) representation if there exists a recursive (positive, negative) system isomorphic to ~2.

216

L?. Khoussainov

Definition 3. Let & be a recursive system. The recursive dimension of &’ is its number of recursive isomorphism types. We denote this number as dim,(&). &’ is recursively categorical if dim,(d)

= 1.

We define some notions

on trees and unary

algebras.

A unary algebra is a system d of signature g = ( fi, . . ,fk), where each fi is one-place function symbol. We refer to unary algebras as algebras. The domain of is A. Let a, b E A. These elements are connected, if there exist a sequence Xl,. . * 3x, E A, and a sequence t,, . . . , t,_l of terms of c such that a = x1, b = x,, ti(xi) = Xi+ 1 or ti(Xi+ 1) = Xi, i, i = 1,. . . , n - 1. In this case the sequence tl, . . . , t,_ 1 &

is called a path. We define a distance

d(a, b) as follows:

1. If a = b, then d(a, b) = 0. 2. If a and b are not connected, then d(a, b) = co. 3. Suppose that a and b are connected. Let N,(a) = {a}. Suppose that N,(a) = (Y Id@, Y) d m} has been defined. Then d(a, b) = m + 1 iff b $ N,(a) and there exist c E N,(a), i < k such that d(a, c) = m, and fi(c) = b for h (b) = c. Let B, C c A. Define d(B, C) = min{d(b, c)) b E B, c E C}. A block is a connected subset of A. Every block is a subalgebra of d. If B and C are blocks, then d(A, B) = co. Let a E x2. The f-root of a is Rf(a) = {y J3n(f”( y) = a)}. Let F c { fi, An F-block is a block of the algebra (A; f E F). Let al,. . . , a, E A of the s-tuple (al, . . , a,) is gl,. . . ,gsEc. The (gl,. . . , g,)-cycle C 91....,9= (al,. . . ,a,) = (Wh), . . . , s34ln Em). Let T be the theory of signature ( < ) with the following 1. 2. 3. 4. 5.

maximal different . . ,fk}. and let the set

axioms:

VX(X < x); Vxy(x d y A y d x + x = y); Vxyz(x < y A y < z +x d z); Vxyz(x < y A x < z -+ y d z v z < y); tlx+(x

<

y

A

vz(X

<

Z d

y

+

X

If &IT, then & is a tree. The notions ~2 are just like those above.

=

Z

A

Z =

J’).

of connectedness,

distance,

block on the tree

2. Strongly homogeneous algebras We fix a signature

cr = (fi,

..,fk).

Definition 4. Let & = (A;f,, . . . ,fk) be an algebra. blocks of ~2, c EC and 4 E o\F. The algebra homogeneous, if the following conditions hold:

Let C be an F-block, & = (A;f,,

. . . ,h)

B1, B, be is strongly

217

Recursive unary algebras and trees

1. The mapping

4 is an isomorphism

between

C and 4(C), and if @‘(c) EC, then

4”(c) = c. 2. If B1, Bz have no finite cycles then for all b, EBB and b2 E B2, there exists an automorphism 3. Suppose

E of d

such that a(b,) = bZ.

that b, E B1, b2 E B2 be elements

Let x E B1, y E B2 be elements

C,,(b,), and y with C,&(b,), respectively. that a(x) = y. j=

4. If b E B1 forms a finiteA-cycle 1,. . . ,k.

Example.

An algebra

(A;f)

forming J-cycles

of the same length.

which have the same set of paths connecting Then there exists an automorphism

for some i, then B1 possesses

with elimination

of quantifiers

x with c( such

finitefj-cycles

possesses

for all

properties

l-4

ClQ We give several properties Property 4,(C)

1. Let

n MC)

This property

41,

of strongly

homogeneous

algebras.

d2 EO\F and let C be an F-block.

= 8 or &(C)

If

$J~ # 42 then either

= 42(C) = C.

follows from the first part of the definition.

Property 2. The unary operations Let A, fi be unary isomorphic mapping

of d

operations. Then from C on J(C).

commute. the restriction

Property 3. Any two blocks of strongly homogeneous same length are isomorphic.

of J on any &block

algebra possessingfi-cycles

C is an

of the

It follows from the third part of the definition. Property 4. Let a be a block. Suppose that g possesses an fi-cycle of length n. Then the length of any fi-cycle in the block B is n. It follows from the first part of the definition. Property 5. Let B be a block such that there are elements x1,. . , x, which form , n, respectively. Then there exists an element a E B f I?..., f m-cycles of lengths nl, . such that for any i < m, the element a forms ani-cycle of length ni. We prove this property by induction on m. If m = 1 then there is nothing to prove. Let x be an element which forms &-cycles of lengths n,,,, m = 1, . , t. Let a be an element which forms a cycle of length n, + 1. Since a and x are connected there exists

218

a sequence

B. Khoussainou

y,, . . . , Y, such that YI = X, a = ypjj,(yi)

= yi+l

orfj,(yi+

1) = yi, where

i < p-

1, fi, E c. By Definition 4, the element yj forms an J-cycle of lengths nl,. ..,n,. It follows that either y, or y, forms anh-cycle of lengths ni, i $ m + 1. Let s$ be a strongly homogeneous algebra. We divide this algebra in two parts as follows. A, = {x 1a block containing

x does not have cycles}

and

Thus d, and &‘rin are disjoint homogeneous. Proposition 1. Let d representable.

subalgebras

of the algebra &‘. They are also strongly

be a strongly homogeneous algebra. Then d;4, is recursively

Proof. Since d, consists of isomorphic blocks, d, is recursively representable, if and only if, a block of dIdz, is recursively representable. Indeed, if &m has a recursive representation, then any block forms an r.e. set. It follows that any block has a recursive representation. Let 99’ be a recursive representation of a block $9. The algebra &, is isomorphic to the union of disjoint recursive algebras pi = (Di; fii, . . . , fik) such that pi is recursively isomorphic to 9?’ for all i. We can construct this union to satisfy the following conditions:

2. The set ((n, i) 1n E Di} is recursive. 3. The function J = ujLj is recursive. . . ,fk) is recursive and isomorphic to ,sl,. The algebra (o;fi, Thus we can suppose that dr$, is connected. We prove that dU has a recursive representation. We prove it by induction on k. The case k = 1 is clear. there exists Let k = n + 1 and dF4, = (A;f,, . . ,fn + 1). By the inductive assumption a recursive algebra (w;f;, . . . ,fi) isomorphic to (A;f,, . . . ,fn). Define an equivalence ye on A by (x, y) EV, if and only if, x and y are {fr, . . . ,f,}-connected. The equivalence q is a congruence on &‘. We have &!z/Y/b Vx(f,(x)

= x A ... A f.(x) = x)

The algebra (A,$;f, + 1) is strongly homogeneous and connected. Hence we can apply induction to this algebra. Let (o;fn(+ i) be a recursive representation of this algebra. On w x w define the following recursive functions: 4ilx, y) = (&h’(Y)),

i G n.

219

Recursive unary algebras and trees

to (A; fi, . . . ,fn), i EW. The algebra Li = ({i} x o; c#I~,. . . , c$,,) is isomorphic Therefore for any i we can effectively define a recursive isomorphism $i from Li onto Lf,‘+,(i). The function A + 1 G-4

Y)

By the construction, d,. This completes Proposition

$ = uj$j =

(II/ (4,

is recursive.

Define

Y).

the algebra (w x o; +i, . . . , & + I ) is recursive the proof. 0

2. Let d

and isomorphic

to

be a strongly homogeneous algebra. Then ZI is recursively

representable ifs the algebras ~4~ and ~9fi, are recursively representable. Proof. Let JYZbe a recursive representation. Then the domain of the algebra ~fin is an r.e. set. Hence the algebra ccgfin has a recursive representation. By Proposition 1, ~9fi” be recursive algebras. Then it is ~2, has a recursive representation. Let d,, obvious that the algebra SZ’ has also a recursive representation. This proves Proposition 2. 0 By these propositions,

the question

of recursive

representability

same question for ~fin. Let JZJ = (A; fi, . . . ,fJ be strongly homogeneous algebra. ~2~ = (A; fi), . . . , dk = (A; fk). These algebras are also strongly call the algebras d,, G!~, . , dk components of SZZ.

of ~2 reduces to the Consider algebras homogeneous. We

Proposition 3. The algebra &tin is recursively representable, ifand only if, the algebras &I,. . . , dk are recursively representable. If ~fin is recursively representable, then obviously, the algebras dl,. . . , dk are also recursively representable. Let ~2: = (0; A’) be a recursive representation of &i, i= 1,. . . , k. On the set mk define operations 41,. . , $bkby h(nl,

The algebra proposition.

. . . , ni,

(ok; 4i,.

. . . , nk)

=

@I,

. . . ,fi’(nih

. . , &) is recursive

. . . , nk)

and isomorphic

to &ri”. This proves

the

Corollary 1. A strongly homogeneous algebra is recursively representable if and only if its components are recursively representable. This corollary can be used to push the question of recursive representability of strongly homogeneous algebras further, by investigating this question with respect to algebras with only one unary symbol. Let ~4 = (A;f) be a strongly homogeneous algebra. Define C = {n 1there exists a cycle of length

n}.

220

B. Khoussainov

Define partial

recursive

s

functions

$i, $2 as follows:

if n E C(d) and there are exactly possessing cycles of length n,

\Cll(4 = ( undefined

s blocks

otherwise,

s

there

exists

such that

$2(& r) = { undefined

a y and

a cycle C of length

d(y, C) = t and

card(f-l(y))

II = s),

otherwise.

Note that, the functions

$i and Cp2are well-defined.

Theorem 1. The algebra d is recursively representable

if and only if

1. The set C is r.e. 2. There are partial recursive functions

g1 (x, y), g2(x, y, z) such that for all n E C:

(4 Il/I(n) = lim,g,(n, x), and ICI&, t) = lim,g,(n, t, x). (b) 1.~1 d x2, then g(n, x1) G g(n, x2), and gi(n, t, x1) < gi(n, t, x2).

Proof. If S! is a recursive algebra, then the set C is r.e. Let PO, Pi,. sequence

of all cycles of the algebra

Stage 0. If the length Stage

t. If the

d.

Define the function

of PO is n, then gI(n, 0) = 1; otherwise, of P, is n, then

length

. . be an effective

gi. gl(n, 0) = 0.

gI(n, t) = gl(n, t - 1) + 1; otherwise

Sl(% t) = g1(n, r - 1). By the construction, the function g1 satisfies the theorem. Using a similar construction one can prove that there exists a function g2 which also satisfies the theorem. Let functions gi, g2 be given. Let no, nl,. . . be a sequence of elements of the set C. We prove that & is recursively representable. Stage t + I. Assume that a finite algebra d, = (A,; f’) has been constructed. By induction, the algebra d, has the following properties: (a) For any i < t the algebra ~2, possesses exactly gl(ni, t) blocks of length ni. (b) For any i < t, if there exists an element y E A, and a cycle C of length ni such that d(C, y) = i, then y has exactly gz(ni, i, t) preimages. Compute gl(n,+l, We can effectively

r + l),

. . . , gl(no,

construct

t +

an algebra

I),

gz(nt+l,

t

+

1, t +

11,.

. . , hb0,

1, t +

1)

STY,+1 = (A,, 1, f ‘+ ‘) such that

(a) For any i < t + 1 the algebra col,+, possesses exactly gl(ni, t + 1) blocks of length ni. (b) For any i < t + 1, if there exists an element y E A,, 1 and a cycle C of length Izi such that d(C, y) = i, then y has exactly gz(ni, i, t + 1) preimages. (c) ~2~ is a subalgebra of d, + 1.

Recursive wary algebras and trees

Let ~2’ = uidi. We have proved Our

next

homogeneous

By the construction, 0

d’ is a recursive

221

algebra

isomorphic

to d.

the theorem.

goal is to give a characterization

of recursively

categorical

strongly

algebras.

Lemma 1. Suppose that JXI = d,. Then, the algebra d is recursively categorical, only if, it consists of a$nite number of blocks. Proof. If d consists

of a finite number

of blocks, then using the definition

ifand

of strongly

homogeneous algebra, one can easily prove that &’ is recursively categorical. Suppose that d has an infinite number of blocks. We also suppose that for any x, 1, it follows that there card(fi-r(x)) = ni, i = 1,. . . , k. From the proof of Proposition exists a recursive representation dr of JZI, such that the block relation in A&‘~is recursive. We construct a recursive representation with an undecidable block relation. Let q be an r.e. equivalence on the set o such that any v-equivalence class is r.e. and not recursive [S]. There is a strongly computable sequence q0 c rI c ... such that q = uitli, and vi is an equivalence on the initial segment (0,. . . , i>, i EO. Using this sequence one can construct recursive functions gr, . . . , gk such that: 1. card(g;’ (m)) = ni, m E O. 2. {(x, y) 1x and y are g-connected, and x, y < i} = V/i. 3. The algebra (0; gl,. . . , gk) is strongly homogeneous. We obtain that the algebra A2 = (0; g1 , . . , gk) is recursive and isomorphic to &. By the construction of dI and J&‘~, these algebras are not recursively isomorphic. This completes the proof of Lemma 1. 0 Lemma 2. If d = dfin,

then the algebra

d

is recursively

categorical.

Proof. For the sake of simplicity we prove the lemma when the signature of d consists of a unary symbol. Let JZZ’and 93 be recursive representations of ~fin. We construct a recursive isomorphism 4 between these recursive representations. Stage 0. Let 0 E A. Consider the finite algebra Cf(0). Let iO be the smallest number in 9? such that C,(i,) is isomorphic to Cs(0). Then for any k < s, define MfS(0)) = f”(i0). Step t + 1. Let m,,, = px(x +! 64,). Suppose that d(m,+,, 64,) = co. Find the smallest isomorphic to C/(m,+ I), and d(m,+ 1, p&)‘= cc Define 4r+lfs(mr+l)

=f%+A

i,+I such

that

Cf(it+l)

is

SEW.

Suppose that d(6&, m,, 1) = s. Let a ~84, be an element such that there is asequencexO=a,xI,..., x,- 1, x, = m, + 1 which connects a with m, + 1. Then there

222

B. Khoussainou

exists the smallest element i,+ 1 and a sequence such that for any j E {O, . . . , s - l}, we havef(xi)

y. = &(a), y,, . . . , y,_ 1, y, = it+ 1 = xi+ r(f(xi+

1) = xi), if and only if,

f(.Yi) = Yi+ 1 (f(Yi+ 1) = Yih Define,

h+ 1 (xi) = Yi.

By the construction, lemma. El As a corollary

r#~= U,~i

is a recursive

of the above lemmas

Theorem 2. Let d

we obtain

isomorphism.

the following

This

proves

the

theorem.

be a strongly homogeneous algebra. The following conditions are

equivalent: 1. The algebra d is recursively categorical. 2. The components of AX!are recursively categorical algebras. 3. The algebra d posseses a jnite number of blocks with no cycles. 4. The component d, is a recursively categorical algebra. We apply

the obtained

results in the next two examples.

Example 1. There exists a strongly homogeneous algebra ~4 with no recursive representations such that for any n > 1 the system ~4” is recursively representable. Let Ci be a cyclic algebra of cardinality i. Let D be a set such that: 1. D c (pq (p # q and p, q E P}, where P is the set of prime numbers. 2. D is not r.e. Consider

an algebra

(A,f)

such that:

1. for any n E P u D, the algebra 2. f is a permutation.

has o copies isomorphic

This algebra does not possess a recursive a recursive representation. Example

2. There

exists

a strongly

representation.

homogeneous

to Ci.

But for any n > 1, &“’ has

unary

recursively

categorical

algebra d such that d2 is not recursively categorical. Let ~2 be a connected algebra, and d b Vxy( f (x) = f (y) + x = y). The algebra ~2 is recursively categorical, and JX?’ has infinitely many blocks. Hence &” is not recursively categorical.

3. Trees and unary algebras Let & be an algebra

of signature

G and let a, b E A. Define:

a < b iff there is a term t such that t(a) = b. The relation

< is a partial

preordering

of A. Let ~2, = (A, d ).

223

Recursive unary algebras and trees

Lemma preodered

3. Let card(a)

= 1. Algebras

& and PJ are isomorphic

if and only if the

sets &, and a0 are isomorphic.

Proof. The following

formula

3nI- . . 3n,(f;’ This proves

Lemma

We construct

expresses

. .fzk(x)

3.

the relation

< :

= y).

0

a partially

ordered

set

which (x, y) E q t-f x d y A y d x. Theorem 3. 1. If d has a recursive (r.e.) representation, then a partially preodered set &, is r.e. representable. 2. if d has a recursive (r.e.) representation, then a partially odered set &,/n is r.e. representable. 3. Suppose that o consists of only a unary function symbol. Ifd has no non-trivial cycles then: (a) The model &, is a tree. (b) Ifdc4, has a recursive representation 9J such that the set Rs = {(x, y)l~~Wx

< 2 < Y,>

is recursive, then r;4 is recursively representable. (c) Ifd has a recursive representation with the decidable block relation, then d, recursively representable. (d) If-Fe,, is recursively categorical, then d is also recursively categorical.

is

Proof. 1. Let 59 be a recursive representation of &. By the previous lemma, d is defined by an El-formula. Hence the relation < on !?8 is an r.e. set. Thus 99’, is an r.e. model. 2. Let 9S be a recursive (r.e.) representation of d. By part 1, &I0 is an r.e. model. Since the relation q is defined by a positive E&formula, B,/n is an r.e. model. 3. Suppose that k = 1. The model d, is a tree. Let W be a recursive representation of & such that Rs is recursive. For any x E o, define f (x) as follows. Step 2. Let y be such that x 6 y. If &JCkl3Z(X

<

z <

y)

then put f(x) = y. Otherwise, Step t + 1. If S?‘,kl3z(x then put f(x)

<

z <

effectively

find an element

y, such that x < y, < y.

y,)

= y,. Otherwise,

find an element

y,+r such that x < Y,+~ < y,.

224

B. Khoussainov

There is a step t + 1 such that at this stepfis defined on x. It follows that the model (B,f) is a recursive representation of d. Let 99 be a recursive representation of S/ with a decidable block relation. If x and y are not connected, then these elements are not d comparable. Let x, y be connected. There exist elements z, t, s such that f”(x) = z and f’(y) = z. By definition of 6 , we have x < y iff t = 0. It follows that a0 is a recursive representation of d,. Let ~4, and ~2~ be recursive representations of &. By the assumption, there exists a recursive isomorphism between corresponding r.e. trees. By Lemma 3, this isomorphism is a recursive isomorphism between dQe,and s’~. We have proved the theorem. 0 Let Y be a tree. If any two elements from different blocks are not comparable then, there is an algebra ~2 such that &, is isomorphic to Y-. We say that ~2 generates Y. The next example shows that the recursiveness of the set RB in the theorem above is essential. Example. There exists a recursive tree Y such that the algebra d generating 9 is not recursively representable. Let g be a recursive function such that: 1. For any x, lim,g(x, n) exists. 2. If n, < n2, then g(x, nr) d g(x, n2). 3. The set G = {lim,g(x, n)lx EO} is not r.e.. We construct a partial ordering $ on w. Step 0. Let
caW{x~&+llbi

< x)) =

g(i,s + 1).

For i = s + 1, let Bs+is+r be isomorphic to the following tree ((1,. . ,g(s + 1, s + 1) + 2}, <) where $ is the smallest partial ordering on (1, . . . , g(s + 1, s + 1) + 2) containing ,the relation

{(1,2),. . .

3

(g(s + 1, s + 1) - 1, g(s + 1, s + I)), (g(s + 1, s + I),

g(s + 1, s + 1) + 11,(g(s + 1, s + l), g(s + 1, s + 1) + 2)).

Recursive

This Bi,+l

procedure nBjs+l

is

Ts+l =

s

F=

F is recursive

that &’ is a recursive

algebras

i. We 1. Define

on

1,. . . ,s+

UBis,

By construction, Assume

uniform

=@,i,j=

wary

225

and trees

suppose

that

for

any

i # j,

IJTi. L and {d(bi, max Bi) 1i EO} = G.

generating F. The sets MI = {xlf(x) = x} Hence are recursive. and M2 = {x~3x,x2(x1 # x2 of =f(x2) = x} {d(x, y)l x E MI, y EM,} is r.e. and is equal to G. Contradiction. 5. T,, is the theory

Definition

vx

i (

s=l

v (

algebra

generated Z(x)

by following

= f;‘(x)

i#j,i,j
We define a partial

preordering

axiom:

. ))

p on the set D, of all finite models

of the theory

T, by: apb iff there is a submodel

of b isomorphic

to a.

Define

gn= (DmPI. Proposition a preodered for ai,,

4. Let o be a signature with only one unary function. set with the following property: any 429.

infinite sequence aO, aI, a2,. . . there . . such that ai,pair+,, s EO.

2. Suppose that o has two function symbols. elements from 9”, such that for all i, j E co:

There

The model 9,, is

is a infkite

is a sequence

subsequence

aO, a,, . . of

Proof. 1. We prove this theorem by induction on n. If n = 1, then part 1 of the proposition is easy. Let n = t + 1. Case 1. Suppose that for any i, the model ai is connected and has a cycle of length 1. Let air, . . . , aipi be elements in ai such that ai = R(air) u . . . u R(aip,). Then any element Ui of the sequence is an element of the set S,(D,) of all finite subsets of D,, 1. We define a relation p’ on the set S,(D,) such that (a, b) E p’ iff there is a one-to-one

function

f from a to b such that for any

x E a, (x, f (4) E p. The preodered set (S(D,), p’) satisfies the property of Lemma 4 [6, p. 3571. Hence there exists a subsequence ai,, ai2, . . . such that for any j, (ai,, Uij+ ,) E p.

226

B. Khoussainov

Case 2. Suppose that for any i, the model ai is connected, and has a cycle of length p > 1, with p < t. In a similar way, defining a partially preodered set S,(l),) as in the previous case, we can prove the proposition. Case 3. Suppose element of the set S,(D,)x.. where

C,,,

proposition,

that the sequence

. WAX

aO, a,, a2,. . . is arbitrary.

Any element

ai is an

{Cc+,>

is a cycle of length their direct product

t + 1. Since all of these preodered also satisfies the proposition

sets satisfy the

[6]. This proves the first

part of the proposition. 2. For any i, define an algebra gi=((al,.

. . ,ai,bl,.

. .7bi-l};Lg)

with s(al)

= 4,

f(aj)

= bj,

j = 1, . . . , i - 1,

g(af) = by _ 1, j’ = 2, . . . , i, f(bj) = g(bj) = bj,

j = 1,. . . , i - 1.

Then, %i I: TI . The sequence gl, Y2, . . . satisfies the second part of the proposition. The proposition is proved. 0 Definition 6. Theory T is strongly V-jinite, if the set of all universal formulas in any extension of T by a finite number of constants is finitely axiomatized by universal formulas. Any r.e. extension of a strongly V-finite theory possesses a recursive model [S]. If for a theory T a preodered set 9(T) satisfies the first part of the proposition above, and the set of all universal formulas of this theory is finitely axiomatized by universal formulas, then this theory is strongly V finite [6]. Thus we get the following corollary. Corollary 2. Let o be a signature with only one function symbol. Then any r.e. extension of the theory T,, possesses a recursive representable model. Indeed, if Ti is an extension of T, by a finite number of constants, then the set of all universal formulas of TA is finitely axiomatized. Repeating the proof of the previous proposition, we conclude that T. is strongly V-finite. Hence any r.e. extension of T,, possesses a recursive representable model. Proposition 5. Let 0 = ( fi, . . . ,fk). 1. Suppose that k = 1. There is an r.e. theory of unary algebras which does not have a recursive model. 2. Let k > 1. There is a r.e. extension of the theory T1 which does not possess a recursive representable model.

Recursit;e

wary

algebras

221

and trees

Proof. 1. Let S be a simple set. Let DO, Di, . . . be a computable sequence of finite sets such that Di n Dj = 0, and S n Di # 0, i, j E co. Define the following r.e. set of axioms of signature

(f):

VieDi(x forms a cycle of length i) ; > ( 2;. 13x(x forms a cycle of length i), i E S. 1. 3x

This

theory

is r.e. Let d

be a recursive

{x ( 3z(z forms a cycle of length

model

x)} is r.e. By axioms

of this theory. this set is infinite

Then

the set

and is a sub-

set of o\S.

Contradiction. 4 above. 2. Consider the sequence 3,,, 9i, . . . from the second part of Proposition Let pi be an 3formula which defines the model Yi. It is obvious that ?Jik Ti, i E co. Repeating the same argument a recursive model. 0

we get an r.e. extension

of T1 which does not have

Corollary 3. The theory of algebras of the signature (f)

is not strongly V-finite.

Using these ideas it can also be shown that the theory of algebras with no cycles is not strongly V-finite.

of signature

(f)

4. Highly recursive unary algebras The notion of highly recursiveness usually arises when we want to impose some conditions of recursive locality, and investigate mathematical systems which satisfy these requirements. In this section we give a definition of highly recursive algebras and investigate

properties

of these algebras.

Definition 7. A recursive

algebra

is highly recursive if there exist recursive

functions

such that 919.. 1. For all x E o, i < k, gi(x) = card({ylfi(y) = x}. 2. The block relation on this algebra is decidable. ., gk

Proposition 6. Let d be a highly recursive algebra. Then Ld = ((2, y)lVa E Aut(&)(+)

# Y)}

is r.e. Proof. We need to prove the following

lemma.

Lemma 4. Let S! be an algebra such that, for all x E A and j E co, the sets Nj(x) are finite. Let a, b E A. There is an automorphism c( of & such that cr(a) = b, ifand only if for any s E o there is an isomorphism cI,from the model N,(a) to N,(b) such that a,(a) = b.

228

B. Khoussainov

Proof. If there is an automorphism which satisfies the lemma. Assume

from a to b, then there is a sequence ao, al,. . . that for any s the models N,(a) and N,(b) are

isomorphic. Since the sets N 1(a) and N1 (b) are finite, there is an infinite sequence kl < k2 < k3 < . . . such that for any i, j E o and for all x E N,(a), Q(X) = clkj(x). Let 60 = ako. Again, since the sets N,(a) and N,(b) are finite, there is a subsequence s1 < s2 < s3 < . . . of the sequence k. < kI < k3 < . . . such that for any x E N,(a) and all i, j E w, a,,(x) = clsj. Let b1 = M,,. Then do c dl. Suppose that 6, has been constructed. OfQ,c(,,.

By the assumption,

there is a sequence

fil, fi2, . . . which is a subsequence

. ’ such that for any x E N,(u) and for all i,j E O, pi(X) = pj(X) and & = PO.

Since the sets N,+ 1(u) and N,+ 1(b) are finite, there is a subsequence

pp,, pm, fi,_, . . . of

the sequence PO, PI,. . . such that for any x E N,+ 1 and for all i,j E o, ppi(x) = ppj(x). Then 6 can be extended to an Let 6 n+l = /?,. Then 6, c fin+l. Let 6 = u,S,. automorphism CIsuch that a(u) = b. 0 By this lemma, for all 2, by A”, IZE CO,there exists an automorphism CIof the algebra such that ~((5) = 6, if and only if, for any s E w we can find an automorphism c(,: N,(G) -+ N,(b) with the property: IX,(E)= g To prove the proposition note that the function (a, n) -+ card(N,(G)) is recursive. 0 Proposition 7. Let JZI be a highly recursive algebra. If the set Ld is not recursive, then d has infinitely many nonequivalent recursive representations. Proof. Let A0 c A1 c tEW,andaEA,,Nl(u) is c E w such that these there is s E o such that

. . . be a computable approximation of d c A,+1 and A, is a finite partial algebra. If elements belong to A,. By Proposition 6, these elements belong to LA,, for all p >

such that for any a, by A, then there if (a, b> E L&, then s.

Lemma 5. Let 4 be a total recursive function. Then there are infinitely many s E w such that for some G, be A,, the following property holds: a, 6E SS4,

@, @ $ LA.,

(a, b, E LA,,

, .

Suppose that the property is not true. Then there is s such that (x, y) 4 Ld, if and only if, for some s’ > s, (x, y)# LA,, and x, y E 4”‘. Hence, the set L& is recursive. A contradiction. We construct a recursive algebra 92 which is isomorphic and not recursively isomorphic to &, Step n + I. Suppose that B, has been constructed, and c(,: A,, + B, is an isomorphism. Compute all functions 4:’ ‘, 4;’ ’ . . . ,&T i. Case 1. Suppose that, there are no elements a, 6~ A,, such that

229

Recursive unary algebras and trees

and forsomejdn+

a, be &p;+ l

1.

B, + 1 and a, + I such that

We construct

B, = &+1,&+1

= A,+19 % = %I+1

let i < II + 1 be the smallest

Case 2. Otherwise,

element

such that:

1. (a,b)4LA,,(a,b)ELA.+,,a,b~6~r+‘. 2. These elements are from the same block in A,. 3. These elements are not marked by 0 j, with j < i. We construct

a finite algebra

B, + 1 to satisfy the following

conditions:

1. B” c &+1; 2. 4;” is not a partial isomorphism from A,, 1 to B,+ 1. 3. We mark the elements a, bby the symbol qi, and if some of these elements is marked by 0 j, where j > i, then we take the symbol j of these elements. is an isomorphism such that for any block T c A,, if 5, b$ T, 4. &l+1:.%+1+&+1 then a,(a) c CI,+ 1(a), for all UE Tn

A,+1.

Case 3. Suppose that a, bare from different blocks and the assumption above holds. If one of these blocks is marked by a j, for some j < i, then e construct B,+ 1 similar to the first case. Otherwise, we construct B,+ 1 similar to Case 2 and satisfy the following conditions: 1. We mark these blocks by n i. 2. We take all symbols nj of these blocks, with j > i. Let

By construction, Remarks.

%9is a recursive

1. Any element

of d

algebra. is marked

by symbols

qi, i E o, only at a finite

number of steps. Let x be marked by 0 i at step t. By our construction, at step t’ > t, this element is to be marked by qj, if and only if, j < i. This proves the remark. 2. Any block of the algebra & is marked by symbols nj, j E w, only at a finite number of steps. 3. For any a E A, The proof follows 4. The algebra B This remark also

lim,cc,(u) exists. from the previous remarks and the construction is highly recursive. follows from the construction of 59 and CI.

of CI,, n E o.

Thus the algebras & and @ are isomorphic. Assume that 4i is a recursive function with the smallest i, which is an isomorphism between & and 9Y. By the construction,

230

B. Khoussainov

we can find a step t at which the function 9 are not recursively isomorphic. Any two highly recursive representations isomorphic.

It follows

types [lo].

that this algebra

is to be violated. of the algebra has infinitely

Hence the algebras &’ are limited

many

recursive

~4 and

recursively

isomorphism

q

Theorem 4. Let W be an algebra. Any two highly recursive representations JZJ’and 69 of %?are recursively isomorphic, $and only i;f, there exists an r.e. set X x Y c w2 such that: 1. For any block in ~4 there exists the unique element x from this block such that (x,y)~XxY,forsomey~B. 2. For any block in ?8 there exists the unique element y from this block such that (x,y)~XxY,firsomex~A. 3. There is an isomorphism a: S@ --f S? such that X x Y c a. 4. The set L& is recursive. Proof. Assume that any two highly recursive representations d and a are recursively isomorphic. By the previous proposition, the set Lid is recursive. Let a be a recursive isomorphism from d to &?. Construct a set X x Y: Step 0. X0 x Y, = {(O, a(O)}. Stepn + 1. X,+1x Y,+l = X, x Y,, u {(x, a(x)}, where x = py (y is not connected with 0, x 1, . . . x,). The set X x Y = u,X,, x Y,, is as desired. Suppose that L& is a recursive set, and X x Y is an r.e. set which satisfies conditions l-3. We construct a recursive isomorphism a by steps. Step 0. Let x0 be an element such that 0 and x0 are connected, and (x0, y) E X x Y for some y. By first condition the element x0 exists. Let a0 = { (x0, yo)}. Step 2n + I. Let y be the smallest element from a such that y $ Paz,. By definition of highly recursive algebra, the block relation is a recursive set. Therefore one can effectively find the element y. Case 1. y is not connected with any element from pa2,,. In this case by the second condition of the theorem, there are elements x and y. such that (x, yo) E X x Y, and y is connected with yo. By inductive assumption, element x is not connected with any element from 6a2,,. Define a,,, 1 = aZnu { (x,yo)}. Case 2. Suppose that y is connected with z E pa2,,. Let x be an element such that a2Jx) = z. By inductive assumption, there is an 6, for any s, there is an isomorphism /? such that p(x) = z and aZn c /I. By Proposition isomorphism /&:N,(x) + N,(z). Let so be the smallest element such that y E N,,(z). Using Proposition 6, we can effectively find a number t > so such that the following holds: for any a E N,(x), there is an isomorphism

p:&

-+ 49

for which /?(a) = b if and only if there is an isomorphism Pr:N,(x) + N,(z) such that &(a) = b.

231

Recursive wary algebras and trees

Let

p be the

smallest

element

from

d

such

that

there

is an

isomorphism

fit: N,(x) -+ N,(z) for which p,(p) = y, and c(~,, c pt. Define c(~,,+1 = CQ,,u {(p, y)}. The next step is symmetric to the previous one. Define cx= U,c(,. By the construction, M is a recursive isomorphism from d to 98. 0 Corollary 4. Let JZI be a highly recursive algebra with n blocks, n E CO.Then thefollowing conditions

are equivalent:

1. The algebra d is recursively categorical. 2. The set L,> = {(a, b) 1there is an automorphism recursive in any recursive representation

cxsuch that cz(a) = b and a, b E A) is

of &.

3. There is a computable sequence 3xO&(a, x0), 3x, &,(a, x,), . . . of 3-formulas of the signature ( fO, . . , fk, a,, . . . , a,) defining the elements of the system & up to automorphisms; namely, (a) For any b E ~2 there is m E w such that dk&,,(a,, . . . , a,,, b). (b) For any m E cc) there is b E & such that &‘b&,(aI,. . . , a,, b). (c) For all m E w and b, c E A, &\4m(a,, . . . , a,,, b) A &(a,, . . . , a,, c), tfand only if there exists an automorphism CIsuch that cl(b) = c. Proof. We can suppose that G! is connected. Any recursive representation of d is highly recursive. Therefore the implication 1 + 2 is easy. The implication 2 -+ 1 follows from the previous proposition. We prove the implication 2 + 3. Step 1. Consider the set N,(a) = {a, cl,. . , c,}. Since Lo> is recursive, one can effectively find a number tl such that for all c, d E N1 (a), we have, (c, d) E L$, if and a formula &a, cl,. . . , c,) of the signature only if, (c, 4 E LN,+). Consider ( fi, . . , fkr a, cl. . , c,) which consists of the conjunctions of all atomic formulas and their negations, and which is satisfied by the model Nt,(aI). Define the formula 3x,. . 3x,g(a,

x1, x2,. . ,x,).

Step n + I. Consider the set N,, 1(a). A s a b ove one can effectively find a number t, + 1 such that, for any two elements a, b E N, + 1 there is an automorphism Mfor which a(a) = b, if and only if, there is an automorphism c(,+ r : N,, r(a) -+ N,, 1(a) such that ~,+~(a) = b. We can construct a formula which defines elements from N,, I(a)\N,(a) up to automorphisms of d. The implication 3 + 1 follows from Proposition 6 and highly recursiveness of Z&!. 0 Definition 8. A constructivizations n E 0.

system .d is strongly constructively stable if for any two v and ,Uthere is a recursive function f such that v(n) = n f (n), for all

If a model & has a continuum constructively stable.

number

of automorphisms,

then & is not strongly

232

B. Khoussainov

Let d be a highly recursive G! such that: 1. The group

algebra.

of automorphisms

2. Any block of dI Define ~2~ = d \&r. Theorem 5. Let d

Let dI

be a maximal subalgebra

of any block of ~2~ is trivial.

is a block of d. Then &‘2 is a subalgebra

be a highly recursive algebra

strongly constructively

of the system

of &. of signature

stable, if and only if; the following

1. The subalgebra &I possesses one of the following (a) The algebra d, is strongly constructively stable. (b) If a is another recursive representation of &, then such that for any block T (T’) in .zdl (in G?l ) there is the such that for some y E B (y’ E A), (x, y) E X x Y ((xl, isomorphism CI:dI -+ B, for which X x Y c CI.

(f ). The algebra

conditions equivalent

JZ? is

hold:

properties:

there is an r.e. set X x Y c o2 unique element x E T (x’ E T’) y’) E X x Y), and there is an

2. The subalgebra d2 has a$nite number of blocks and possesses one ojthe following equivalent properties: (a) The system d2 is strongly constructively stable. (b) For any a E JZ?~the group of automorphisms of the system (&,, a) is3nite. (c) The set R = ((x, y) 1f(x) = f (y) and there exists an automorphism CIsuch that M(x) = y} is finite. Proof. Let d be a highly recursive strongly constructively stable algebra. Then dr has a finite number of automorphisms. Otherwise, G? would have a continuum number of automorphisms. Moreover, &‘2 consists of only a finite number of blocks. Otherwise, we could construct a continuum number of automorphisms of &. Hence, in any recursive representation of ~2 subalgebras dr and d2 are recursive. Therefore the algebra d is strongly constructively stable, if and only if, the algebras &r and JZZ~are strongly constructively stable. We have to prove parts 1 and 2 of the theorem. If ~4~ is strongly constructively stable, then d is recursively categorical. Hence we can apply Theorem 5. We prove the second part. Let ~2~ be strongly constructively stable, but R has infinitely many elements. Case 1. There is an infinite subset B = {(xi, yi) 1i E w) of R such that, for all i # j, elements xi, xj, yi, yj are not comparable with respect to the relation < defined on the previous section. By definition of R, there exists a continuum number of automorphisms of -01,. Contradiction. Case 2. There is (x, y) E R such that the set {(a, b) 1a < f(x) A b Q f (x)] n R is infinite. In this case, the group of all automorphisms of the binary tree is embedded into the group of all automorhisms of d2. Again we have a contradiction. These two cases show that the set R is finite. Let R be a finite set.

233

Recursiae unary algebras and trees

Case 1. Suppose

that a does not form a cycle. Consider

the following

sequence

of

sets: J& = R(a),

Pi = WW)\W4

R, = R(f”(a))\R(a)

u R(f(a))

u.

. . u R(f”-‘(a)),

...

Any automorphism of the algebra (dz, a) is defined by automorphisms of models Ri, i E w. If Ri has an infinite number of automorphisms, then the set R is also infinite. Hence the number

of automorphisms

of Ri is finite. If CIis a non-trivial

of Ri, then there is a pair (x, y) E R n Ri. There is an automorphism cc c /s. Thus the group (J&‘~,a) is finite. Case 2. Letf”(a) = a. Consider the model (A2; P.r), with P/(x,

y) tf (f(x)

= y

A (x #

automorphism fi of d2 such that

a A . . . A x #f”-‘(a)),

Applying the previous considerations to the models R(a), . . , R(f”-‘(a)), we obtain that the group of automorphisms of (J&‘*,a) is finite. If the group of automorphisms of (d2, a) is finite, then L(d,) is recursive. Using the same method as in Theorem 5, we obtain that JZZ~is a strongly constructively stable algebra. 0

5. R.e. uuary algebras and a specification In this section

we mostly

investigate

Definition 9. An equivalence relation finitely generated algebra over q.

Let P be a subset (x,y)~q(P) The equivalence investigate algebras

r.e. unary

v] is jnitely

of w. Define the following

algebras. generated

equivalence

if there exists a recursive

relation

v(P):

if x=yorx,yEP. relation y(P) is r.e., if and only if, P is an r.e. set. First, over equivalence relations of the type q(P).

Lemma 6. Let P c o. If the set o\P generated algebra over q(P). Proof. Let B be an infinite bO < bl -c.

problem

recursive

and

is not immune, then there exists

subset of w\P. Let

a, < a, <. . ’

we

a jinitely

234

B. Khoussainov

be effective sequences of all elements two recursive functions:

in B and o\B, respectively. Define the following

The relation u](P) is a congruence on the algebra (o;A g). Hence this algebra is represented over r(P). This algebra is generated by bO. This completes the proof. Lemma 7. If o\P

is a hyperimmune

set, then any algebra over q(P) is locally Jinite.

Suppose that there is a recursive algebra d = (CO;gl,. . . ,fk) over q(P), and an infinite subalgebra 33 of d such that &3is generated by elements no, . . . , n,. Define the following strongly computable sequence of finite sets: & = {no,. . . , ns}, and i=n

D n+l=

fi(X)lXE

i

i=n

UDi,i=l,...,

k

i=O

\UDi. I

i=O

The subalgebra 93 is finitely generated and infinite. Therefore for any i E o,

For all i # j, we also have that Di n Dj = 8. Since &?is infinite, we obtain that

The set w\P is hyperimmune. A contradiction. We formulate the following questions: 1. Let P be a simple and not hypersimple set. Does there exist a finitely generated algebra over q(P)? 2. Does there exist a simple and not hypersimple set P such that every recursive algebra over q(P) is not finitely generated? Theorem 6. There exists an algorithm G with the following property. Let W, be a simple set. Let Do, D1,. . . be a strong table for o\ W,. Let (y, z) be numbers such that: (a) For any i E co, 4,(i) = card(Di). (b) W, = ((n, m)ln ED,>. Then WCC, y. z) is a simple set, and the equivalence generated. Proof. Let x, t be natural

numbers

v](WG(~,~,z~) is finitely

x > 1 and t > 1. We define a finite to the algebra ({ 1,. . . , xt);f; g), where

such that

algebra A,,. This algebra is isomorphic

relation

235

Recursive wary algebras and trees

f, g are unary

operations

defined

as follows: ify=ax

and

if y&(x,2x,.

g(y)

y + x

=

. . ,tx},

- 1)x + 1,. . . , tx},

if y E {(t - 1)x + 1,. . , tx}.

i Y

By the definition, cycles of length

if y${(r

a~{l,...,t-1},

f is a permutation

x. The number

of the set (1,. . . , xt}

of these cycles is t. The function

such that f forms g connects all these

cycles. Let PI be a simple set. Let FO, Fr, F2, . . . be a strong table for w\Pr . We construct a recursive algebra & of signature (f; g, h), wheref; g, h are unary symbols. Stage 0. Let F, have exactly m elements. Let FO = {b, . . , c}. At this stage we construct a partial algebra A,. This algebra satisfies the following conditions: 1. The wheref( 2. All by *. 3. The

algebra A, is isomorphic to the disjoint union Abm u . . u A,, u { - l}, - 1) = g( - 1) = - 1. elements of the set MO = {g”(xJ Ix, = min A,,, a E FO, IZE Z} are marked function

h(x) =

h is defined

as:

undejned

if x E MO,

X

otherwise.

Thus, the operations f; g are defined on the domain of the algebra AO. By the construction, the operation h is partial. The set FO is coded by the lengths of the f-cycles in do. The number off-cycles of length a is the number of elements in F1, where a E F,,. Stage n + 1. Let M, = (x 1x has been marked by * at stage PI}. Say that elements ml, m2 E M, are g-connected if there exists s E Z such that gs(ml) = m2 or g”(m2) = ml. By inductive assumption, this is an equivalence relation on M,. The function h is not defined on the set M,. Let M, = L1 u L2 v . . u Lk, where Li is a g-equivalence class, i = 1, . . , k. By induction, each Li has exactly m elements, where m is the number of elements of F, + 1. Let F, + 1 = {b < . . . < c}. Suppose that F n+2 has exactly t elements. For all i, i = 1, . . , k, x E Li, we construct an algebra A(Li) as follows. 1. The algebra A(Li) is isomorphic to the disjoint 2. For anydEF,+r, Adti is isomorphic to Adr. 3. A&) n A, = 0.

union

We have to define the function h on A(Li). Let Li = {xi, g(xi), the mark * off all elements in Li. We mark all

Abri u .

u Acti.

. . , gm-l(xJ}. elements of

We take the set

236

Min+l

B. Khoussainov

=

(gn(Xd) (It E 2, xd = min A&i, d E F,+ 1}. Define h(Xi)

=

max A&i,

h(x) = i

. . . , g(g”-

‘(Xi))

if

h(x)

if x E A,,

X

otherwise.

Let A(Li) n A(Lj) be the empty A n+l = A,, u A(L,)

max Acti,

Mi, + 1,

undejned

x E

=

set, i

#

j.

We define A,+ i:

u . . .u A(L,J

The construction at this stage is completed. Thus in this stage the operationsf; g are defined on any element of the algebra A,,+ 1. The operation h is partial. Moreover, the set F,,, is coded by the lengths of the S-cycles in A,, ,\A,. The number of f-cycles of length a is equal to the number of elements in Fn+2, where a E F,+ 1. Let & = UneoA,,. Lemma 8. (i) The algebra ~2 is recursive. (ii) The algebra d is finitely generated. Proof. Part (i) follows from the construction generators. The lemma is proved. 0 Define a subalgebra

of &‘. It is clear that the set A0 is a set of

(P;f; g, h) of zd which is generated

{-l}u{x~3nEPl(f(x)#x

A . ..Af”_‘(X)#X

by the set A\fn(x)=x}.

The set w\P is infinite. Indeed, by the construction, there is an x E A0 such that the S-length of the cycle formed by x does not belong to P1. Hence x $ P. The element x belongs to a subalgebra which is isomorphic to some A,,. By the construction, A,, n P = 0. Since F1 n o\P, # 8, there exists y E A,, such that h(y) forms anf-cycle the length of which does not belong to PI. Hence x1 = h(y)$P and x1 E AI\Ao. Continuing this procedure by induction on n, we see that for any n E co, there exists an element x, such that x, $ P, x, E A,\ A,, _ 1. Thus w\P is infinite. Lemma 9. (i) The set P is simple. (ii) Let Ro = Ao, Ri+ 1 = Ai+l\Ai,iEo. W\P. (4

Then Ro,R,,.

. . is a strong

table for

I;or any p E P, f(p), g(p), h(p) E P.

Proof. It is obvious that P is an r.e. infinite set. We have already proved that o\P is an infinite set. Let S be an r.e. infinite subset of o\P. Consider the set x A f”(x) = x}. By the construction of S’={n~3xES(f(x)#x A ... Af”_‘(X)# algebra &, iff”(x) = x A . . . A f(x) # x and x E S, then n $ PI. Moreover, in this

Recursive

unary algebras

231

and trees

case, if x E &\A,_ 1 and f”(x) = x A . . . A f(x) # x, then n E R,\P1. Since the set S is infinite, S’ is also infinite. This set is an r.e. infinite subset of w\P,. A contradiction. The first part is proved. By the construction, y belongs to P, if and a

term t of A ... Af “-l(x)

guarantees

that,

signature

(f; g, h)

and

only

if, there

exists

an x E@‘,

an integer m such that f(x) # x and g”(t(x)) = y. The construction

# x A f”(x) = x, It E PI, for any i there exists an element

from (Ai+ ,\Ai)\P.

Indeed,

it is

1)\o # 8. We suppose that x E Li and at the stage x = Xi for some i (see stage n + 1). Since F, + 1\ P # 0, by the construction, obvious

that Ao\P is not empty.

Let (A,\&_

n + 1 there exists an m E Z such that h(g”(xi)) E R,+ l\P. The third part of the lemma follows from the fact that the algebra a subalgebra

of &. The lemma

is proved.

Lemma 10. There exists a jnitely

generated

q(P)-algebra.

Proof. We show that J&’is an q(P)-algebra. Suppose that (a, unary operations, and P is closed with respect to these (g(a), g(b)), (h(a), h(b)) E u(P). Hence the operations f; g, h a finitely generated algebra and d/q(P) is its homomorphic generated.

The lemma

G? = (P;fT g, h) is

0

b) E v(P). Sincef; g, h are operations, (f(a),f(b)), admit q(P). Since d is image, d/q(P) is finitely

q

is proved.

To finish the proof note that the construction of r(P) has been uniformly effective on (x, y, z), where x, y, z are numbers stated in the theorem. Theorem 6 is proved. q Theorem 7. There exists a simple and non-hypersimple algebra over q(P) is not finitely generated.

set P such that any recursive

Proof. Let M be a simple set, and let D = (DO, D,, . . .) be a strong table for u\M. A function defined by f(x) = card(D,) is called a characteristic function for D. Lemma 11. Let P be a simple set. Zf there is ajinitely generated algebra over q(P), then there exists a strong table D for w\P, a characteristic function of which is primitive recursive. Proof. Let fi, . . . ,fs be recursive functions such that (w; fi, . ,fs) is a finitely generated algebra over u](P). Let mo, ml,. . . , mk be generators of the algebra. Define the following sequence of finite sets: MO = {mo,. . . , mkj,

J+fk+l = FOfd

y EF(M~), if and only Mi+ l\Mi # 8. Hence the sequence

where

M,\Mo,

Mz\Ml,

...

if,

u Mk,

3x E Mk3i < s(fi(x)

= y).

By

definition,

238

B. Khoussainov

is a strong

table for w\P. Define a functionf: f(t + 1) = (s + l)f(t).

f (0) = k This following

function

is primitive

two strong

recursive.

For

t, card(M,)


Consider

the

tables for o\P: M1\Mo,

Mz\MI> Mzt\Ms. . . and Using the first strong sequence, withf(2x) as characteristic

o\P

M3\M2,.

..

one can change the second one to a strong function. This proves the lemma. 0

Let G be the class of all primitive universal function for G. Define: f, = AxF(x,

any

recursive

functions,

table for

and let F be a recursive

y).

Letdb,4,,. . . be a standard Kleene numeration of all partial recursive functions. Let k + Fk be the canonical numeration of all finite sets. For any pair (f,, pi), we define a partial recursive mapping D&f,) from the set o to the set of all finite sets. The value O+,(f,)(j) is not defined if one of the following conditions holds: 1. There is k < j such that +i(k) is not defined. 2. There is k Q j such that f,(k) # card(F,,(,,). 3. There are different i, j < k such that FgiCkj n F4JCkj # 8. If none of these conditions

is satisfied,

then we put

By the definition of O+i(f,), this operation defined for any j, then the sequence

~&J(O)> D&)(l),

recursive,

and if O+,(f,)

is

...

is a strong table of pairwise disjoint proves the following lemma: Lemma

is partial

finite sets withf,

12. There exists an efSective procedure mappings of the form D4i( f,).

as characteristic

un$orm

on pi and f,,

function.

This

computing

all

partial recursive

Using

the effective procedure

constructed

above we prove the following

lemma.

Lemma 13. There exists a simple and non-hypersimple set P such that the set CO\P does not possess a strong table with a characteristic function from G.

Recursioe

Proof. partial

Consider recursive {(Oj,

is recursive. Die, Dil,.

unary algebras

and trees

239

an effective procedure computing a sequence D,,, III, mappings of the type D4i(jY). We suppose that the set D+i(f,))IDj

=

. . of all

D+i(f,)>

If Dj = O&,(f,), then let ~j =f,. the following set C:

Any

sequence

Di has

the

form

. . . Consider

C = ((it Di,)lVZ E Dix(2($()(0) + ' ' ' + $i(i))< Z)}. Define C’ = {(i, Oix) 1(i, Oi,) E C and for any y # X, if (i, D(v) E C, then the element (i, Oi,) appears in C before (i, Diy) appears in C during some fixed computation of C}. Define P =

IJ Di,. (i,D,,)

The set P is r.e. For any x consider . 2Ph(0)

(0, 1,2,.

+ .

the following

set

+ $x(x,,>.

By construction, card(P

. . ,2($,(O)

n (O,l,.

. . + $x(x))})<($o(O) +. . . + $Jx)).

+.

the set o\P is infinite. Suppose that there is a strong table functionf, = $i from G. There exists t such Ro,Rr,. . . for w\P with a characteristic that for all ZER,, z < 2($,(O) +. . . + t,bi(i)). Hence for some k, a pair (i, Rk) E C’. Therefore Rk c P. Contradiction. We obtain that P is a simple set. Using the recursive function defined as

We conclude

that

It/(x)= W(O) + . . . + tix(x)) one can construct hypersimple. 0 Lemmas

a strong

computable

table for o\P. It follows that the set P is not

11, 12 and 13 prove Theorem

As a corollary finitely generated

we obtain algebras.

the next theorem

Theorem 8. Let P be an r.e. set. Then: 1. If P is not simple, then there is a jnitely 2. If P is hypersimple,

7. which classifies

generated

algebra

then any algebra over q(P) is locally

r.e. sets in terms

over q(P).

jinite.

of

240

B. Khoussainov

3. The set P can be simple and non-hypersimple, and possess a finitely generated algebra over r(P). 4. The set P can be simple and non-hypersimple such that v](P) is not finitely generated. Definition 10. An algebra d is$nitely approximable if for any two different elements of & there exists a homomorphism on a finite algebra which separates these two elements. The next theorem is a characterization approximability of algebras over partitions

of simple of CO. The

sets in terms of finite proof follows the proof

from [26]. Theorem 9. Let P be an r.e. set. Then the following two conditions are equivalent: 1. The set P is simple. 2. Any algebra over q(P) is$nitely approximable. There is a recursive infinite Proof. Suppose that P is not simple. B = {bO < bl < bz <. . .> c co\ P. Define the following recursive function f:

set

The algebra (w/u](P); f) is represented over y(P). The elements bO and b, are not separable by any homomorphism on a finite algebra. Suppose that d = (w; fi,...,fk)be a recursive algebra over q(P). Let x and y be different elements of d. There exists an q(P)-closed finite recursive set D which separates x and y. Let 8 be a maximal congruence relation on this algebra separating all elements of D from all elements of w\D. The relation 8 is a co- r.e. equivalence relation. It is known that, if q is a co-r.e. equivalence relation, then the set {WY((%Y)~Yl+x~Y)~ is r.e. [S]. Hence the set (xI\JY((x,Y)~~+x

GY))

is a subset of w\ P up to a finite number and separates x and y. This completes Definition 11. An equivalence is not recursive. Precomplete

equivalence

relation

relations

Theorem 10. There exists a finitely

of elements. the proof.

Therefore q

r;4/8 is a finite algebra,

q is perfect if any proper non-trivial

are examples generated

of perfect equivalences

q-closed set

[S].

perfect r.e. equivalence relation.

Recursive wary algebras and trees

241

Proof. Let A, B be disjoint recursively inseparable sets. Suppose that, the set o\(A u B) is not immune. (Note that there exist such sets [39].) We define the following equivalence:

Lemma

14. Let q be an equivalence

relation. Let k be a recursive function

1. Zf i # j, then (k(i), k(j)) $ V, i,j E co. 2. The set [{k(x)lx E co}], = {yl!lx((k(x)x, 3. For any i, [k(i)],

= (yl(y, k(i))Eq}

Then the relation q is a finitely

generated

y)~?)}

such that:

is recursive.

is an r.e. set. equivalence.

Proof. By conditions 1-3, the set [k(i)], is recursive, i E o. Let cl < c2 < . . . be an effective sequence of all elements from w\ [ (k(x) I x E o}]~. We define the functions f; 9 as: f(x)

=

g(x) =

k(i + 1) x

if x E [k(i)],, if

x$C{Wlx~~)l,,

ci

if x E [k(i)],,

x

if x$[{k(x)IxEm}],.

It is easy to see that the algebra (w;f; g) is an q-algebra. This algebra is also a finitely 0 generated algebra with generator k(0). This completes the proof of the lemma. By assumption,

the set o\(A u B) is not immune. Let k be a recursive function such {k(x) 1x E co} c o\(A LJ B). We apply the previous lemma to the equivalence tl(A, B). The algebra (40, B);J 9) is well-defined over q(A, B). The functions J g satisfy the following condition: that

for all x E A, y E B,

(*)

f(x),

g(x) E A and f (y), g(y) E B.

We define:

v = sup{{(x,~)IDA4 Lemma

= A},

Vl((4D.VnA

z S>)>.

15. The relation y is a perfect equivalence.

Proof. Suppose that there is a non-empty proper q-closed recursive set R. The sets A* = {x(D, c A) and S(B) = (x10, n A # 8) are q-closed. Moreover these sets are q-equivalence classes. Indeed, let T c w. Then: (* *)

T is q-closed

iff T is q*(A) and q(S(B))-closed.

Using this property we can verify that A* and S(B) are r]-closed sets. By the definition of q and the sets A*, S(B), it follows that C(O)], = A* and S(B) = [{b}],, where Db = {b’}, b’ E B.

242

B. Khoussainou

We suppose that S(B) is a subset of R. Otherwise, we could take o\R. If x 4_R, then D, n B is the empty set. Moreover, for any D c A, the set (Ox u D) n B is also empty. Hence, if a E A and y E w are such that D, = D, u {a>, then y I#R and (y, x) E q. Define the set R’: y~R’iff

D,=D,u{y)andz$R.

Since R is a recursive set, R’ is also recursive. If b E B and D, = D, u {b}, then z E R. Consequently, b #R’. We prove that the set R’ is recursive and separates A, B. A contradiction. 0 Lemma 16. The relation q is a finitely Proof. Let L g be recursive functions

following 1. The 2. The Define

generated

equivalence.

such that the system (w; f; g) satisfies the

conditions: algebra (o/q(A, B),f, g) is finitely generated. functionsf; g have the property (*). recursive functions F, G, U: F(x) = the canonical index of the set ,f(Dx+ 1), G(x) = the canonical index of the set g(Dx+l), U(x, y) = the canonical index of the set D,+ 1 u D,+ 1.

Consider the recursive algebra M = (0; F, G, U). We prove that this algebra is an q-algebra. Let (a, b), (c, d) E q. Case 1: D, n B # 8. In this case, Db n B is also non-empty. By property 2 of the functionsf; g, we obtain that the sets f(DJ n R, g(Db) n 4

f(Db) n R, (DOu D,) n 4

g(Da) n 4 (Db U DA n R

are non-empty. Thus in this case by the definition of ‘I, we obtain

W4, W), (W), G(b)),(u(a, 4, UP, 4)

E VI.

Case 2: D, n B = 8 and D, n B = 0. In this case by the definition Db n B, Dd n B are empty. If D, is a subset of A, then the set Db also has to be a subset of A. we obtain that RD.), g(D,),f(Db), g(&) are subsets of A. It F(a), G(a),F(b), G(b) E A*. By the definition of q, since D, c [DO u D,], = [Db u D& = [DJ,. Thus, if D, is a subset of A, then

of q, the sets Hence, follows A,

we

that have

(F(4, R(b)), (G(a), G(b)), P(a, 4 U(b, 4) E V. Suppose that D, n (o\A) and D, n (co\ A) are non-empty. In this case, since = [Db], and [DJ, = [DJ,,, we have CD. u D,], = CDb u DA,. Hence,

[DJ,

Recursive

unary algebras and trees

V-J@,4, UP, 4) E r. Sincef(4, d4 and Cd~Jl, = CsVhJl,. Hence

are subsets

243

of the set A, [f(DU)],

= [f(DJ],

F’(4 F(b)),(Wh G(b))Erl. Thus we have proved Now we prove

that

that the system (w; F, G, U) is an q-algebra. the algebra

(0; F, G, U) is finitely

generated.

0 Let z E w and

D, = {a,. . . , c}. Then D, = D, u . . . u D,, where D, = {a}, . . . , D, = {b}. We suppose that 0 $ A u B and k(0) = 0, where k is a function from Lemma 14. Using that (o;f, g) is finitely generated with the generator 0, we find terms t,, . , tb of signature (f, g), such that t,(O) = a, . . . , q,(O) = b. Replace the functionsf, g in these terms by the functions of F, G, respectively. Thus we form terms T,, . . . , T, of signature (F, G), such that T,(l) = x,. . . , T,(l) = y. We know that, D1 = (0). Applying the function U we find a term T of signature (F, G, U) such that (T(l), z) E q. Thus, we have proved that the algebra (w/v; F, G, U) is finitely generated. If A and B were r.e. sets, then the relation ‘1 would be an r.e. equivalence. This completes the proof of Theorem 10. 0

the algebra

Definition symbols.

12. Let g be a finite signature consisting of a finite number Let b,, . . . , b, be constants which do not belong to G.

of functional

(i) An algebraic specijication is a pair (I, R) such that I is a finite set of identities of signature 0, and R is a finite set of formal equations on symbols bI, . . . , b,. (ii) An algebra d is specified by (I, R) if d is the initial system defined by (I, R). (iii) An r.e. representable system & has a positive solution for a specijication problem if there exists an r.e. representable finite enrichment d* of s$ which is specified. By the definition, it follows that if & is specified, then & has a finitely generated representable enrichment.

r.e.

Definition 13. An r.e. representable algebra d is absolutely locally finite if any r.e. representable finite enrichment of d is locally finite. Note that, if an r.e. representable system d is absolutely a negative solution for the specification problem.

locally finite, then it has

Theorem 11. There exists an r.e. representable absolutely locally finite algebra. Therefore this algebra has a negative solution for the specijication problem. Proof. Let C, be a finite algebra sequence

which forms a cycle of length n. We suppose

Ci, Cz, C3,. . . has the following

1. The set ((x, y)lx E C,} is r.e. 2. Foranyi#j,CinCj=@. 3. W = UjCj.

properties:

that the

244

B. Khoussainov

Hence an algebra set. Define

99 = (o;f)

the relation

(X,Y)EI]

defined

by uiCi

is recursive.

Let P be a hypersimple

q: v 32(2EP

t-) (x=y

A X,YEC,)).

Lemma 17. (i) q is a congruence on $8. (ii) The algebra d = B/q is recursively enumerable. The proof of this lemma Lemma

follows from the definition

of q.

18. Algebra d is absolutely locally$nite.

Proof. Let &‘* be an r.e. finite enrichment of d. Let c* be the signature of d*. Suppose that there is a finite subset (a,,, . . . , a,} such that the subalgebra of ZZ?* generated by this finite set is infinite. Since &‘* is an r.e. algebra, there is an equivalence 8 such that &* is an d-algebra. Let a, = 8(p,), . . . , a, = O(p,), for PO,. . . >pn. Let T c CO.Then F(T) = {yj3x,.

. .x,~T3g~a*(g(x~,...,x,)=y)}

and F,(T)

= {f’(x)lx

E F(T)}

We define an effective sequence SO =

maxFl((p0,.

u T. so, sl,.

. . . Let

..,P,)),

let F’ = F, ((PO,. . . , p,}), and let Sn+1- -

max{F,(F”)}.

Using the construction of d, and the definition of the sequence so, sl,. . . , we with obtain that the sequence so, si, . . . majorizes the set o\ P. This is a contradiction 0 the fact that P is hyperimmune. Lemmas In [30] constructed.

17 and 18 prove Theorem using

these

ideas,

11.

examples

of non-specified

groups

and

rings

are

6. Algorithmic dimensions of algebras Let J&’be a recursive model. A relation R on JZZis stable if it is invariant with respect to the automorphisms group of &. Let St(&) be the set of all stable relations on d.

Recursive

unary algebras

245

and trees

For each S E St(d) we formulate a problem Ps which is called the algebraically correct algorithmic problem for S [7, 151: Find an algorithm that, given nl, , nk E co, decides whether (nl,. . , nk) E S. Definition 14. Let d be a model and let di,

d2 be recursive

d. Let S E St(&) and let Si, S2 be images of S in d1 and d2, the characteristic function for D. 1. The recursive representation s4i is algebraically

(r.e.) representations respectively.

of

Let ch~ be

reducible to d2 (&r Galgdz)

if

the problem Ps2 is decidable, implies the problem Ps, is also decidable. These representations are algebraically equivalent (szfI -alg~2), if di
Gal&l. 2. The recursive representation d, is program reducible to dz (dl dPdz) if there exists a partial recursive function $ with the property that if $,, a partial recursive function with Kleene number x, is a characteristic function of the set S2, then &+) is a characteristic function These representations

for S, are program

equivalent

(szI1 mpd2),

if &i

dPd2

and

dz

G&l. 3. The recursive representation dl is uniformly reducible to d2 (&‘i Qud2) if 9 such that F(ch,,) = chs,. These there exists a computable operation representations are uniformly equivalent (szII -U&z), if &‘I du&2 and AZ
E {R, alg, p, u>)

is the algorithmic dimension “stable”, if dim@(d) = 1. The dimensions (*)

dim,,,(&)

defined

of d.

The model

d

is O-categorical,

above satisfy the following

< dim,(d)

< dim,(d)

or alternatively

inequality:

d dim,(d).

V.A. Uspensky and A.L. Semenov formulated the problem for the class numerated models which contain both the recursive models and r.e. models

of as

246

B. Khoussainov

[15]: Is there a model for which one of the signs d

subclasses by < ?

We investigate

this problem

in (*) can be replaced

in this section.

Theorem 12. For any cardinal number n E w v {co), there exists an algebra XI such that dim,(&)

= n.

Proof. Let S be a family one-to-one

numberings

of r.e. sets with

[lo].

exactly

n non-equivalent

computable

Let c( be one of them. Let i E w. We construct

a partial

algebra -01i = (A u o u (ai};fi). Suppose that A n w = 8, card(A) = W, ai#A The partial operation5 is defined by the following rule: Let C#I : A + o be a one-to-one mapping and p4 = cc(i). Then

u

CO.

if a = ai, if n E w, n > 0 and

J(a) =

a = n,

if a = 0, if aEA.

Lemma 19. 1. -c4i possesses a recursive representation. 2. The set cc(i) coincides with the set {t

1!lx(J;:‘+l(x)

=

ai

and x has two preimages}.

The proof follows from the definition of di. Let G3i = (Bi, di), i E w, be a computable sequence of partial algebras such that 2Yi 2 &i, Bi CI Bj = 0, w = u Bi. Let a $ u,Bi. Define the following unary operation: if x~(ao,aI,...}u{a}, f(x) =

ii(X)

if

X E Bi\{U.}

I .

Let &(cL) = ( UiBi U {a);f ). Lemma 20. (i) The algebra G!(M) is recursive. (ii) Let fi be a computable one-to-one numbering of S. Then 1. Algebras &(or) and d(p) are isomorphic. 2. If cIand /3 are equivalent, then algebras d(a) and d(p) are recursively isomorphic. 3. Let g be a recursive representation of d. There exists an algorithm which constructs a computable one-to-one numbering y of S such that 9Y and d(y) are recursively isomorphic. Part (i) follows from the construction the numberings CIand j3 are one-to-one. Let f be a recursive function such isomorphism from &(cI) to &‘(/I).

of A!‘(M).Part (ii) 1 follows from the fact that that

c1= /?f: Then

there

exists

a recursive

241

Recursive unary algebras and trees

Let a be a recursive

representation

R, = {x(Bkf(x)

# x A f2(x)

of &. The sets = x}

and R, = {x 1x has two preimages} are recursive.

For any n E RI we construct

an r.e. set y(n):

y(n)={tl3x(f’+‘(x)=n,x~R~}. The mapping

y is a one-to-one

computable

~8 and d(y) are recursively isomorphic. Lemmas 19 and 20 prove Theorem 12.

numbering

of S. By the construction,

0

Let a,, . . . , a, E A, n E co. Then, dim,(&)

d dim,&&, a,, . . . , a,).

Corollary 5. For any n > 3, there exists an algebra SZ! and a,, .

, a, E A, s < n such

that dim,(d)

= n < dim,(d,

a,, . . . , a,) = 2”(n - s).

Let dl,. . . , dn- 1 be pairwise disjoint to &(a). Let

Let aI EAT,.

algebras.

Suppose

that they are isomorphic

. . , a, E A,. Then it is easy to show that

dim,(&)

= n

and

dim,(&,

aI,.

. . , s) = 2”(n - s).

In the next result we use the language of constructive systems. Let v be a constructivization of a model d and let s;, qy2, relations on J&’ such that the following conditions hold:

be a sequence

1. For any j, rJ is an equivalence relation on &j such that, if x E dj, nj(x) E St(A); where Ilj(x) is an equivalence class containing x under ‘lj.

of

then

2. The set {(n, m, k) 1(v(n), p(m)) E yky} is recursive. 3. There is a recursive function f;: u oi -+ o such that for any n E oi, we have A”(n) = card(ql(v(n))). 4. There is a p.r. functionf; : ( u coi)2 + o such thatf;(n, m) is defined if and only if (v(n), v(m)) E ye; for somej, and if cxE Aut(A), then ctv(n) # v(m). Moreover, iff;(n, m) is defined, then f;/(n, m) is a Giidel number of an j-formula F(x) such that Wc4,v) bF(v(n))

A 1 F(v(m)).

Proposition 8. Let (A, v) be a constructive system, and let ‘I;, II;, . . . be a sequence which sutisjes conditions l-4. If the constructivization p is program reducible to v, then ,a is uniformly reducible to v.

248

B. Khoussainoo

Proof. Suppose that p is a p.r. function which program reduces the constructivization p to v. By condition 2, there is a recursive function t: uo’ + w such that for any x E oi, function of v -’ yy(v(x)). Since the function p program reduces #Jf(,, is a characteristic p to v, the function function p and recursive.

c#&, is a characteristic function of the set pL- ’ vr(v(x)). Using the condition 2, we obtain that the set f(n, m, s)I (p(a), p(m)) E yl,Y}is

We define an enumerable operator v.LetD c Uw’x{O,l)beafinitesetandp,,..

which uniformly

reduces constructivization

p to

. , pt be all natural numbers such that D n copI x (0, l} # 8,. . . ,DnePx{O,l}#@. Let D,=Dn~Psx{O,l), where s= 1,. . . , t. It is sufficient to consider the case when s = 1 and p = p1 . We define the following sets: (s, E) ED*, if and only if, (s, E) ED and E = 1. Let D** = D\D*. The equivalence relation q; defines disjoint subsets RF, . . . , RF and R:*, . . . , R:* of D* and D**. The canonical enumeration of all finite sets and properties l-4 allow us to compute canonical indices of D*, D** and RT, . . . , R:, R:*, . . . , Rz*. Let D* = ((sl, 1,. . . , (slz, l)} and D** = ((n,, 0), . . . , (nk,, O)}. We define the following sets: RF = V-‘(viV(Si)) Rj** =

v-’

x

(yz(v(nj))

PI = {Sl,

n D*,

(1)

x

. . , a,>,

P2 =

=

P2 n

X

{

l}\D*,

= v-l (Vi(vnj)) x l\D**,

. . . 2 Q’>,

p* = v lPT*

V-‘(yi(V(tlj)))9

fori=l,..., Iandj=l,..., k. Let F, be a formula of the signature the following

{n,,

l

V-‘(y~(V(Si))

TR?* .I

(0) n D**,

P* = P1 n V-‘(v]i(V(Si))), Pj**

1 R* =

1 (Y~(v(si)))\

p1

2

= V-‘(V]i(V(Hj)))\P2

of the model d with Giidel number

n. Define

formulas: undefined

if n E Pt? or f;(n, s) is not defined for some s E 1 Pi”,

Y(1, i, n) = G Y(l, n) =

Ff; (n,k)

otherwise if Y(l , i, n) is defined

VY(l,i,n) i undefined

for all n E P,?,

otherwise.

For each P*, i = 1,. . . , I’, define F’(PT): if Y(1, i) is not defined, F’(P*)

= { ;(x,

l)l&,,{,(X)

Let F’(D*) = UfL 1 F’(PT). Y(2, j),j = 1,. . . , k, we put:

I n a similar

if Y(2,j) F’(PT*)

=

= 1 A AkY(1,

i)p(x)}

way defining

otherwise. &formulas

Y(2,j, n) and

is not defined,

;(x. O), $Jptn,)(X) = 1 A A 1 Y (2, j)p(x)}

otherwise.

Recursive unary algebras and trees

the

T c uw'x

function (0, 1). Define

condition l(D)

a characteristic

definition of F, constructivization

F(T)

249

4, if D E St(&) is a finite set, then F’ transforms to a characteristic function of p-‘(D). Let

= u L c rF’(L),

where

L are finite sets. By the

all

F is an enumerable operator which p to v. This proves Proposition 8. 0

Let JZZ’ be a constructivizable

uniformly

model and r] = (9 1, q2, . . . ) be a sequence

on & such that for every i E o and x E -02’, rli is an equivalence rlitx)

relation

reduces

the

of relations on d’

and

E st(d).

Definition 16. A pair (JZ!, v) is Iocally constructive hold:

stable if the following

1. There is a constructivization v such that: (a) The set {(n, m, i) 1(v(n), v(m)) E vi} is recursive. (b) There is a recursive function fr”: uw’ + w such fi”(x) = card(qi(x)). 2. For each constructivization

that

p there is a p.r. functionff(n,

defined if and only if (p(n), p(m)) E vi for some i, and a(p(n)) # sr(p(m)). Moreover, F’;(,,) is an j-formula such that (d, ~)~F~;t,,m)(0))

for all x E &“, i E CO, m) such thatf$(n, if E E Aut(A),

m) is then

A iFf;(n,m)(Or)).

3. For any two constructivizations that P/t(V(n)) = ?/t(/l(fyp(n))

Theorem 13.

properties

v, /J of d

there is a recursive

functionf,,

such

for all n E oi, i E W.

Zf(d,ye) is locally constructive stable, then the model d is program stable.

Proof. Let v be a constructivization

from condition

1. Let ,u be any constructivization

of d. We prove that they are program equivalent. By condition 3, there is a recursive function fvpsuch that for all i E o and x E wi, V]i(p(n)) = qi(vfpy(n)). Hence (p(n), v(m)) E r/i, if and only if, (v( fpY(n)), v( f,“(m))) E vi. Thus for each constructivization p the set C(n, m, i) IM4, Am)) E Vi> is recursive. Using Definition 16, it can be shown that for each constructivization ,u there is a recursive function f/‘: u oi + COsuch that for all i E CO,x E oi, we have f/‘(x) = card(qi(p(x))). We construct a recursive function which program reduces the constructivization p to v. Let 4; be a p.r. function. We define a p.r. function 4 which effectively depends on x. A computation of this function on any input z occurs according to the following rules step by step: 1. Compute f@“(z). 2. Find all elements of the set qi(v( fpY(z)). Let q(v( fVy(z)) = {v(zl), v(z2), . . , v(z,)}.

3. Compute &(zi), . . . , c$i(zk). If there is t d k such that 2 d &.(z,) or &z,) defined, then let 4(z) be undefined. 4. If &(zi), &(z, + 1) = . .

. . . , &(zk) . =

&(z,)

=

je{p+ 1,. . . ,k}. 5. IfforsomeiE(l,...,

are 0,

computed,

then

compute

p),andjE(p+

l,...

&(z,) f;(zi,

= . . . = &.(z,) = 1, zj),

where

is not and

i E (1,. . . , p},

, k}, f;l(zi, Zj) is not defined, then let

4(z) be undefined. Let f;(zi, zj) be defined for all i E { 1, . . , p), and j E {p + 1, . . . , k}. We construct the following formulas:

@l =

A F&“(z,,;,)

v . .

j=p+l

v

j=&+ FfJ% 1

2,)

and

@z = j=lAFg(z,,zp+j)v

P

... ” A

6. Let Q1 and GZ be constructed.

j=

FfT(zj9zk)

1

Then:

These instructions are uniform in x. Consequently, there is a recursive function fsuch that d(z) = $),X,(z) for all z E oi and x E o. Therefore, if 4X is a characteristic function for v- ’ (S), then cbf (Xjis a characteristic function for pL- ’ (S), where S E St(&). Hence ,u is program reducible to v. Similar arguments show that the constructivization v is program Corollary

reducible

to ,u. This completes

the proof.

6. lf a pair (,Oe, q) is locally constructive

0 stable,

then

the model

x2 is

a uniformly stable model.

Applying Proposition 8 and Theorem 13 to the pair (&, q), we conclude that & is a uniformly stable model. Many known examples of recursively non-categorical models possess recursively categorical enrichments by relations invariant with respect to Aut(&). Theorem 13 above can be applied for constructions of models which do not have recursive categorical

enrichments

in the sense pointed

out above.

Corollary 7. Let (&, q) be a locally constructive stable, but not a recursively categorical c St(&). Then the enrichment (~4; Si; i E w) is also not recursively

model. Let {Si)ioo categorical.

Indeed, let v be a constructivization of &* = (&; Si; i E CO), and let p be a constructivization of &. There is an effective sequence x0, xi, . . . such that for any i,

251

Recursice unary algebras and trees

&., is a characteristic function for v -i (Si). Since the model & is program stable, there is an algorithm which program reduces the constructivization v to p. Hence, p is also a constructivization

of d*. If v and p are not recursively

for &, then they are not recursively We give an application

equivalent

of Theorem

Let a E A. a is particular

equivalent

constructivizations

constructivizations

13 to unary

for d*.

algebras.

if ~(a) = a for any ti E Aut(d).

Proposition 9. If a highly recursive unary algebra has afinite number of blocks and each block has a particular

element, then this algebra

is uniformly stable.

Proof. It is sufficient to prove this proposition when the algebra d is connected. d be a particular element. Define the following sequence of subsets:

Let

N,+I(~) = {h(d)Ii = 1,. . ,n}

No(d) = (4,

u aG~cd,{.f’;1(4i= 1,. . . ,n> m

By the definition, we have A = UiNi(d). Since d is highly recursive, there is a recursive function h such that for any i E co, h(i) = card(Ni(d)). It is obvious that for any i E co, Ni(d) E St(d). Thus we have the following sequence: N,(d) c N,(d) c N,(d) c . . . . By Lemma 4, for any two elements b, c E A there is an automorphism a such that cc(b) = c, if and only if, for any m whenever b, c E N,,,(d), then there is an automorphism cc,:N,(d) + N,,,(d) such that cc,(b) = c. We conclude that the set # p(m)}

R = ((n, m) I Vx E Aut(,d)(av(n))

is r.e. There is an algorithm L such that (n, m) E R, if and only if, L(n, m) is defined. Moreover, if L(n, m) is defined, then L(n, m) is a Giidel number of an 3-formula F(x) such that & bF(v(n)) Let

A iF(v(m)). C onsider

~1 = {(x,y)13i(x,yENi+I(d)\Ni(d))}.

rl = (Ye, q2, ~3,.

the

following

sequence

.) where

vm = {((a,, . . . 2ant), lb,, . . ,b,)l(al,b,),...,(a,,b,)E~l}, The pair (&, v]) is locally

constructive

stable. Hence d

2
is uniformly

Corollary 8. There exists an algebra ~4 such that 1 = dim&‘)

stable.

< dim,(&).

Proof. There exists a highly recursive connected algebra ~2 with a particular such that the set Lid is not recursive and dim,(d) = w [33]. By Proposition dim”(&) = 1. This proves the corollary. 0 Theorem categorical.

14. There exists an algebraically

categorical

algebra

0

element 9 above

which is not program

252

B. Khoussainov

Proof. Let P be a simple set. Define the following

family of r.e. sets:

SP = (01 u #+$P} A computable

numbering

v(i) # CO,v(j) # CO,i fj,

v: CO-+ SP is good

Lemma 21. There exists a countable family

if for

any

i, jE o

such

that

we have v(i) # v(j). number of non-equivalent

good numberings

of the

Sp.

Proof. Define

the computable

numbering

y:

if i E P, otherwise.

;]

Then y is a good numbering of SP. Let P,, c PI c . . . be a computable sequence such that P = u,Pi, Pi c (0, 1,. . . , i} and card(Pi\Pi,) = 1. We define an approximation of y. Namely: if t < i, y’(i) =

0

if iEP,, if i d t and

e {i} We construct

a good numbering

i $ P, . CC

Step t + I. Suppose that we have a set D,, a number /?,:{O, 1,. , t} -+ (0,. . . ,rr} such that: 1.

R = (0,. . . >4>, Pt{O,. . . 9t\Pt)

P,(PJ = D,

= (0,

rl, and

a mapping

and

. t r,}\R;

if Y, < i, 2.

a’(i) =

if iED,,

0 e y’(j)

if p’(j) = i.

WedefineD;+,,~+,,r,+,,cc,+,. Case 1. Suppose that P,+ 1\ P, = 8. We put fir+ 1 = D,, ft

cC,+I(i) =

e

r = r1 + 1 and

if Fr+I < i,

0

if iED,+,,

y”‘(j)

if i = P;+I(j).

Case 2. Suppose that is P,+l\P,. We define fit+, = D, u {m}, with m = rl + 1, if i = t + 1; or m = /If(i) if i < t. We put Ft+ 1 = rt + 1. The functions fit+ 1 and E,, 1 are the same as above.

Recursine mar-y algebras

253

and trees

from (0,. . . , t + l}\P,+, . Compute Let nb’ 1 < . . . < n:+ 1 be all elements t+1 40 ,‘. , ?&+l. Subcase 2.1. Suppose that i0 = ~y(y16&+’ = Bf+, and _#b”($+‘) 1) is not defined. In this case we set r,+l = i;t+l,Dt+l = fi,+l,j,+l = fit+l and CA’+~= $+I. Subcase 2.2. The number i0 is defined. Then let D,, 1 = fit+ 1 u { bt+ 1(n&+‘)} and rftl

= F;+, + 1. Define 13,+,(i) B,+l(i)

if

= i

rt+

i

n:o+l,

otherwise,

1

if rl+l < i,

8 R’+‘(i) =

i #

cc)

if iED,+l,

y’+‘(j)

if fl,+l(j)

= i.

We set cc(i) = U,“‘(i). By the construction, c( is a good numbering of Sp. This numbering to y. Any two non-equivalent good numberings of SP are limitedly the lemma

follows from known

results

[lo].

is non-equivalent equivalent. Hence

II

Using a good numbering

of SP, we construct an algebra JZZ== (A u o;f; g), with A n w = 8. Let a,, a,, u2,. . be an effective list of all elements of the set A. Let f(a) = g(u) = a for all a E A. Let FF?~be the algebra isomorphic to the cyclic algebra with exactly n elements We construct an algebra d, by steps. Step n + 1. Let {jl,j,,

. . . ,A> = cr”“f”(l(?l

where s0 is the largest

+ l))\P(l(n

+ 1))

s < n + 1 such that l(s) = I(n + 1). We also suppose

aO(l(t))c cc’(&))

c c2(l(t)) c

. .

is a computable sequence such that @(l(t)) = U,c~(l(t)), On the first j, + . . + j, numbers of the set o\A,, disjoint algebras

to Vj,, . . . , %‘jk, respectively. rules:

which are isomorphic J&?n+1 by the following 1.

A,+l=A,UBj,U.“UBjk;

2.

f”“(j)

g”+l(i)

This completes

=

=

f”(i)

if i E A,,

fj,(i)

if i E Bj,,

g”(i)

if ig A,,,

u~(,+I)

if iEBj,

the step.

that

1 < r < k,

u

’

u

B,.

I, m E co. effectively

We define

define

pairwise

a finite

algebra

254

B. Khoussainov

Define

Lemma 22. 1. The algebra &‘, is recursive. 2. If CIand b are good numberings of Sr, then algebras d, and d, are isomorphic. 3. If a and /I are equivalent good numberings, then recursive algebras &‘, and &, are recursively isomorphic. 4. For any recursive representation .!A9of algebra d,

one can eflectively construct

a good numbering u such that the algebras %7and JZZ,are recursively isomorphic. The proof of this lemma

is similar

to the proof of the Theorem

12.

Lemma 23. The recursive dimension of &, is equal to w. The proof follows from the previous lemma and the construction. Let & be an abstract algebra isomorphic to da. Define the following set B(A) = {x 1f(x) = x, g(x) = x}. We call the elements of this set nodes. The set of nodes is intrinsically recursive. An element a E A realizes o if the block containing a is infinite, and a realizes k if 1. a does not realize o. 2. There exists b such that g(a) = g(b), and b forms anf-cycle

of length

k.

Lemma 24. Let &I and -01, be recursive representations of &. Then ~2, and -Qzzare recursively isomorphic, tf and only if, &I and &‘* are program equivalent. Proof. If d1 and d2 are recursively isomorphic, then they are program equivalent. Let ~&‘i dpAz via $0, and let A2 dpAl via $i. Let B,(A), B,(A) be images of B(A) in -c4r and JS!~, respectively. We construct a mapping &B,(A)

+ B,(A).

If we construct 6 effectively, and there is an isomorphism from di 6, then there exists a recursive isomorphism extending 6. Consider B”(A) = (x 1x E B(A) and x “realizes

to AZ extending the set

o”}.

Let By, B$ be images of B” in SZZ~and dZ, respectively. These sets are r.e. and non-recursive. Let tl and t2 be one-to-one recursive functions such that ptl = By(A) and pt2 = B,“(A). Let n,, n,, . . . be an effective sequence of all elements from B1 (A), and let be an effective sequence of all elements from &(A). mo,ml,. We define 6 by a “shuttle” method.

Recursiae wary algebras and trees

255

Stage s. Let n, be the first number in B,(A) on which the function 6 has not been defined. The computation of J(Q) occurs by the following algorithm: Step k. Suppose

that &(n,) has not been defined on the step k - 1. If t,(k) = ns then

Put

6Cn.T) = t2 Suppose

S-l PLY /j &Cni) i=O

((

t2(Y)

+

>)

that t,(k) # n,. In this case compute

where: 1. rE{xIx
of the characteristic

functions

of the sets (ns}

and {I), respectively. Suppose

that r. is the least number &~ti,,(ro)

= 1

and

such that

4&&s)

= 1

Define &n,) = yo. If there is no such Y, then move to step k + 1. At some step k, $(n,) is defined. Due to the fact that &‘1 -p&Z via $. and $1, we obtain that there is a recursive isomorphism LXfrom &I to -c42 which extends 6. El Lemma 25. The algebraic

dimension of d

is 1.

Proof. We prove the lemma for those SE St(&) which are subsets of A. For other stable relations, the proof of the lemma is similar, but with more cumbersome combinatorial arguments. Let S E St(&) and S c A. Define S1 = {xIxEB(A),xES}

and

SZ =S\S1

The sets S1 and S, are recursive in dl, if and only if, S is recursive in dl. Therefore it is sufficient to consider two cases. Case 1. Let S = S1. Then B”(A) c S or B”(A) n S = 0. The set P is simple. Therefore, if S is recursive in dl, then either S is finite or B”(A)\S is finite. Hence the set S is recursive in any recursive representation of &. Case 2. Let S = Sz. Suppose that there does not exist an element which realizes o and belongs to S. Suppose that S is recursive in G?‘~. Using the simplicity of P, we obtain that S is a finite set. Hence in any recursive representation of d the set S is recursive. Let T be an infinite block. Then T is recursive in any recursive representation of &. If S is recursive in &‘, then A n T is also recursive. Let Hs = {n 1there exists x which forms a cycle of length

n and x E S n T}.

256

B. Khoussainov

If S is recursive thef-cycle

in &, then the set Hs is also recursive.

formed

by x belongs

If x realizes o, and the length of

to Hs, then x E S. Define the following

RI = (x E S( the length

of theS_cycle

Rz = {x $ S 1the length

of the f-cycle formed

formed

sets:

by x does not belong

to Hs)

and by x belongs

to Hs}.

If S is recursive in di, then by simplicity of P, the sets RI and R2 are finite. Hence we can suppose that RI = Rz = 8. We obtain that x E S, if and only if, the length of the 0 f-cycle formed by x belongs to Hs. The proof of Theorem

14 follows from Lemmas

22-25.

Cl

Theorem 15. There exists an algebra which possesses direrent program and uniform dimensions. Proof. Let N, = (2kj k E N, k # 0} and N2 = {2k + 1 I k E N}. For every recursive one-to-one functionffrom N1 to Nz such that card(N,\rang(f)) = o, we define the following algebra A(f) = (0; h,.):

h,(x) =

0

if x E Nz u {O},

f(x)

if XE Ni.

It is clear that A(f) is a recursive algebra, and for every injection f’ from Ni to N, such that card(N,\rang(f’)) = o, the algebras A(f) and A(f’) are isomorphic. Let d be an abstract algebra which is isomorphic to A(f). Lemma 26. The algebra d is recursive, and has a countable number of non-uniformly equivalent recursive representations. We prove the second part of the lemma. Let S = {x E Al hrx # x and 3y( h,( y) = x)> and let B, C be disjoint r.e. sets which have different Turing degrees. Suppose that rang(f) = B, rang(f’) = C andi_/-’ are recursive functions. Then, there are recursive representations &i, J@‘~of d such that S1 = B and SZ = C. Suppose that J$, and -tpZ are uniformly equivalent. There are enumerable operators F, F’ such that F(c(S,)) = c(S,) and F’(c(S2)) = c(S,), where c(R) is a characteristic function of R. Hence the sets B, C have the same Turing degree [39]. This contradicts our assumption. It is well known that there is a countable number of r.e. Turing degrees [39]. recursive Consequently, s$ has a countable number of non-uniformly equivalent representations. Lemma 27. The representations.

algebra

d

has exactly

two non-program-equivalent

recursive

25-l

Recursice wary algebras and trees

sets from St@‘). Let x = (x1,. . . , x,) E A”. For any

Proof. We need to characterize

xi (1 d i < n) we define tl (Xi), t2 (Xi, t3 (Xi), tb(Xi): t,(Xi)

=

0 iff

t,(Xi)

=

1 iff t(xi) # 0 and xi$S;

t,(Xi)

=

2 iff

Xi E

tl(Xi)

=

3 iff

tl(hf(Xi))

tz(Xi)

=

0 iff Vj (j E { 1,2,.

tz(Xi)

=

(PI,

hf(Xi)

=

Xi;

S;

p2,.

2;

=

. , Pk)

iff

. . , ll}\(i}

(Xi

=

A

tj(Xi)

=

0 iff

(tl(Xi)

=

0

V

+

Xpl

Xi

A

#

Xj);

’ ’ ’

A

Xi

=

XJ

Vj(xi=xj+xj=x,,

t,(Xi)

=

1

v

tl(Xi) = 3

V

..*

(tl(Xi)

V

-Vj(i tJ(Xi)

=

(Y1,

. . . , Tk)

iff

(tl(Xi)

=

2 A

tb(Xi)

tb(Xi)

=

=

0 iff

(k,

(tl(Xi)

3. . . ,

=

0

k,) iff

V

tl(Xi)

(t,(Xi)

Vj

h,(X,,)

Vj

3

=

(h,(Xj

1

=

= A

A

V

xi

tl(Xi)

V

(tl

A

(h,(Xi)

(hf(xi)

=

(Xi)

=

A +

=

2

# j-+hf(xi . .

j

=

xj=x,);

Y1

A V

h,(X,,)

...

# =

V

j

Xj));

Xi =

i-J);

#

Xj));

2

=

3 -+ Vj (i # j +

=

xj

Xi

v

. .

A

hf(Xi)

h,(Xi)

=

Xkl

=

xki

A

+

j

k - 1 v . . . v j = k,)).

=

We denote t(xi) = (t,(xi)y tZ(Xi), ts(Xi), td(Xi)) and t(x) = (t(xl), . . , t(x,)). For all x, y E A”, we write t(x) = t(y), if and only if, there is an automorphism tl of the algebra d such that U(X) = y. Let t(x) be an n-type of x E A”. Since, for every n the set of all n-types is finite, we obtain that A” = A 1 u . . u A,, where A,, . . . , A,,, are orbits. Hence any set from St(&) is a union of a finite number of orbits. If di and dZ are recursive representations and Si, S2 are recursive sets, then recursive algebras d1 and d2 are recursively isomorphic. Consequently, these representations are program equivalent. Let &i and dZ be recursive algebras such that Si, S2 are not recursive sets. We define

Ro =

{xIh&J

R, = {xlhfx R2 = A\R,

= =

x>,

h;(x) A

h,x

#

x>,

u Rz.

Let R,(l) and Ri(2) be images of Ri in ._M’~and &, respectively. We construct an algorithm T which program reduces &‘i to A?~.

258

B. Khoussainov

Step 0. Let nPE (R,)(2), p = 0, 1, 2. Define

VU, n, x) =

This function

6: (x)

if x E (R,)(2),

&(x)

if x E (R,)(2),

I 4: (x)

if x E (R,)(2). TI(n) such that

is p.r., and there is an algorithm

!P(l, n, x) = &-,,n,(x) If 4” is a characteristic function for E2.

for all x, n E 0.

function

for Ez and E E St(&), then $bl(,,) is a characteristic

Step N + 1. Suppose that TI(n), . . . , T,(n) have deen defined, and if E c A”, E E St(A), 1 < s d N, and & is a characteristic function for Ez, then 4;, is characteristic function for E, . Let E E AN+‘. Denote El = {(xi, xz,.

. .,~~+~)13n,m(n#m

&

=

{@1,x2,.

. . 9 xN+l)

E3

=

{(XI,+,.

A

EE\EI

. >x N+l)EE\El

b={h,xz,...,

x,=x,)},

Igi(t,(xi)

=

O)>,

“E~l3i(t1(xi)=3

XN+l)EE\El

U

EZ

A

‘JE~II~(~I(X~)

t3(hf(X))=O)},

=

3

A

t3(hf(xi)

f

O)},

E5 = E\EI v Ez v E, v Eq. Let E,(l), Ei(2) be images of Ei in &i and dzr respectively. For any i E { 1,2,3,4,5}, we have Ei E St(&), if and only if, E E St(A). Consider the following sets: E’ =

{(x1,x1,.

. . ,

E2 = {(x,, x2,.

E4 =

{(x,,.

. . ,Xi,.

. . ,Xi,.

. . ,xN)E

El)},

. . , XN)I~~(X)=X~(X,XI,...,XN)EE~ V

E3 = {(XI,%,.

xN)/3i(x1,.

(X1,X,.

..,XN)EEZ

. . ,~N-1)~~~~(~I,~.

. . ,xN)13x(x1,X2,.

V

. 3%. . . ,X,.

‘..

. . ,Y,.

V

(X1,X2,.

..,XN,x)EEz)},

. . ,x)~E3)},

. . ,XN)E&}.

Again, we define E ‘( 1) and E ‘(2), respectively. According to these definitions, for all i E (1, 2, 3, 4}, we have E’ E St(&) and E’(2) is recursive if and only if E E St(&) and Ei(2) is recursive. Suppose that (x,, . , x~+~)E E, and E5 E St(&). Then for each PER,, if y#x,,. . . ,y#xN+l, then (y,xl,. . ,xN+~)EE~. In the other case Sz where E, = Ri\id(A, N + l), would be a recursive set. Consequently, we obtain that E, = 8 or id(A, n) = ((x,, . . . , x,) E Al Vij(xi = Xj)}. Thus E, = R,\id(A, N + 1). Let ml,. . . , mN+ 1 be different numbers from RI, and let E = &#$“. The above considerations show that there is an algorithm which transforms any natural number

259

Recursive unary algebras and trees

n to a 5-tuple (x1, x2, x3, x4, x5) such that SC&, = E’, . . . , Sqb,, = E4, S&, = ES, and if 6”NC1 is a recursive function, then & is also a recursive function i = 1, . . . , 5. is a characteristic function of the set E,, then for Moreover, if E E St(d) and 4:” function of E’(2). By each i E (1,2,3,4}, E’ E St(d) and 4X, is a characteristic are characteristic &,VCXI), . , 4~~~~~~~ p.r. function:

induction, following

y(N

+ 1, x5,

=

tl,.

for E’, . . . , E4. Define the

. . , tN+l)

&Q(~I,. i

functions

. ,mN+l

)

if trz R,\id(A,

N + l),

otherwise.

0

There is an algorithm

TN+ 1 such that:

4 NTN+,:(X,)(rl>. 1t N+l)=Y(N+l,x5,tl,...,tN+l). If E, E St(A) and 4_ is a characteristic function N+l is a characteristic function of 4 T,+,cx.j

of E,(2), then by definition which program

d2.

completes unary

the dl

to

proof of Lemma algebra

computations

[36]

reduces

of TN+ 1,

such that @ has exactly (id}. 9? has exactly [7]. Let d* = JY u V. Using combinatorial prove d* has exactly

program

r.e.

models

are

constructed.

trees Definition tree is jnitely blocked if any block of this tree is a finite set. (b) A tree is strongly jnitely blocked if there exists a number n E COsuch that a number of elements in any block of the tree is less than n. Proposition 10. Let 9 be a tree. Iffor any n E w there is a finite than n elements, then 9 is not recursively stable. Proof. To prove this proposition

we need the following

block in 9 with more

lemma.

Lemma 28. Let B,,, B1, . . . be a sequence ofjnite trees such that for any n E w there is a block Bi,, with more than n elements. Then there is a subsequence Bj,,, Bj,, , . . such that card(Bj,) < card(Bj,+,), and B, is embedded into Bj,,, for all i E CO.

260

B. Khoussainov

Proof. Define

a partially

ordered

(a,b)d(c,d)iff For

any sequence

set P = (w2;

(0; d)bad~

< ), with

A bdd.

aO, al, . . . of this partially

ordered

set, there

is a subsequence

UiO,Ui, , . . such that Ui, d Ui~+,, for all k E co. The partially preodered set S(P) = (S(P); < ) possesses the same property [6]. Any finite tree can be embedded in the partially

ordered

set P. This proves the lemma.

q

There is a sequence of finite blocks CO, C1,. . . of 9 such that for any i E co the number of elements in Ci+ 1 is more than the number of elements in Ci, and Ci is embedded into Ci+ 1. of 9. The set q = {(x, y) ( x and y are LetDo,D1,. . . be an effective approximation connected} is r.e. For any x E w, define (x}(n) = {x} u {yl (x, y) appears in r] during n steps of computation of II}. For a set Y c w, put Y(n) = u {y>(n). Let C be a block in 9 such that there is a sequence

of blocks of the model 9, with CO E C, card(Ci) < card(Ci+r), and Ci n Ci+l = @Iand Ci+i has a submodel which is isomorphic to Ci. We construct recursive models & and &? which are isomorphic Do c Di c 02 c . . . be an approximation of 9. Step 0. Let A0 = BO = Die, where iO is the smallest i such that C c Suppose that dN and BN have been constructed. Step N + 1. Let N + 1 = (k, t). Let AN z DsN and BN be isomorphic Case 1. There is no sequence TO 4 T1 4 . . . G T, of blocks of the model card(TO) < card(T,)

< . . . < card(T,),

and

Subcuse that

l)n&=(b,

2.1. There is no sequence

T,, “=C,

card(T,)

andTk4{Ij)(N+1)foranyj=1,...,N+1. In this subcase we set &+I

i=

= AN+~,BN = &+I.

1,. . .,N+

AN+ 1 = A,,

and

Let

Di. to A,. AN such that

AIV such that

1.

To 4 . . . 4 T, of blocks

< . . . < card(T,)

to 9.

TO z C.

In this case we set AN+ 1 = AN, BN+ 1 = Bslv. Case 2. There is a sequence To 4. . . 4 Tk of blocks of the model card(TO) < . . . < card( T,), and C E T,,. Let lo, I,, . . . , l,, 1 be a sequence of first numbers such that {Zi}(N+

CiG Ci+r for

< card(lj(N

we

of the model

AN such

+ 1))

construct

BN+l

such

that

261

Recursive unary algebras and trees

Subcase

2.2. There

To% T1 4 . . . 4 Tk of blocks

is a sequence

T, E C,

card(T,)

< card(T,)

. < card(T,)

<

model

< card( { lj}(N + 1))

and Tk 4 {lj} (N + 1) for some j E { 1, . . . , N + l}. Let T,i~ Tli4 . 4 Tki, i = 1, . . . , Y, be the list of all sequences AN

of the

with the property:

A N+l

such

that

for

T z

card({ li,}(N + l)), < . . . -c minTkr. we set AN+1 =

any

AN

U

i~fl,.

. . , rj

C, Tki4 { li,}(N (_):=

1 {li>

(N

+

we

have

card(Tio)

+ 1) for some

of the model

< . . . < card(Tki)

i’, and

<

min T,, < min Tkl

1).

We construct BN+ 1. Compute 4:’ ‘, . , . , &t,‘:. If any of these functions is not a partial isomorphism from AN to BN, or for any i E (1,. . . , r>, Tki #S#+‘, with BN+l such that BN+i E AN+l, and j= 1,2,. . . , N + 1, then we construct BN

C

BN+~.

Let i be the smallest natural natural number such that: (a) (b) (c) where

number

which is not marked

by 0; let j, be the smallest

Tkjo c 84”’ ’ ; Tkjo is not marked, or marked by El,, where i < s; the block in BN which contains 4i( Tkj,,) is not marked i < s’.

We construct

BN + 1 to satisfy the following

or is marked

by

q,,,

properties:

1. BN c BN+l and BN+l = AN+l. 2. The function 4i is violated on Tkj,, We mark contains

the number

4i( Tkjo) by

4i(Tkjo). Let S? = U,B, Remarks.

q,>,

of i by sign 0, the block

Tkjo by lJi, and the block which

qi. We take the signs q,, 0; off Tkjo and the block containing

and d

= u, A,. The model 99 is a recursive

model.

0

1. For any two blocks T from d and T’ from 98 the sets (s) T is marked

by

{sl T’ is marked by q,} are finite. Let T (T’) be marked by El, on step t. If T is marked by sr on step t’ > t, then by construction s’ < s. Hence these sets are finite. By this remark, the models & and J% are isomorphic. 2. For any i E co, i is marked by q only on a finite number of steps. 3. The models JZZand 9 are not recursively isomorphic. Let 4, be a p.r. function with smallest x which is a recursive isomorphism from A’ on @. By remarks above there is a step t such that on any step t’, where t’ > t, none ofthe blocks of the models A,.\A, and B,.\B, is marked by qo, q1, . . , qx-l, and none of the numbers 0, 1,. . . , x - 1 is marked by 0. Since 4x is a total recursive function, there is a step N + 1 = (k, i) > t, a sequence Tom T1 cs . . . 4 Tk 4 { l}(N + 1) of blocks of the model AN+1 such that Ti c AN+l\At, Tk c c#$+‘, and

262

B. Khoussainov

To 2 C. By instructions

of the step N + 1, the function

4X has to be violated.

A contradiction. Corollary 9. Anyfinitely blocked recursively categorical tree is stronglyjinitely blocked. Let & be a model. Consider S define the preorder < :

the following

B1 < B2 iff B1 is isomorphic

set S = {B 1B is,a block of &‘}. On the set

to some submodel

of B2

Let Bi - B2 iff B1 < B2 and Bz d Bi. The model (S/ - ; < ) is a partially ordered set. If the model d is strongly finitely blocked, then (S/ - ; < ) is a finite model. In the proof of the next theorem we use the language of constructive models. Theorem

16. For a strongly jinitely blocked tree d

the following properties are

equivalent: 1. The tree d is recursively stable. 2. The tree d is an algebraically stable system. 3. If B,l - < B2/ - and B,/ - # B2/ - , then either B,/ -

or B,/ -

is ajinite

set. Proof. The implication

&I-

1 + 2 is clear. We prove the implication

d&l-,

and both B,/ - , B2/ A,=A\B1/-

B,l-

2 + 3. Suppose

#&I-

are infinite

sets. Let

uB,l-

and

AZ=B1/-

vB,/-

The models Al and A2 have constructivizations vO, po, respectively, ~0 ‘(B,l - ) is recursive. sequence of finite models such that: Let Lo, L1,. . , be a computable 1. Foranyi,j,ifi#j,thenLinLj=@. 2. For any i, Li is isomorphic

that

such that

to B1 and u Li = (2n 1n E w};

Letfbe a recursive function such that rang(f) is not a recursive set. It is clear that u LrCi, is an r.e. and non-recursive set. Let k = card(B*) - card(B,). Consider a computable sequence Ro, RI,. . such that for any i, card(Ri) = k and u Ri = (2n + 11n E co}. By our assumption we can construct a sequence Go = (L,,,, of finite models

u Ro; Pf,“, . . . , Pin),

such that the following

G1 = (L,,,,

holds:

1. For any iEW, we have Gin B2. 2. For any i E co, the model Lrci, is a submodel

of Gi.

u RI; P”o,. . . , Pin), . . .

Recursive

Consider

263

mar): algebras and trees

the model

By construction

this model

is recursive

and isomorphic

to AZ. The set of all blocks

which

are isomorphic to Br is not recursive. Thus, the model A2 has is not recursive. Let a constructivization vr such that the set v-l BJ c( = vO @ vr, fi = vO @ pLo[4]. Then c( and /I are constructivizations of the model d. the set fl-‘(B,/

By construction Consequently,

d

- ) is recursive,

is not an algebraically

and the set cc-‘(B1/

- ), is not.

stable model. A contradiction.

3 + 1. Let B,/ - , . . . , B,/ - be all elements of the set be finite sets. Suppose that B,/ - , . . . , Bk/ - are ..,B,l-

We prove the implication

Sl - , andletB,+,/-,. infinite sets. Using the assumption,

we suppose

that B,, I/ -

= . . . = B,/ -

= 8. Let

F,(x), . . . 3FL(x) be atomic formulas such that for any i the set of all models satisfying ~Fi(x) possesses the minimal element ,4(3x F,(x)), i E (1, . . , k}, and E B1,. . . , A@xF,(x))

A(3xF,(x))

cc Bk.

For every a E A there is b such that a and b are connected, Moreover, if &‘bFi(a) for some i, then by assumption & &t

Fj(a)

and d+Fi(b)

for some i.

we obtain that any two constructivizations This completes the proof of the theorem.

v, ,LJof the

for all j, j # i.

Thus using these properties, model d are auto-equivalent.

Corollary 10. If a strongly jinitely blocked recursive dimension of this tree is CO.

tree is not recursively

stable,

0 then the

Proof. Let 4 be a recursive function such that rang(4) and rang(f) have different T-degrees, wherefis the recursive function used in the proof of the theorem above. If we repeat the proof of this theorem with respect to the function 4, then we construct a constructivization y which is not auto-equivalent to 8. 0

8. Subalgebras, automorphisms Define the following

and homomorphisms

two sets P, and Qm:

q E P, iff q is an r.e. equivalence relation each class of which is infinite

on w

and p E Q. iff p is recursive permutation without finite cycles.

on w

264

B. Khoussainov

Let p E Qw. Define an equivalence (4 4 E p -

relation

p_:

iff 3s(p”(n) = m v p-“(n) = m).

p _ E P,.

By the definition

Lemma 29. yeE P,, if and only iJ there is p E Qw such that p _ = q. The proof of this lemma

is easy.

Let p E Qw. We define an algebra

d,

= (A;&).

There exists an effective procedure which for any n E o constructs partial algebra dE4,= (A,,;f#) which is isomorphic to ({a,,

1,.

a finite unary

>n + 1, b,, c,, 4,);f)

where f(a,) is not defined, and f(1) = a,, f(2) = 1,. . ,f(n) = n - 1, f(bJ = n, f(dn) = n,f(c,) = d,. In the algebra d, only one element has to preimages, and the elements b,, c, do not have preimages. We define an algebra &, as follows: 1.

A,,=UA,,

where

AinAj=~

foralli#j.

Lemma 30. The algebra &, is recursively representable. The proof follows from the construction. Lemma 31. In any recursive representation of d,

the set {a,,, a,, . . . } is recursive.

Proof. Let 99 be a recursive representation of d,. By construction of dP, there is an algorithm which for any s gives a sequence of numbers tk, tk_ 1, . . . , to such that: 1. fj(ti = ti+l), and ti has two preimages, 2. f;(s) Therefore

i = 0,. . . , k - 1;

= tk. s E {a,, aI,.

. . }, if and only if, to = s. This proves the lemma.

0

Lemma 32. For any recursive representation W of &,,, the set {(s, m) 1s = a, and s E B} is recursive. Otherwise, we could effectively find a number Proof. Ifs+z{ao,ul ,... },thens#a,. t in the recursive representation 93 such that fp(t) = s, and the root K(t) of t is finite.

265

Recursioe wary algebras and trees

Let y be an element f”+‘(y) = s. 0

of K(t) which has two preimages.

11. 1. For

Corollary

effectively “recover” 2. The algebra

any recursive

the recursive

representation

permutation

&,, is recursively

complete

B of .d is r.e. if in any recursive

Moreover, recursive

11. For any v] E P,,

r.e. subalgebras

representation

subalgebras

there exists an algebra

of SI! is isomorphic

this lattice coincides algebra

-Pe,, one can

lemmas.

For any algebra ~2 the set of all r.e. complete to the operations of union and intersection. Proposition

of the algebra

p.

93 of d is complete if each block of g is a block of &‘.

18. 1. A subalgebra

2. A subalgebra 99 is an r.e. set.

g

s = a, if and only if

stable.

The proof follows from the previous Definition

Then

of ~2 the subalgebra

is the lattice with respect

d

suck that the lattice of all

to the lattice of all r.e. q-closed

with the lattice of all complete r.e. subalgebras

which is isomorphic

sets.

of any

to d.

Proof. Let p E Qw such that p _ = v. Let d be the abstract algebra isomorphic to d,. The map q(n) + [a,], where [a,,] is the block of ~4, containing element a,, induces the desired isomorphism. 0 Corollary complete

12. There subalgebra

exists

an algebra

with infinite

of which is recursively

number

representable,

of blocks

any proper

if and only if; it has a finite

number of blocks.

Let p _ be an r.e. equivalence such that each proper p _ -closed r.e. set is the union classes [S]. Then the algebra dP satisfies the of a finite number of p _ -equivalent corollary. Theorem 17. 1. There exists an algebra d suck that: (a) The cardinality of the group Aut(&) of all automorpkisms (b) dim,(&) = CO. (c) Let 93 be a recursive automorpkisms

representation

The

group

of all recursive

of &I is trivial.

2. There exists an algebra

SJZ suck that:

(a) The cardinality of the congruence (b) SI is recursively stable. (c) Let 93 be a recursive trivial.

of ~2.

is continuum.

lattice of ,c4 is continuum.

representation

of &‘. Every recursive

congruence

of g

is

266

B. Khoussainou

Proof. 1. Let ‘1 be a perfect

equivalence

relation

[S].

Consider

the

following

equivalence: (x, Y) E Y iff (lx, [Y), (TX, ~YJ E rl. This equivalence relation is perfect [S]. Define the set S by IZE S iff (In, 0) E v]. This set S is r.e. y-closed and consists of an infinite number of equivalent classes. Let p E Qw be such that p _ = y. Consider the algebra &,. Define a complete subalgebra /3 = (B; f) of the system d,, with B = Ui,,R(Ui). By the previous proposition, 93 is an r.e. subalgebra of d. Consequently, the system W is recursively representable. Let -c41 = &, u &?‘, where 93’ z &I and the intersection of the domains of $3’ and d, is empty. Then &I is a recursively representable algebra. There are infinitely many disjoint isomorphic blocks of &I. Hence the cardinality of the group Aut(d,) is continuum. For any automorphism p and for any a E AI, we have either /3(a) = a or /?(a) # a A P’(a) = a. Let 93 be a recursive representation of zzI1 such that there is a non-trivial recursive automorphism /I of 39’.Then we can divide the domain 93 in two parts B1 and B2, with B1 = {x 1p(x) = x} and B2 = {x 1b(x) # x}. Define subalgebras WI and 3%?*with the domains as B1 and B2, respectively. Since /3 is recursive, these algebras are also recursive. By the last two lemmas, it follows that the sets ~1 = {ilUiEB,}andy, = (ilUiEB2) are r.e. These sets are y-closed. By construction, y1 u y2 = o and y1 n y2 = 8. We obtain that y1 is a recursive set. But y is a perfect equivalence. A contradiction. Let S be a family of r.e. sets possessing infinitely many non-equivalent computable numberings. Let c( be one of them. As in Section 6, we construct the algebra d(a). From the construction, it follows that the group of automorphisms of this algebra is trivial, and dim,(&((cr)) = w. Let d = &I u d(x), with &I n A(a) = 0. Then J&’possesses a recursive representation. From the previous arguments, it follows that this algebra satisfies the first part of the theorem. 2. Let q be a perfect equivalence, and p E Q. such that q = p _. On the algebra &‘, define the following operations g1 and g2: g,(x) =

g2(x)

=

y

if f,(y) = x and

X

otherwise,

Y x

iff(y)=x

x$ {b,, bI, .

. . . },

and x~{co,~o,~l,~l,. . .>,

otherwise.

The algebra & = (A,;f,, gl, g2) construction, this algebra is recursively Define

has a recursive representation. By the stable. Let B be a complete subalgebra of &.

(x, y) E q iff x = y or x, y E B. This equivalence is a congruence on z&‘.Since the number of complete subalgebras of & is continuum, the cardinality of the lattice of all congruences of & is also

Recursiw unary algebras and frees

267

continuum. By the definitions off,, gi, g2, we obtain that if y is a congruence on d, then every y-class is a union of suitable complete subalgebras. Hence, if there was a non-trivial recursive congruence y, then it would be possible to find a partition of COon r]-closed recursive subsets. It is a contradiction with the fact that y is perfect. This completes the proof of the second part. q

Acknowledgement

The author acknowledges many very helpful discussions with his colleagues Yurii Ventsov, Nadim Kassimov, Sergey Fedoryaev and Ruzmat Dadajanov. He thanks Professor Djavat Hadjiev, who has been supporting his research in logic. He expresses many thanks to Professor Anil Nerode who encouraged him to write this paper, read the manuscript carefully, pointed out a numerous number of mistakes and suggested possible improvements. He is indebted to Professor Sergey Goncharov, for this paper would not be written if he did not lead the author’s research.

References [1] [2] [3] [4] [5] [6] [7] [S] [9] [IO] [11] [12] 1133 1141 [15] [16] 1173 [18] [19]

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