Annals of Pure and Applied Logic 25 (1983) 47-58

North-Holland

47

AUTOMORPHISMS OF SUBSTRUCTURE LATTICES IN RECURSIVE ALGEBRA

David R. GUICHARD

Vassar College, Poughkeepsie, NY 12601, USA

Communicated by A. Nerode

Received 10 July 1983

Metakides and Nerode [4, 51 introduced the study of the fully effective vector space V, and the lattice L(V,) of r.e. subspaces of V,. Further study has shown L(V,) to be quite different from 8, the lattice of r.e. sets, although notable similarities have also been discovered. Kalantari and Retzlaff [3] showed that there are maximal subspaces of V, with the property that no automorphism of L(V,) takes one to the other; this is in contrast to Soare’s result [8] for maximal subsets of iV. Kalantari and Retzlaff discovered infinitely many classes of maximal subspaces (the ‘k-thin’ spaces, for all integers k) with the property that spaces from different classes have lattices of superspaces which are clearly non- isomorphic. For the class of O-thin spaces they coined the term ‘super-maximal’.

A. Nerode and our advisor T. Millar suggested to us the question, “Given any two super-maximal spaces V and W, does there exist an automorphism F : I.( Vm) += L( V,> with F(W) = V?“Initially we conjectured that the answer was “Yes”. Kalantari [2] had already provided a proof that there are continuum many automorphisms of L(V,), which made this plausible; it also appeared that a proof would be at least as difficult as Soare’s proof.

In fact, there was a mistake in Kalantari’s proof. As we show in Theorem 1, every automorphism of L(V,) is recursive in a strong sense. It is then relatively simple to construct supermaximal spaces W and V such that no automorphism exists taking W to V. We do this (and somewhat more) in Theorem 2, and extend the result to the k-thin spaces, for any k, in Theorem 3.

Terminology

The space V, is a vector space of countably infinite dimension over a finite or countable field. By fully effective we mean that V, is presented as a recursive subset of N, that the underlying field E is likewise recursive and that vector addition and scalar multiplication are recursive. In addition, we require that the

0168-0072/83/$3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)

48 D. R. Guichard

dependence relation be recursive, that is, {x 1 D, is a dependent subset of V} is recursive (recall that D, is the finite set with canonical index x). The r.e. subspaces are simply those subspaces which are r.e. subsets of N. We say that an r.e. subspace W is recursive if there is an r.e. subspace V such that W + V = V andWnV={O}.(W+Vistheweaksumof WandV,i.e.,{w+v]wEW,uEV}, where of course ‘w + 2)’ is the vector sum of w and v.)

For any vector spaces, XC3 Y = Z will mean that X+ Y = Z and that X n Y = (0). For any set I we will denote by I” the smallest subspace of V containing I, i.e., the span of 1. We will also write (x,, . . . , x) for {x0, _ . . , &}*. In section two we use (x,, . . . , x,,) to denote the smallest subalgebra (of a given boolean algebra) containing x0, . . . , x.

We write W =* V (W is almost equal to V) if there is a finite dimensional space F such that W + F = V + F.

We fix a recursive, l-l and onto pairing function on N which we denote ( , );

the context should keep this from becoming confused with the use of angle brackets to denote span. We usually write the customary ej instead of Q,. We will often construct sets and spaces in stages and refer to fixed enumerations of r.e. spaces and sets. If I is a set, we denote by I(s) or p the subset of I constructed by stage s or enumerated after s steps in some fixed enumeration. For a space W, W” will usually mean (IS)* for some basis I of W. In particular we fix a simultaneous enumeration {W, 1 e E w} of all r.e. subspaces of V, given by We = It, where {I, 1 e E o} is a simultaneous enumeration of the r.e. independent subsets of V,. Fix also an enumeration cf, ( e E o} of all partial recursive functions on N.

Our main construction will involve the use of markers or windows. That is, at stage s in the construction we may have some integer (often thought of as a vector) denoted by cS. One should imagine c to be some particular ‘window’ or ‘marker’ which at stage s ‘contains’ or ‘marks’ the integer c’. If the lim, cS exists, we will also denote the limit by c, confusing the ‘window’ with its final contents.

A note about our presentation of priority arguments: We usually stipulate that something is to be done if there is a requirement which needs attention. In general this is a C, question since it requires a potentially unbounded search. We may assume all searches to be bounded by the stage of the construction, i.e., search only those requirements with number less than the current stage of the construction. This will not affect the constructions or the proofs, but they are easier to follow without the extra detail.

A classical theorem (‘The Fundamental Theorem of Projective Geometry’; see [l]) states that every automorphism of the lattice of all subspaces of V, is induced by a semilinear transformation on V,. (A semilinear transformation is a map

Automolphisms of substmcture lattices 49

F: V,+ V, with an associated automorphism f : E + E such that F(v + w) =

F(v) + F(w) and F(av) = f(a)F(v).) It is easy to see that every automorphism of L(V,) is the restriction of an automorphism of the lattice of all subspaces (L(V,)

contains all one dimensional spaces; any permutation of these which preserves dependence induces an automorphism of the full lattice of subspaces). In fact, we can say more.

Theorem 1. Every automorphism of L(V,) is induced by a recursive semilinear

transformation.

Proof. The essential feature of the proof is that vector addition serves as a recursive pairing function which commutes with semilinear transformations.

Let 923 = {vi 1 i E w} be a recursive basis for V,. Suppose F is a similinear transformation inducing a given automorphism of L(V,). Denote by f the field automorphism associated with F, and enumerate the underlying field E of V, as

(0 = uo, Ul, u2, . . .}, where E may be finite or infinite. Suppose that under the given automorphism

b2i I i E W> + WI,

(v2i+l I iEw)+ W2,

tv2i+V2i+l I iEo)-+ W3,

Cv2i+l + V2i+2 1 i E 0) -+ W4.

for some r.e. spaces W,, . . . , W,.

We will describe algorithms for computing F(q) and f(4) for all i. Then to compute F(v) for any v E V,, we effectively write v = C qvi, q E E, and then F(v) = C f(q)F(v,). In describing the algorithms we will use the recursive enumer- ations of the spaces WI, W,, W,, W, and also the single vector w. = F(Q); denote

F(G) by w,. To compute wl, search for a y E W2 such that w,+ y E W,. Then for some ai

and b,

wO+Y=wO+~aiw2i+l since y E W,,

= C bi(w2i + wzi+l) since w,+ y E W,.

Recall that the representation of a vector in terms of a given basis is unique. Comparing these two representations of w,+ y, we see that

wo = bowo, so bo= 1,

aowl = bow,, so a, = 1

and for i>O,

0 . wai = biw2,, so bi = 0,

qw2i+l= biwzi+l, so ai =o.

Hence y = wl.

50 D.R. Guichard

In general, suppose we know wZkcl. Then we search for and find a y E W, such

that W2kfl +yE W,. Then

(*) WZk+lf~ aiwZi = c h+l(W2i+li w2i+2)

and as above we have

W2k+l = bk+lW2k+l> SO bk+l= 1,

ak+lW2k+2= bk+lW2k+2, so ak+l = 1

andfor ifk

0 * wzi+l= bi+lW2i+l, SO bi+l=O,

ai+lwzi+2 = bi+lw2i+z, SO Ui+l= 0.

Of course a, = 0 since w0 does not occur on the right in (*). Hence y = u&+2. In case we know w2k, we proceed to find W2k+i in similar fashion. In this way we can enumerate in order wO, wl, w2, . . . .

If the field E is finite, then f is of course recursive. Otherwise, suppose that under the given automorphism of L(V,)

(2)2i + UiZlZi+lI i E 0) ~ W,.

Search for a y E W, and a u E E such that y = w2i + uwzi+i; this is effective since we can compute w2i and wZitl. Then u =f(q). q

Metakides and Nerode defined maximal subspace analogously to maximal subset:

Deiinition. A c V, is a maximal space if (1) [Vm:A]=m, (2) VWEL(V,)(WI>A implies W =*A or W =* V,).

Recall that Soare [8] showed that for any two maximal subsets of N there is an automorphism of 8 which takes the first to the second. In contrast to this, Kalantari and Retzlaff [3] have shown that there are maximal subspaces of V,, A and B, such that no automorphism of L(V,) takes A to B. To do this they introduce supermaximal and k-thin spaces. The definition of k-thin below is only slightly different from theirs.

Definition. A c V, is a supermaximal space if (1) [Vm:A]=a, (2) VWEL(V,)(WIA implies W=*A or W=V,).

Definition. A c V, is a k-thin space if (1) [V,:A]=m, (2) VWEL(V,)(W=JA implies W=*A or [V,:W]sk), (3) LlWEL(V,)(W~A&[V,:Wl=k).

Automorphisms of substructure lattices 51

The supermaximal spaces are exactly the O-thin spaces. It is clear that no

automorphism can take a k-thin space to a j-thin space unless k = j. We will show

below that k-thin spaces exist for all k, so there are infinitely many orbits among

the maximal spaces.

It is natural to ask whether for every A, B k-thin there is an automorphsim of

L(V,) taking A to B. Remmel [6] has constructed supermaximals in every r.e.

degree and so we have the following.

Corollary. There exist supermaximal spaces A, B such that no automorphism of

L( V,> takes A to B.

A stronger version of this corollary can be proved by constructing supermaxi-

ma1 spaces A and B while meeting requirements Q, which assert that if f, is a

recursive semilinear transformation of Vm, then f,(A) # B. By mixing this strategy

with one similar to Remmel’s we get Theorem 2. For W c V, let D(W) = {x 1 D,

is dependent over w. The dependence degree of W is the (Turing) degree of

D(W).

Theorem 2. Let d be any non-zero r.e. degree. Then there exist supermaximal

spaces A and B such that (1) The degree and dependence degree of both A and B are d. (2) No automorphism of L(V,) takes A to B.

Proof. Let D be an r.e. set of degree d with 0 $ D, g :N+ N a l-l recursive

function with range D. Let D(s) = {x j3y s s (g(y) = x)}. We may assume that the

basis 93 is enumerated so that u,,< u1 < vu2 <. . . .

We will construct independent sets I and J such that I* = A and J* = B have

the desired properties. At each stage s we will have r.e. subsequences of B

a~=Ca~<a~<. . . , bs,<b;<b;<. . .

such that I(s) U{af 1 i E CO} and J(p) U{bf 1 . 1 E o are bases of V,. We let supp,&x) }

denote the support of x with respect to I(s) U {af 1 i E w}, and similarly for J. When

no s is indicated, we mean the support with respect to ~U{C+ ( i E CO), where ai is

the limit of the at if it exists; similarly for J with no s. It will be convenient to

think of the af and bt as occupying two towers of windows (af occupies the ith

window of one tower at stage s; similarly for bf). When we remove an af or by from a list, the remaining elements must be reindexed to become the a;+’ and

b,S+‘; to visualize this process think of the remaining elements ‘falling’ down the

tower to fill any empty windows.

To ensure that A and B are supermaximal we will meet the following

requirements:

PC,,,: W,3A and [W,:A]=mimplies II,EW,),

pz,,,: W, 1 B and [W, : B] = M implies 21, E W,.

52 D. R. Guichard

We also have the requirements

Qe: of f’ is a semilinear transformation of V,, then 3c E A

such that f=(c)$B or 3cgA such that fe(c)~B.

If the Q, are met, then fe(A) # B for every recursive semilinear transformation f,. We order the requirements: P$, PO”, Q. . . . P$, PF, Q, . . . .

In the construction we attempt to meet all requirements while guaranteeing that [V, : A] = [V, : B] = 00 and we use a permitting argument and some other coding to insure that A zT B END. At each stage in the construction we will have defined some finite number of elements ci, di which are meant to witness fi(A) #B. It will be useful to define the restraint function

r(e, s) = max {m I 3 <e (a”, E suppI&> or b; E supp,&id}.

We say that PC,,, requires (or needs) attention at stage s + 1 if

(1) %$ W:+(I(s))“, (2) 3x E WZ (xti (I(s))* + (n,, a;, . . . ,4>),

where r = max[n, g(s), r((e, n), s)]+ 1. Similarly, P& requires attention if

(1) %k WZ-t (J(s))“, (2) 3x E W: (x$ (J(s))* + (u,, K, f 1 * 9 ml

where r = max[n, g(s), r((e, n), s)]+ 1. If PC,,, or Pg+) is the least requirement which needs attention at stage s + 1, denote by x~+~ the least x which satisfies clause (2). The construction will guarantee that v,, E I(s)+(a& . . . , ai) so that clause (2) is equivalent to 3x E Wi (x$ I(s)* + (a& . . . , af)) and similarly for J. This makes the proof of Lemma 6 easier.

Finally, we say that Q, requires attention at stage s + 1 if (1) c, is undefined, (2) 3j ((j > r & fz(a;) y! B”) or fz(af) E B”)

where r = max[g(s), r(e, s)]. We denote by j(s + 1) the least such j. We say that Q, is injured at stage s + 1 if c, and d, are defined and c, cAS+l if and only if d2EBS+l.

COlNhlKtiOn. Stage 0. Let a; = bf = 2)i ; let I(0) = J(0) = { }; all ci and di are undefined.

Stage s+ 1. Look for the least requirement which needs attention. If such exists, there are three possibilities.

Case 1. P&, requires attention. Let I’ = I(s) U {x,+~ + II,,} and remove from the list {a:} the largest element of s~ppr~~,(x,+,+v,,).

Case 2. P&, requires attention. Similarly define .I’. Case 3. Q, requires attention. Let c, = a;(,+,, and d, = f=(c,). In case fz(ce) 6 B”,

let I’ = I(s) U{c,} and remove afcs+lj from {ai : i E w}.

Whether or not one of cases 1, 2, 3 pertained, we now want to ensure that a;(,,

Automorphism of substructure lattices 53

or 4h+l (rev. b”,bb b”,(sj+l) is in I(s + 1) (resp. J(s + 1)). In adding an ai to I’, we might injure some requirment Q,, namely, if af is the last element in supprCs)c, still in {a; 1 i E CO}. Obviously, no Q, can be injured by two a:; hence pick which- ever one of a:,,,, a&)+-r injures the Qe of least index, or no Q, at all if possible. We will never injure the least Q, which is ‘threatened’ by this maneuver. Adding a by to J’ may also injure some Q,, if bf is the last member of supp,&, still in the list {b:: i E CO}. Again no Q, will be injured by more than one choice of bf. Hence, for J(s + 1) and the b’s, make the choice in the same way as above, injuring the Q, of least priority if one must be injured at all. Let the elements fall in both towers to fill the windows.

Finally, some Qe’s may have been injured by actions taken at this stage. We declare as undefined the c, and d, for all such e.

We now prove the usual series of lemmas about the construction, which will finish the theorem.

Lemma 0. ‘tin Vs (u,, E I(s) + (a& . . . , ai) and II, E J(s) + (b& . . . , bz)).

Proof. We have n, E (a:, . . . , a:) since 0, = a:. It now suffices to prove that

SUPPrcs+l)(4) = as + 1) Wa;” : j s i}. This is true because af is removed from the array only if i is the maximum of G ( ai E suppr&y)} for some y E I(s + 1). Similarly for u,~J(s)+(b&. . ., bi).

Lemma 1. The q and bi exist, and hence [V,: A] = [V,: B] = 00.

Proof. By the construction, ai;+’ # a: only if i 2 g(s). Since g(s)< i for only finitely many s, ai exists; similarly for bi.

Lemma 2. If f, is a semilinear transformation of V,, then f,(A) # B.

Proof. Suppose on the contrary that f=(A) = B, with e the least such index. We first show that no 0, can be injured infinitely often. Using this, we show that D is recursive, a contradiction.

Q, can be injured at most once for each PA and I’s of higher priority (if a Pi requires attention at some stage, it will be satisfied at that stage by the inclusion of some element in I or J; this Pi will never again require attention; in fact it will have no further influence on the construction). Thus Q, will be injured at most 2e +2 times in this way. Q, can also be injured due to the demand that one of

G(s) and a&j+l enter AS+l, or the similar demand on B. We show by induction that each Qi can be injured only finitely often in this way also, i.e., by the ‘coding’ part of the construction.

Consider Q,,. Q0 can never be injured in this way, since it will always be

54 D.R. Guichard

possible and hence mandatory to pick whichever of a&), &)+I and b”,,,,, biCs)+l does not injure Qo.

For Q,, let t be a stage after which no Qi, i <j, is ever injured or ever requires attention, and such that for all u > t, g(u) > r(e, t). Then Q, will never again be injured by the coding demands of the construction, for e is now the least index of all the Qi which could possibly be injured in this way, and so we always choose not to injure Q,.

To show D is recursive, let t be a stage after which Qi, i =Z e, never requires attention or is injured, and after which no PA or Pf, Ike, requires attention. Since 0, will never require attention, there can be no stage u > t and i such that f,“(ar) E B”. To decide x ED, wait for a stage u > t at which 3j >max(x, r(e, t)) + 1 such that fF(ay)$ B”. By choice of t, notice that Vu > t (r(e, v)~r(e, t)) and so Vu > u Cj > max(x, r(e, v)) + 1). Now since Q, never requires attention it must be that Vvau (g(v)+l>j); since jsx+l we haveVv>u (g(v)>x) so xgD if and only if x E D”.

Lemma 3.

D +A sT{ai 1 HEW}; D +I3 ST{bi 1 iEO).

Proof. For the inequality D +A: With an oracle for A, we wish to decide x ED for any x. By the construction, (zz+’ # a: only if 3i <x such that al E I(s + 1). Thus uniformly in x and recursively in A we can find a stage t(x) such that Vi== x + 1 (ai’“‘+!A), and hence Vs 3 t(x) Vi <x (cx~‘~‘= a:). Moreover, also by the construction, Vs > t(x) (g(s) > x). Thus x ED iff x E Dtcx). D s,B is similar.

For A +. {q : i E CO}: Given v E V,, we can find recursively in (4 ( i E o} the representation of v in terms of the basis I U {q 1 i E co}. If some q appears in this representation with non-zero zero coefficient, then v+ A, otherwise v E A. Simi- larly for B.

Proof. Notice that a, < a, < a2 < * * * so that Vv(v E{LI+ 1 i E o} if and only if VE{&J,..., u,,}); also Vk Vs at+ # at only if g(s) < k. Uniformly in k and recur- sively in D we can find a stage t(k) such that Vs 2 t(k) (g(s) > k) and hence such that Vss t(k)Vi<k (ufCk)= a:= a,). Then given VE V,, v~{% 1 iEco} iff v E

{a,, . . . , u,,} = {a~“‘, . . . , a$“}. Similarly for B.

Lemma 5. A =,D(A), B =,D(B).

Proof. A + D(A) is trivial. Suppose, for the other direction, that we want to know whether some set {wi 1 i < m} is dependent over A. Express each wi as c ci+ where {xi 1 j E w} is the basis IU{q ( i E w}; this is recursive in {q 1 i E w} and so in

Automotphisms of substructure lattices 55

A. Let WY = CisG Cj$, where j E G iff 3 $ A. Then {wi ( i < m} is dependent over A iff {w; ) i > m} is dependent. Similarly for B.

Lemma 6. All requirements PA and Ps are met.

Proof. Suppose not, and suppose P& is the least not met (a similar argument will

work if the least is a Pf). That is, suppose W, IA, [W, : A] = 00, and u, & W,. Let t be a stage after which all earlier requirments never need attention and are never injured. We give a procedure for deciding x ED, in contradiction to the choice of D non-recursive.

Given x, let k = max[n, x, r((e, n), t)]+ 1. Since [W, : A] = 00, there exist y in W, with ygl*+(a,,..., ak). Thus we can effectively find u > t and y E W,U such that

y&I(u)*+%. . . , a;). The only thing which can keep P&,, from requiring attention at stage u is that y is in the space I(u)*+(a& . . . , aiC,,,J. Suppose m is the largest index such that a: is in ~upp,~,,(y). Then since P&, never requires attention it must be the case that ttv 2 u (g(v) + 12 m), for as soon as g(v) + 1< m, P&, will require attention and be met at stage v + 1. But now Vv 2 u (g(v)+l>m>k 2x+1), so g(v)>x and xgD iff xED”. q

Kalantari and Retzlaff construct k-thin spaces for all k as follows. Let V, = (vi ( i Z= k). The construction of a supermaximal A may now be carried out in the space V,. Then in V, A is k-thin, for suppose not. Then there is a W IA with [V,: W] = j > k and V, f~ W contradicts the supermaximality of A in V,.

It is also easy to see that a k-thin space has a unique superspace of codimension k. Suppose V and W are distinct codimension-k superspaces of the k-thin space A. Then VII W is a superspace of codimension j > k, a contradiction. This fact makes the following theorem an easy consequence of Theorem 2.

Theorem 3. Let d be any nonzero r.e. degree. Then for all k there exist k-thin

spaces A and B such that (1) The degree and dependence degree of both A and B are d. (2) No automorphism of L(V,) takes A to B.

Proof. Let V, be as above and carry out the construction of Theorem 2 within V,. Then A and B are k-thin spaces of degree d, and no automorphism of L(V,) takes A to B. We will use the following little lemma.

Lemma. Any automorphism of L(V,) which takes A to B fixes V,.

Proof. Suppose H is such an automorphism. Then B = H(A) c H( V,). Since any k-thin space is contained in a unique space of codimension k, and since B c V,, we have H( V,) = V,.

Now any automorphism of L(V,) which takes A to B induces an automorph- ism of L(V,) taking A to B, but no such automorphism exists. 0

56 D.R. Guichard

2.

J. Remmel has examined the lattice of r.e. substructures for a variety of algebraic structures, including extensive work with boolean algebras. He asked how many automorphisms of I@,) exist, where B, is a recursive presentation of the free boolean algebra on countably many generators. Using methods similar to those for Theorem 1, we have answered this question: every such automorphism is induced by a recursive automorphism on B, itself.

The Fundamental Theorem of Projective Geometry was crucial to the method of proof used in Theorem 1; we need a similar result for boolean algebras. Let Lat(B) denote the lattice of all subalgebras of a boolean algebra B. Sachs [7] has shown that any automorphism on Lat(B) in induced by one on B. As in the vector space setting, this implies also that every automorphism of I(&) is induced by an automorphism on B, (L(B,) includes all the four element subalgebras; the permutation of these subalgebras by the given automorphism can be extended in the obvious way to an automorphism of Lat(B,)).

Let {vi ) i E o} be a recursive independent subset of B, which generates B,, i.e., a set of free generators for B,. The proof is made much easier by thinking of B, as a Lindenbaum-Tar-ski algebra and (by some abuse of notation) the vi as the propositional variables. In particular we use the interpolation theorem for propos- itional logic (if s, t are formulas and s -+ t is valid then there is a formula p, called an interpolant, which contains only those variables common to s and t, and such that s -+ p and p -+ t are valid). Roughly, we use this to get around the lack of unique representations in B,.

Theorem 4. Every automorphism of L(B,) is induced by a recursive automorphism

of Boa.

Proof. We adapt the vector space argument to B,. Let F be the given au- tomorphism of L(B_); let f be the automorphism on B, which induces F. Suppose that under F

lv2i I iEw) + Cl,

G-h+1 I i E 0) + C2,

h2i & V2i+l I i E w> + G

bZi+l & v2i+2 I i E W> + Cd.

As before let w,, = f(v,,); note that {wi 1 i E w} is also a generating set for B,. We will use w0 and enumerations of the Ci in describing an algorithm for computing each w, in turn. Then to compute f(v) for any v, we write v as a boolean combination of vi, v = s(vO, . . . , vk) and then f(v) = s(wO, . . . , wk).

To compute wl, search for y E C, such that w0 & y E C, and w0 & y # 0. AS mentioned above, we abuse notation by thinking of the w,, as propositional

Automorphisms of substructure lattices 57

variables and also as elements of B,. Thus we may write w0 & y as s(wO, wl, w3,. . .) and as t(w, & wl, w2 & w3,. . .), and obviously bs* t. Let s even= S(wzi+lWzi ) i >O), i.e., replace each w2i+l in s by wzi for i>O. Since t is symmetric in wzi and wzisl for i ~0, we have !=seven ++ t and hence ks+= seven. Now let p be an interpolant between s and seven, i.e., l=(s + p) & (p + seven), where p mentions only the letters w0 and wl. Hence also l=st, p, !=t@ p.

Recall here that a (propositional) model A is a subset of the propositional letters. We say A != wi if and only if wi E A, and by induction define A l=q in the obvious way for any sentence.

Now bp + w0 because i=s + wO. Also, p is symmetric in w0 and w1 because t is, so kp + w1 and hence i=p 4 w0 & wl. Suppose f(wO & wJ + p. Then there is a model A 3 (w,, wl} and A l=lp. But then any model containing w0 and w1 is a model of lp since p depends only on w0 and wl. Hence l=(w, & wl) + -IP and so t=-~p and l=ls, a contradiction. Hence it must be that b(w, & wl) + p and so b(wo & WI) f-, s.

Finally, we now have bwO-, (yew,), where y mentions only Wzi+l for iS0.

Hence l=y ts, w1 (this is actually another use of the interpolation theorem), which is to say y = w1 in the boolean algebra. Now wz is found similarly using C1, C, and the just computed wl, and the rest are computed in turn, just as in the V, construction. 0

3.

We may form a second lattice, related to L(V,), which is analogous to %*, formed by taking the lattice g mod the finite sets. The appropriate operation here is to mod out the finite dimensional vector spaces to form L*(V,), hereafter simply L”. In [2], Kalantari presented a proof that every automorphism of L” is induced by an automorphism of L(V,). There was a mistake in that proof; were the theorem true it would follow that every automorphism of L” is induced by a recursive semilinear transformation. In fact, there is a simple example of an automorphism of L* which is not induced by any semi-linear transformation.

Example. Let f: V,-+ V, be given by f(vi) = vi+1 (the ui are the standard basis of V,). Then f induces an automorphism on L *. It is not difficult to prove that no semi-linear transformation can induce the same automorphism on L”.

C. Ash conjectured that the correct theorem is: every automorphism of L” is induced by a semi-linear map with finite dimensional kernel and whose image is co-finite in V,; we have not been able to prove this theorem. We also do not know the number of automorphisms of L*.

58 D.R. Guichard

References

[l] R. Baer, Linear Algebra and Projective Geometry (Academic Press, New York, 1952).

[2] I. Kalantari, Automorphisms of the lattice of recursively enumerable vector spaces, Z. Logik

Grundlagen Math. 25 (1979) 385-401.

[3] I. Kalantari and A. Retzlaff, Maximal vector spaces under automorphisms of the lattice of

recursively enumerable vector spaces, J. Symbolic Logic 42 (1977) 481-491.

[4] G. Metakides and A. Nerode, Recursion theory and algebra, in: Algebra and Logic, Springer

Verlag Lecture Notes 450 (1975) 209-219.

[5] G. Metakides and A. Nerode, Recursively enumerable vector spaces, Annals of Math. Logic 11

(1977) 147-171.

[6] J. Remmel, On r.e. and co-r.e. vector spaces with nonextendible bases, J. Symbolic Logic 45

(1980) 20-34.

[7] D. Sachs, The lattice of subalgebras of a boolean algebra, Canad. J. Math. 14 (1962) 451-460.

[8] R.I. Soare, Automorphisms of the lattice of recursively enumerable sets. Part I: maximal sets,

Annals of Math. 100 (1974) 80-120.