Finite fixed point algebras are subdiagonalisable

Algebra Universalis, 25 (1988) 210-222 0002-5240/88/020210-13501.50 + 0.20/0 �9 1988 Birkh/iuser Verlag, Basel

Finite fixed point algebras are subdiagonalisable

DAVID HOBBY

Introduction

Fixed point algebras were introduced by Craig Smoryfiski in [10]. They have also been studied by Montagna in [7] and Solovay in [11].

Fixed point algebras are a generalization of the diagonalisable algebras of Magari and others. (See [1], [5], [6] or [8].) This paper establishes some connections between fixed point algebras and diagonalisable algebras.

DEFINITION 0.1. A fixed point algebra is a structure (A, B) where A is a Boolean algebra and B is a set of unary operations on A such that:

i) B is closed under composition. ii) B is closed under v, ^ and ' (defined pointwise).

iii) B contains all of the constant functions. iv) Every function in B has a fixed point.

The functions in B form a Boolean algebra under the pointwise v, A and ' Thus one can also consider a fixed point algebra as a pair of Boolean algebras.

DEFINITION 0.2. A diagonalisable algebra is a structure (A, r} consisting of a Boolean algebra with an additional unary operation r satisfying:

i) r (1) = 1 ii) f ix ) ^ r(x=>y) <- r(y)

iii) l:(x) <- 7:(r(x)) iv) r ( , ( x ) ~ x ) <- r(x).

An important example of a diagonalisable algebra is obtained from the

Presented by Joel Berman. Received November 30, 1984. Accepted for publication in final form September 22, 1987.

210

Vol. 25, 1988 Finite fixed point algebras 211

Lindenbaum algebra of a sufficiently strong theory, T. (Recall that the Linden- baum algebra of T is the Boolean algebra of classes of provably equivalent sentences.) The Lindenbaum algebra is the Boolean algebra, and 3 is the operation defined by:

3([q~l) = [Prr ( [q~] )].

(Here Prr ([4~]) is the sentence of T that asserts that 4) is provable. T must be strong enough for this to be expressible.) Conditions (i)-(iii) are obviously true, and (iv) is equivalent to L6b's Theorem.

In this paper I will use Pol (B) to denote the clone of polynomial functions of B. This is the clone generated by the basic operations of B and the constant functions. I will use Pol, (B) to denote the set consisting of those functions in Pol (B) that are n-ary.

DEFINITION 0.3. Suppose that B = (A, v) is a Boolean algebra with the additional unary operation 3. Define F(3) to be the set consisting of the f (x ) ~ POll (B) in which x only occurs within the scope of 3. Specifically, F(3) is the closure under ^ , v and ' of {vof :f e Poll (B)}.

The following was proved independently by DeJongh (published in [9]) and Sambin (in [8]).

THEOREM 0.4. Let (A, 3) be a structure that satisfies all of the conditions in Definition 0.2 except (iv). Then (A, 3} satisfies (iv) iff every function in F(v) has a fixed point.

Note that F(3) contains the constant functions and is closed under composition and the Boolean operations. Thus every diagonalisable algebra (A, 3} has associated with it the fixed point algebra (A, F(3)). Fixed point algebras of this form will also be referred to as diagonalisable algebras.

DEFINITION 0.5. A fixed point algebra (A, B ) is subdiagonalisable iff there is a set of functions C such that B c_ C and (A, C ) is a diagonalisable algebra.

DEFINITION 0.6. A fixed point algebra (A, B} is closed iff whenever g e B and h ~ Poll (A), g o h e B.

An equivalent condition is that whenever g 6 B, the functions that send x to

212 DAVID HOBBY ALGEBRA UNIV.

g(x ^ c) and to g(x ') are in B. Note that every diagonalisable algebra (A, F( r ) ) is closed.

These notions were introduced by Smoryfiski in [10], where he proved that all finite closed fixed point algebras are subdiagonalisable. He in fact conjectured that every finite fixed point algebra was subdiagonalisable and that every finite closed fixed point algebra was diagonalisable. I prove these conjectures in sections 2 and 3 of this paper, respectively.

1. Notation and preliminary theory

I will use standard Universal algebraic notation. As was noted before Definition 0.3, by the polynomial functions of an algebra I will mean those in the clone generated by the constant functions and the algebra's basic operations. Congruences will be considered to be sets of pairs of elements of an algebra. The congruence generated by a set ~p will be denoted by Cg0p). The least and greatest elements of the congruence lattice of A, Con (A), will be written A and V, respectively.

Boolean algebras will be used extensively. Recall that if A = (A, v , ix, ') is a Boolean algebra then there is a natural isomorphism between Con (A) and (A, v, ^ ). If A has a lattice structure upon it, Atoms (A) will denote the set of atoms of that lattice. I will write the symmetric difference function of a Boolean algebra as 4-.

(i.e. x 4 - y ~ ( x A y ' ) v (x' Ay) ) .

One can use the work of J6nsson and Tarski in [3] to obtain a representation theory for diagonalisable algebras. This was first noted by Magari in [6]. Since this paper is concerned only with finite algebras, I will state a simpler version of the theory that is valid for them. Proofs can be found in the papers of Magari [6] and Smoryfiski [10].

THEOREM 1.1. Let A be a finite Boolean algebra with atoms ao, a l, o . . , a,. There is a canonical one-to-one correspondence between functions ~ that make (A, 3) a diagonalisable algebra, and strict partial orders on Atoms(A). Furthermore, i f < is the order that corresponds to r:

ai < aj iff (Vx ~ A)(ai <- v(x) ~ a j <-x),

and v(x) = V {ai~ Atoms (A): (Vaie Atoms (A))(ai < a i ~ ai <- x)}.

Vol. 25, 1988 Finite fixed point algebras

If we define o by o(x) = (~:(x'))', then we can simplify this. For,

ai "< aj iff (Vx e A)(ai <- (o(x ')) ' ~ a t <<. x)

(Vx __ a;)(a, (o(x'))') r

iff (Vx -< at)(a, <- o(x'))

iff (Vx -> at)(ai <- o(x))

iff ai <- o(at),

where the last step uses that o is monotone. We also have that:

213

o(x) = (V {aie Atoms (A)" (Va i e Atoms (A)) (a /< aj::~aj --- x ' )}) '

= V { a i e Atoms (A)" (3a t e Atoms (A))(ai < a t & a i :~ x')}

= V {aie Atoms (A)"(3aj e Atoms (A))(ai < a j & a t -< x)}.

A useful fact about any finite fixed point algebra (A, B) is that all the functions in B have unique fixed points. This was proven by Smorytiski in [9]. I give a proof here for the reader's convenience.

L E M M A 1.2. Let (A, B} be a finite fixed point algebra. Then every function in B has a unique fixed point.

Proof. Let f be a function in B that has distinct fixed points a and b. We may as well assume that the cardinality of the interval [a/x b, a v b] is minimal for all such f, a, and b.

Let a ^ b and a v b denote the constant functions with values a A b and a v b, respectively. Let g ~ B be the function defined by: g ( x ) = ( f ( x ) v ( a"Ab) )^ (a'--~b). It is clear that a and b are fixed points of g, and that the range of g is contained in [a/x b, a v b]. Thus a and b are the only fixed points of g, by minimality.

Let n be such that gn is idempotent. (Taking n equal to IAI! works.) Again by minimality, a and b are the only fixed points of g". Since g~ is idempotent, the range of g~ is {a, b}.

Now let d and 6 be the functions with constant values a and b, respectively. Let h E B be defined by:

h(x) = gz 4- g~(x) 4- b.


Then the range of h is {a, b}, h(a) = b, and h(b) = a. Thus h has no fixed points, a contradiction.

2. Finite fixed point algebras are subdiagonalisable

DEFINITION 2.1. If A is a Boolean algebra, and f : A - ~ A , define a mapping 1~ from Con (A) into itself as follows:

f(O) = Cg (( f(x) , f ( y ) ) : (x, y) e 0}.

Note that f is monotone (satisfies (VO, r r Con (A))(O - q~jZ(O) ---~(r and that ](A) = A, for any function f.

DEFINITION 2.2. Let a Boolean algebra A and a set C of unary functions from A into itself be given. Define C* to be the least set of functions from A into A that contains C and the constant functions and is closed under v , A, ' and composition.

LEMMA 2.3. Given any a, b c A and f :A--~A, we have:

Cg (f(a), f (b)) = {(g(a), g(b)):g ~ {f}*}.

Proof. This is an immediate consequence of the description of principal congruences in permutable varieties given by H. Lakser in [4]. This lemma can also be proved directly by showing that { (g (a) ,g (b ) ) :ge{ f }*} contains {(h(f(a)), h( f (b)) ) :h e Poll (A)} and is symmetrically and transitively closed.

LEMMA 2.4. Let <A, B) be any finite fixed point algebra. Then there is an atom ol of Con (A) such that

(vf B)( (oO = AA).

Proof. Suppose (A, B) is a finite fixed point algebra for which the Lemma fails. Then for every atom cr of Con (A) there is a f~ ~ B such that fo~(c 0 q: AA.

Since Con (A) is finite, there must be a "circuit", that is, atoms or1 . . . . . or, of Con (A), and fl . . . . . f,,, ~ B such that )~,(o:n) -> or1, and f~(cri) - aq+l for all i < n.

Since f,(cr,) = V {Cg (f,,(x), f , (y)) : (x, y) e or,} and or1 is an atom, there must be (a, b) e crn such that oq ~ Cg (f~(a), L(b)) . Thus Lemma 2.3 implies that

(V(x, y) e oq)(3h e B)(x = h(a) & y = h(b)).


Likewise, ?n _l( Ogn _ l) = V {Cg ( f , , - l (x ) , f , - l ( Y ) ) : (x, y) e or,_1} implies that there are (a~_~, bn-~)eo t~- i and g n - l e B such that g,,_~(a,,_t)=a and gn-~(b~-l) = b. Similarily, there are (a,-2, b~_2) e o:,_2 and g , -2 e B such that an-1 =gn-2(a,,-2) and b,,_~ =gn_2(b,,_2). Continuing in this manner, we even- tually get (al, bl) s or1 and gl E B such that g~(a~) = az and g~(bl) = bz.

Let h e B be such that h(a) = aa and h(b) = bl. Now gn-1 ~ . . . . gl oh has 2 fixed points, since

g , -1 ~ . . . . gl o h(a) = g~-I . . . . g1(a~) . . . . g , - l ( a , - l ) = a

and gn-1 ~ gn-z ~ �9 �9 gl ~ h(b ) = b. This contradicts Lemma 1.2.

D E F I N I T I O N 2.5. Given a fixed point algebra (A, B ), define the mapping q~ from Con (A) into itself by:

e ( 0 ) = V {,(O):f eB}.

The mapping q~ depends on the fixed point algebra, of course. Since there is little danger of confusion, I will refer to it simply as q~.

Note that q~(A) is always z~, and that q~ is monotone. The next few lemmas present other properties of q~ that will be needed.

L E M M A 2.6. For any fixed point algebra (A, B ), cp is a join-endomorphism. That is, (p(o: v /3) = c#(o:) v cp(fi) for all congruences ol and ft.

Proof. Since q3 is monotone, it is clear that

v/3) -- r v q,(/3).

To prove that q~(cr v/3) --- q~(oc) v q~(/3), it suffices to show that ( f (a ) , f ( b ) ) is in 4~(cr) v 9(/3) whenever f e B and (a, b) e oc v/3. So suppose that f e B and (a, b) ~ a~ v/3. There is a chain ao, al, a2" �9 �9 an of elements of A, with a~j = a, an =b , (ai, a i+a )~x for i even, and (ai, a i + l ) ~ for i odd. Now (f(ai), f (a i+l ) )s (p(oc) for i even, and ( f (a~) , f (a i+l ) )e (p( f l ) for i odd. Thus ( f (a ) , f ( b ) ) = (f(ao), f (an)) is an element of q~(cr) v q~(/3). This shows that all of the generators of q~(o: v/3) are in q~(a:) v q5(/3).

L E M M A 2.7. I f (A, B ) is any closed fixed point algebra, q~(Cg (a, b)) = { ( f ( a ) , f ( b ) ) : f ~ B}.


Proof. By the argument used in the proof of Lemma 2.3, { ( f (a ) , f ( b ) ) : f B} is a congruence of A. It is obviously contained in O(Cg (a, b)), so the proof will be complete once it is shown that {(g(x), g(y)):g ~ B & (x, y )~ Cg (a, b)} is contained in {(f(a), f ( b ) ) : f e 13}.

Now if (x, y) e Cg (a, b), there is an h in Poll (A) with x = h(a) and y = h(b). This is because A is in a congruence-permutable variety. (It is easy to show that if (x,y) ~ Cg (a, b), then x=x4-[ (a4-a) A (X-i-y)] and y =x- i - [ (b 4~ a )A (x 4-y)]. This gives a direct proof.) So i fg is any element of B, (g(x), g(y)) is (f(a), f(b)), where f =goh is in B since (A, B) is closed.

C O R O L L A R Y 2.8. If (A, B) is' a finite closed fixed point algebra, then dp2(O) <-- r for every 0 e Con (A).

Proof. Since A is finite, O=Cg(a, b) and q~(0 )=Cg(c , d) for some a, b, c, d e A. By Lemma 2.3, (c, d ) = (f(a), f(b)) for some f e B. Then 4~2(0) = {(g(c), g(d)):g ~ B} = {(gof(a), gof(b)):g ~ B}, which is contained in q~(0).

L E M M A 2.9. If (A, B) is any finite fixed point algebra, there is an n such that 4:( v) = a.

Proof. By induction on [At. If IAI = 1, VA = AA, SO the result is true in this case. Now assume that the result is true for all fixed point algebras of cardinality less than IA].

Using Lemma 2.4, let cre Atoms (Con (A)) be such that q~(cr) = ziA. For each f e B , define f/cr:A/o:--*A/a" by: f /ol(x/o 0 = f ( o 0 / m Since (x ,y )~ o: implies f (x) = f ( y ) , f/ol is well defined.

Now (A ,B} /or=(A/o t , { f / o L : f e B } ) is a fixed point algebra. This is because A/a~ is a Boolean algebra and each f/or has fixed point c/cr, where c is the fixed point of f.

Let Z0, Zt, Z2, . . �9 be V, q~(V), q~2(I7) . . . . in (A, B)/ol. There is an n such that Z, = A, by assumption.

For all i, let :'g21(Zi) ECon(A) denote { (x , y ) eA• Suppose f e B and ( a, b) e ar 21( Zi ). Then ( a / o~, b / o~ ) ~ Z, , so

( f / o~(alol), f /ol(blo:)) = (f(a)/o~, f (b )lol) ~ Z, . ,.

That is, ( f ( a ) , f ( b ) ) e ~ ' ( Z , + O. So we have f(=~(Z,))<_-er2~(Zi+:) for every f 6 B .


Now z-dl(Zo) = VA and each .f is monotone, so (p(VA)<-;rT,'(Z,) and by an easy induction, q)i(VA)-< Jr-dl(Zi) for all i.

Thus q)"(VA) --< z21(Z, ) = o~. But )~(~) = AA for every f e B, so q~n+l(VA) = AA. This completes the induction.

D E F I N I T I O N 2.10. Let A be an algebra and let ~p be a mapping from Con (A) into itself. Let f be a unary function from A into A. Say that f preserves ~p i f f f <- % (i.e. (V0 e Con (A))(V (x, y) ~ O)(( f (x) , f (y)) e ~p(0))).

Say that a set of unary functions preserves v2 iff all of its elements do, and let FOp ) be the set of unary functions that preserve ~p.

L E M M A 2.11. Suppose that A is a finite algebra, and that

~p : Con (A) ~ Con (A) is monotone and satisfies tp 2 - %

Then (FOP))* = FOP).

Proof. The constant functions are clearly in F(~0). Thus we need only show that FOP) is closed under v , ' and composition.

Suppose that g and h are any elements of FOP), and let 0 be an arbitrary element of Con (A).

It is easy to see that goh(O)---~(/~(O)). But g( /~(0))-<g0p(0))-< ~p2(0)-- Vd(0), so goh ~ F(~O).

Now for every (a, b) ~ O, (g'(a), g'(b)) = (g(a)', g(b)') ~ g(O), so g'(O) <- g(O). Thus g' is an element of F ( ~ ) .

If (a, b) ~ 0, (g v h(a), g___v h(b))= (g(a) v h(a), g(b) v h(b)) ~ Cg (g(a), g(b)) v Cg (h(a), h(b)). Thus g v h(O) - - g ( 0 ) v/~(0) -< vd(0), so g v h ~ F(V).

COROLLARY 2.12. l f A is a finite Boolean algebra and f :A--+ A is such that ~n(V) = z~ for some n, then (A, {f}*) is a f xed point algebra.

Proof. Let A a n d f b e as in the statement. Let V= Zo, and let Zi+l = f ( Z i ) for all i. Let n be least such that fn(V) = A. Then 3~n(v) = Zn = A and Z0, Z1 �9 "" Zn is a strictly decreasing sequence in Con (A).

Define ~p :Con (A)--* Con (A) by setting ~p(0) = f (Z i ) , where i < n is maximal so that 0 - Zi. Now f preserves % and it is clear that ~p is monotone and satisfies ~p2< % Thus Lemma 2.11 implies that {f}*c_ FOP)* =FOP) . The proof will be completed by showing that every g e FOP) has a fixed point. So suppose g preserves %

218 DAVID HOBBY ALGEBRA UNIV,

Then g"(x) is constant. For let x and y be any elements of A. Now g(V)= Z1 so (g(x), g(y)) ~ ZI, thus (g(g(x)), g(g(y))) e Z2, and an easy induction gives (g"(x), g"(y)) e Z, = A.

Let c be the constant value of g"(z). Then g(c) = g(g"(c)) = g~(g(c)) = c, so g has a fixed point.

THEOREM 2.13: Let A be a finite Boolean algebra. I f V2:Con(A)--* Con(A) /s monotone and satisfies ~z<_~ and V2"(V)=A for some n, then (A, FOP)} is a closed fixed point algebra.

Proof. Lemma 2.11 implies that FOp ) is closed under v, A, ~ and composition. And the proof of Corollary 2.12 gives us that each g e FOP) has a fixed point. Finally, if g e F(~0) and h e Poll (A), goh(O) <- g((/~(0)) -<g(0) -< v2(O ), so gob e F(V2). Thus (A, FOp)) is a closed fixed point algebra.

COROLLARY 2.14. Given any finite fixed point algebra (A, B}, there is a closed fixed point algebra ( A, C ) with B c_ C.

Proof. Let q~ be the mapping on Con (A) for (A, B), defined as in Definition 2.5. Then q~ is monotone, and satisfies 9 2 - q~ and q~"(V)= A for some n, by Lemmas 2.8 and 2.9. Thus (A, F(q~)) is a closed fixed point algebra. But B preserves 9, so B c__ F(qS) and we are done.

Smoryfiski proved in [10] that every closed fixed point algebra was subdiagonalisable. Thus Corollary 2.14 implies that every finite fixed point algebra is subdiagonalisable. We will prove in Section 3 that a finite fixed point algebra is closed if and only if it is diagonalisable, giving us a self-contained proof of this result (Corollary 3.9).

3. Finite closed fixed point algebras are diagonalisable

Theorem 3.8 contains the main result of this section. It follows easily from Lemmas 3.6 and 3.7. Some preliminary Definitions and Lemmas must first be presented.

DEFINITION 3.1. Let (A, B) be a finite closed fixed point algebra with 4~ as in Definition 2.5. Define the binary relation < on Atoms (A) by:

a < b iff Cg (a, O) -< q~(Cg (b, O)).


L E M M A 3.2. I f {A, B) and ~ are as in Definition 3.1, then < is a strict partial order on Atoms (A).

Proof. Lemma 2.9 implies that < is irreflexive and antisymmetric. Suppose a, b, c e Atoms (A) are such that a < b and b < c. Let a" = Cg (a, 0), fl = Cg (b, 0) and 7 = Cg (c, 0). Then o: <_. ~(fi) and fl -< ~p(y), so oc _< q52(7). Now Corollary 2.8 gives a: -.< q~(~,), so a < c and < is transitive.

D E F I N I T I O N 3.3. Let (A, B) and < be as in Definition 3.1. Let r be the function on A that is induced by <, as in Theorem 1.1. Define the function ~ on A by o ( x ) = (~(x')) ' .

Note that r(x) = (~r(x'))'. Thus if (A, B) is closed, cr~ B iff r ~ B.

L E M M A 3.4. Let (A , B) and o be as above. For every x and y in A, we have:

a(x v y) = a(x) v o(y).

Proof. Using Theorem 1.1, we have that o(x) is ~ / { w E A t o m s ( A ) : ( 3 z e Atoms (A))(z ---< x & w < z)}. The desired equality follows from this characterization.

L E M M A 3.5. With (A, B), q) and o as above:

Cg (o(x), 0) = q)(Cg (x, 0)) for all x e A.

Proof. Since a and q~ both preserve joins, it suffices to prove the equality when x e Atoms (A). (Note that x--~ Cg (x, 0) is a lattice isomorphism between A and Con (A).)

Let a be an atom of A. As in the proof of Lemma 3.4,

a(a) = xff {w ~ Atoms ( A ) : w < a}

= V {w e Atoms (A) :Cg (w, 0) -< ~p(Cg (a, 0))}.

Thus

Cg (o(a), 0) = ~ / { u ~ Atoms (Con (A)): u - e ( C g (a, 0))}

= q~(Cg (a, 0)).


LEMMA 3.6. If (A, B) is a finite closed fixed point algebra and ~ is as in Definition 3.3, then o �9 B.

Proof. For each a e Atoms (A), let ha be a function in B such that ho(0) = 0 and ho(a) = ~r(a).

To see that there are such functions, note that (o(a), 0 ) s (p(Cg (a, 0)) by Lemma 3.5. Now Lemma 2.7 implies the existence of ha.

For each a ~ Atoms (A), let go E B be defined by g,(x) = h~(x ,\ a). Thus

0 if a :~x go(x)= or(a) if a-<x,

and go(x) <- a(x) for all x. Now let f � 9 be V { g o : a ~ A t o m s ( A ) } . It is clear that f ( y )=~r(y ) for

y e Atoms (A). Since both functions are join-endomorphisms, we have f = ~r.

Note that if ( A , B ) is as in this lemma, then r � 9 and since B is closed, B _~ F(~). Theq~ of {A, F( r ) ) is the same as that of {A, B).

LEMMA 3.7. Let (A, B ) be a finite closed fixed point algebra with associawd mapping 4). Then every function that preserves (p is in B.

Proof. By induction on the cardinality of A. If IAI = 1, this is clear. Assume the lemma is true when ]AI -<2 n, and let (A, B) be a closed fixed point algebra with [A[ = 2 ~+1. Choose a �9 Atoms (A) so that (p(Cg (a, 0)) = A, and let a, denote Cg (a, 0). F o r f �9 B de f ine f /a :A/a- - -~A/a by:

f /o~(x/ a) = f (x ) /a .

Let ( C , D ) = ( A , B ) / a = (A/o~, { f / o l : f � 9 As in the proof of Lemma 2.9, this is a fixed point algebra. Note that it is closed.

If fl �9 Con (A), define/3/ol �9 Con (A/a ) to be {(x/a, y /a) : (x , y) s ~}. Let (PA and (Pc be the mappings of Definition 2.5 for (A, B) and (C, D) respectively. Given ~ �9 Con (A), let x, y �9 A be such that fl = Cg (x, y). Now

(pc(/~/oL) = (pc(Cg (x/a, y /e ) ) = {( f /a~(x/o~), f /o~(ylo~))'f /a �9 D},

the latter equality by Lemma 2.7. But

{( f /a(x/ol), f /o~(y/ol)): f /a c D} =

{( f (x) /a , f(y)/oL): f �9 B} = CPA(Cg (x, y)) /m


Thus (pc(fl/ol) = I~)A(/3)/O{.

Now let OA and Oc be the functions of Definition 3.3 for (A, B) and (C, D), respectively. Using Lemma 3.5 and the fact that q)c(/3/o~) = CpA(/3)/O~, we get that OA/el= OC. Recall that every unary function that preserves q~c is in D, by hypothesis.

Let f be any unary function on A that preserves q~A. Since q)c(/3/o:)= CPm(/3)/a, f /ol preserves q~c. Let g be some element of B such that f / a = g/oc.

The proof will be completed by showing that f 4- g is an element of B. Some observations are in order. The function f4- g preserves ~PA, since f and g do. Since f/ol = g/oi, the range off4- g is contained in {0, a}. Lastly, if 13 e Con (A) is such that q~(/3) ~: of, then (f4-g)(x) = (f4-g)(y) whenever (x, y) ~/3, since q~(fl) A 0r = ZX.

Let 7 = V {/3 e Con (A): q)(/3) ~: ol}. Note that y is maximal so that q~(~') ~: o~, and so f4 -g is constant on the blocks of ~'. For any b e A, define hb by:

hb(x) = {~ if (x, b) E y if (x, b) ~ y.

Now f4 -g = V {hb:b e A & (f4-g)(b) =a}, so it suffices to show that hb e B for every b.

So let b be given. Then hb is the function that sends x to [cr(x 4- b)]' ~, a. For suppose (x, b) e y. Then o(x 4- b) dy a, since 4~(7) ~ a~. And if (x, b) ~ V, then c~(x 4- b) >- a, since q~(Cg (x, b)) > ol. Thus hb is an element of B.

THEOREM 3.8. Let (A, B) be a finite fixed point algebra and let 4~ be the associated mapping on Con (A) as in Definition 2.5. The following are equivalent:

1) (A, B) is closed. 2) B contains every unary function that preserves 4). 3) ( A, B) is diagonalisable.

Proof. Let (A, B) be a finite fixed point algebra with associated mapping q~. First assume that (A, B) is closed, and let o and z be the functions on A

given by Definition 3.3. Lemma 3.6 gives us that B _D F(z). Obviously, B ~_ F(q~). Now ( A , F ( z ) ) is closed, so Lemma 3.7 implies that F(cp)=F(v). Thus B = F ( r = F(T) .

Finally, (A, F0:)) = (A, F(4~)} is closed, so we have the converse as well.

COROLLARY 3.9. Every finite fixed point algebra is subdiagonalisable.

Proof. Let (A, B) be a finite fixed point algebra, with q~ as in Definition 2.5.

222 DAVID HOBBY

Let C be the set of all unary functions on A that preserve q~. It is clear that B is contained in C. And (A, C) is diagonalisable by Theorem 3.12. Thus (A, B) is subdiagonalisable.

BIBLIOGRAPHY

[1] C. BERNARDI, The uniqueness of the fixed point in every diagonalisable algebra, Studia Logica 35 (1976), 335-343.

[2] S. BURroS and H. P. SANKAr'PANAVAR, A Course in Universal Algebra, Springer-Verlag, Graduate Texts in Mathematics, 1981.

[3] B. JONSSON and A. TARSKt, Boolean Algebras with Operators I, Am. J. of Ma~h. 73 (1951), 891-939.

[4] H. LAKSER, Principal Congruences in N-permutable varieties, Alg. Univ. 14 (1982), 64-67. [5] R. MAGARI, The diagonalisable algebras, Boll. Un. Math. Ital. (4) 12 (supp. fasc. 3) (1975).

117-125. [6] R. MAGARI, Representation and duality theory for diagonalisabfe algebras, Studia Logica 34

(1975), 305-313. [7] F. MONTAGNA, A completeness result for Fixed-point Algebras, Preprint - to appear. [8] G. SAMBIN, An effective fxed-point theorem in intuitionistic diagonatisable algebras, Studia

Logica 35 (1976), 345-36I. [9] C. SMORY~SKI, Beth's theorem and self-referential sentences, Logic Colloquium "77, North-

Holland, Amsterdam, 1978. [10] C. SMORYr~SKI, Fixed point algebras, Bull. of the Amer. Math. Soc., 6 (1981), 317-356. [11] R. SOLOVAY, lnfinitefixed point algebras, Preprint- to appear.

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Finite fixed point algebras are subdiagonalisable

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