free algebras in discriminator varieties

Algebra Universalis, 28 (1991) 401-447 0002-5240/91/030401-4751.50 + 0.20/0 �9 1991 Birkh/iuser Verlag, Basel

Free algebras in discriminator varieties

H. ANDREKA I, B. J6NSSON 2 AND I. NI~METI 3

Abstract. We investigate V-free algebras on n generators, F n = Fr(V, n), where V is a discriminator variety and, more specifically, where V is a variety of relation algebras or of cylindric algebras. Sample questions are: (a) Is F ,§ embeddable in F~? (b) Does Fn contain an n-element set that generates it non-freely? The answer to (a) is affirmative in some varieties of relation algebras, but it is negative in every congruence extensile variety in which some nontrivial finite member is an absolute retract. The answer to (b) is affirmative in every variety of relation algebras that contains the full algebra of relations on an infinite set.

Introduction

The results presented here evolved f rom a s tudy o f relat ively free re la t ion

a lgebras , and the p r ima ry app l i ca t ions are to re la t ion a lgebras and to cyl indr ic

algebras. A m o n g the quest ions tha t we ask, concern ing a var ie ty V o f a lgebras , are

the following:

(a) Is Fr(V, n + 1) e m b e d d a b l e in Fr(V, n)?

(b) Is Fr(V, n) non-freely genera ted by some n-e lement set?

Here Fr(V, n) is the V-free a lgebra on n genera tors , i f the genera t ing set X needs

to be specified, then we write Fr(V, X). F o r many famil iar varieties, the answer to

(a) is k n o w n to be affirmative, while the answer to (b) is negative. This is the case,

e.g. for the variety o f all g roups ( for n -> 2) and for the variety o f all latt ices ( for

n >-- 3). F o r many d i sc r imina to r variet ies and, more specifically, for m a n y variet ies

o f re la t ion a lgebras and cyl indric a lgebras , the answer to (a) turns out to be negat ive

and the answer to (b) affirmative. There are, however , variet ies o f re la t ion a lgebras

for which bo th quest ions have an aff irmative answer, and obvious ly there are o ther

variet ies where the answer to bo th quest ions is negative, since every locally finite

Presented by E. Fried. Received March 10, 1989 and in final form February 13, 1990. ~,3 Research supported by Hungarian National Foundation for Scientific Research grant No. 1810. 2 Research supported by NSF grant DMS-8800290.

401

402 H. ANDRI~KA, B. JONSSON A N D 1. NEMETI ALGEBRA UNIV

variety is such. The only combination that we have not encountered for relation algebras is the one that applies for the variety of all groups and the variety of all lattices: yes for (a) and no for (b). The same applies to finite-dimensional cylindric algebras. Complementary results on the structure of free algebras is obtained in Berman and Blok [ 1987].

In a sense, the following theorems could be regarded as the main contributions of the paper: Theorem 2.6, Theorem 3.1, Theorem 6.6, Corollary 8.5, Theorem 9.1, Theorem 10.3, Theorem 10.4.

We list here some of the notation and concepts that will be used below. An algebra A will be identified with its universe, except in the rare situations where this is likely to create ambiguities. We denote by Con(A) the congruence lattice of A, and if R is a binary relation on A, then con(R) will be the congruence relation generated by R. If R is a singleton, R = {(u, v)}, then the principal congruence con(R) will be written con(u, v). A congruence relation R on A is said to be compact if it is generated by a finite relation, this means that R is a compact element of the algebraic lattice Con(A). We say that A is congruence extensile if every congruence relation on a subalgebra of A can be extended to a congruence relation on A, and we say that A is semisimple if A is (isomorphic to) a subdirect product of simple algebras. A class K of algebras is said to be congruence extensile if every member of K is congruence extensile, and K is said to be semisimple if every member of K is semisimple.

For a class K of algebras, H(K), S(K), P(K), Ps(K) and Pu(K) are the classes consisting of all algebras that are isomorphic, respectively, to homomorphic images, subalgebras, direct products, subdirect products, and ultraproducts of members of K. Var(K) is the variety generated by K, and Si(K) is the class of all subdirectly irreducible members of K.

1. Discriminator varieties

The discriminator played an important role in the work of Alfred Foster on primal algebras, starting in the early 1950's, and it has since then found extensive use in a variety of contexts in the work of many mathematicians, including Stan Burris, Stephen Comer, Alden Pixley, Robert Quackenbush and Heinrich Werner. After the pioneering work of S. Comer and others motivated by algebraic logic, it was H. Werner who founded the general theory of discriminator varieties, and an excellent survey of the subject can be found in Werner [1978]. The omnibus theorem stated below is essentially Werner's Theorem 2.2, and we refer the reader to his article for a more detailed treatment.

Vol. 28, 1991 Free algebras in discriminator varieties 403

The (ternary) discriminator on a set U is the o p e ra t i o n f : U 3 ~ U such that, for

all x, y, z ~ U,

f ( x , y , z ) = { z x if x ---y otherwise.

A variety V of algebras is called a discriminator variety if V is generated by a class K of algebras with the property that some term t in the language of V represents the discriminator on each member of K. The term t is called a discriminator term for K.

It is often convenient to use, in place of the ternary discriminator f , the quaternary discriminator, or the normal transform. This is the operation g of rank four such that

g ( x , y , u , v ) = { ~ i f x = y if x Cy .

The two operations generate the same clone, for

g(x, y, u, v) = f ( f ( x , y, u),f(x, y, v), v) and f (x , y, z) = g(x, y, z, x).

Consequently, if there exists a term representing the ternary discriminator on a class K of algebras, then there also exists a term representing the quaternary discriminator on K.

The quaternary discriminator can be used to define operations by cases. Given n-ary operations h', h", k' , k" on a set U, if we let

h(x_) = g(k'(x_), k"(x_), h'(x_), h"(x_))

for _x e U ", then

Sh'(x_) if k'(x) = k"(x) h(_x)

[h '(x) otherwise.

This patching technique will play an essential role in Section 8.

T H E O R E M 1.1. Suppose V is a discriminator variety generated by a class K, with t a discriminator term for K. Then the following statements hold:

I. V is congruence permutable, congruence distributive, congruence extensile and semisimple.

404 H. ANDRI~KA, B. JONSSON AND 1. NI~METI ALGEBRA UNIV

II. For every non-trivial algebra A m V, the following conditions are equivalent: (i) A is subdirectly irreducible.

(ii) A is simple. (iii) A is directly indecomposable.

(iv) A �9 SPu(K). (v) t represents the discriminator on A.

III. For every algebra A in V: (i) For all u, v �9 A, con(u, v) = {(x, y) : t(u, v, x) = t(u, v, y)}.

(ii) Every principal congruence on A is a factor congruence. (iii) Every compact congruence on A is principal.

(iv) The principal congruences on A form a sublattice on Con(A).

IV. HSi(V) = HSPu(K). HSi(V) is a universal class, and the map M ~ Var(M) is an isomorphism from the lattice of all universal subclasses of HSi(V) onto the

lattice of all subvarieties of V. The inverse of this isomorphism is the map U ~-} HSi(U) = U m HSi(V), for U a subvariety of V.

V. There is an effective way of associating with each open Horn jormula ~t in the language of V an equation ct* in that language in such a way that ~t and ct* have the

same truth set in every member of Si(V). VI. For any open Horn jbrmula ~t in n variables, and for any n-ary terms u and

v, there exists an n-ary term w with the property that, for all A �9 Si(V) and ~ �9 A ~,

~u(~) i fA, 6~ W(~) ~x

[v(~) otherwise.

2. Locally finite varieties

A variety V is called locally finite if finite sets generate finite subalgebras in every member of V. The following simple observation is certainly known.

LEMMA 2.1. I f A is a finitely generated algebra of a finite similarity type, and if R is a congruence relation on A such that A / R is finite, then R is compact.

Proof. Choose a finite generating set X for A, and a transversal T for R, i.e., a set consisting of just one member of each R-block. For x �9 A, let f ( x ) be the member of T that is R-related to x. Let Q be the set of all ordered pairs (x , f (x)) with x either a member of X, or else an element that is obtained by applying one of the basic operations of A to a sequence of elements from T. Then R = con(Q).

COROLLARY 2.2. Suppose V is" a discriminator variety of a finite similarity

type. I f B is a finite homomorphie image of a finitely generated member A of V, then B is isomorphic to a direct factor of A.

Vol. 28. 1991 Free algebras in discriminator varieties 405

Proof The kernel R of the homomorphism is compact by the preceding lemma, and is therefore a factor congruence by Theorem l.l.III(iii), (ii).

COROLLARY 2.3. Suppose V is a discriminator variety of a finite similarity type, and V' is a locally finite subvariety of V. Then, for every positive integer n, Fr(V', n) is isomorphic to a direct factor of Fr(V, n).

Proof By Corollary 2.2, with A = Fr(V, n) and B = Fr(V', n).

If V is a locally finite discriminator variety, and n is a positive integer, then the finite algebra Fr(V, n) is a direct product of finite simple algebras. The factors are the members S of Si(V) that are n-generated (are generated by a set of order n or less). In order to describe how often each such factor S occurs in the direct product, we need the following concepts.

DEF INI TI ON 2.4. Suppose S is an algebra, and X is a set. (i) Two mappings f and g from X onto generating sets for S are said to be

equivalent if g = hf for some automorphism h of S. (ii) If IX[ = n, then we denote by /~(S, n) the number of non-equivalent map-

pings from X onto generating sets for S.

As indicated by the notation, the cardinal/~(S, n) is completely determined by S and n, and does not depend on the set X. If S is finite, which is the only case that concerns us here, then #(S, n) is a natural number; it is positive iff S is n-generated.

An automorphism of S is completely determined by its values on a generating set for S. Consequently, i f f is a mapping of X onto a generating set for S, then distinct automorphisms h of S yield distinct mappings hf The number of mappings in each equivalence class is therefore equal to the order of the automorphism group

of S. If S is an algebra in a variety V, then each homomorphism from Fr(V, X) onto

S maps X onto a generating set for S, and each mapping from X onto a generating set for S can be uniquely extended to a homomorphism from Fr(V, X) onto S. Consequently, #(S, n) is equal to the number of homomorphisms from Fr(V, n) onto S, divided by the number of automorphisms of S.

The following observation will be used later with A = Fr(V, n).

C OR OLLARY 2.5. Suppose f and g are homomorphisms from an algebra A onto an algebra S, X is a generating set for A, and f" and g' are the restrictions o f f and g to X. Then the following conditions are equivalent:

(i) f and g are equivalent

406 H. ANDRI~.KA, B. JONSSON AND 1. NEMETI ALGEBRA UNIV.

(ii) f ' and g' are equ&alent (iii) ker f = ker g. Thus the number of non-equivalent homomorphisms from A onto S is equal to the

number of congruence relations R on A with A/R ~-S.

Proof For an automorphism h of S, the homomorphisms g and hf are equal iff they agree on the generating set X, i.e., iff g ' = hf'. Hence (i) and (ii) are equivalent. Obviously f and hf have the same kernel, and (i) therefore implies (iii). Conversely, if k e r f = ker g = R, t h e n f a n d g induce isomorphisms f* and g* from A/R onto S, and h = g ' f * - 1 is therefore an automorphism of S. For x ~ A,

hf(x) = hf*(x /R) = g*(x /R) = g(x),

so that g =hf.

THEOREM 2.6. I f V is a locally finite discriminator variety and n is a positive integer, then

Fr(V, n) ~- I ] ( S'~s' =~ : S e Si(V)).

Proof The finite algebra A = Fr(V, n) is isomorphic to the direct product of all the algebras A/R with R a co-atom in Con(A). Each of these quotients is an n-generated member S of Si(V), and by the preceding corollary, each such S will occur precisely/~(S, n) times as a quotient AIR.

Observe that only finitely many members S of Si(V) are n-generated, and that for the remaining members kt(S, n) = 0. The above direct product therefore has only finitely many non-trivial factors.

LEMMA 2.7. Suppose V is a congruence distributive semisimple variety, and A ~ V. For any subvariety U of V, let F(U) be the smallest congruence relation on A such that A/F(U) ~ U. Then F is a homomorphism from the lattice of all subvarieties of V into the dual of Con(A).

Proof. Since V is semisimple, every member of Con(A) is the intersection of co-atoms in Con(A). It therefore suffices to show that, for any subvarieties U and U' of V, and any co-atom R in Con(A),

(1) R ~_F(Uc~U')

(2) R ~ F ( U + U ' )

iff R _~ F(U) + F(U'),

iff R ~ F(U) c~F(U').

Vol. 28, 1991 407

The statement (1) is obvious:

R ~ F ( U c ~ U ' ) iff A / R ~ U n U ' ,

iff A/R ~ U and A/R ~ U',

iff R ~_ F(U) and R _~ F(U'),

iff R _~ F(U) + F(U').

To prove (2), we use the fact that V is congruence distributive, whence Si(U + U') = Si(U) u Si(U'). Also, from the distributivity of Con(A) it follows that the co-atom R of Con(A) is above F(U) n F(U') iff R is above either F(U) or F(U').

Consequently,

R _ F(U + U') iff A / R ~ S i ( U + U ' ) ,

iff A / R ~ Si(U) or AIR ~ Si(U'),

iff R ~ F(U) or R ~_ F(U'),

iff R _ F(U) n F(U').

T H E O R E M 2.8. Suppose V' and V" are subvarieties of a discriminator variety V of a finite similarity type. I f V' c~ V" is locally finite, then for every positive integer n,

Free algebras in discriminator varieties

Proof We may assume that V = V' + V'. Let A = Fr(V, n) and, defining F as in

the preceding lemma, let R ' = F ( V ' ) and R ' = F ( V " ) . Then A/R', AIR" and A / ( R ' + R") are free algebras on n generators in the varieties V', V" and V ' n V',

respectively. The relation R'c~ R" = F(V) is the trivial congruence relation on A, i.e., R ' and

R" are disjoint in Con(A). Also, by Lemma 2.1 and Theorem 1.1.III(iii),(ii), R' + R" has a complement Q in Con(A). Letting

B = A/(R" + R"), C = A/(R' + Q), D = A / (R"+ Q)

Fr(V' + V", n) • Fr(V' c~V", n) ~ Fr(V', n) • Fr(V", n).

408 H. ANDRI~,KA, B. JONSSON A N D 1. NC:METI ALGEBRA UNIV.

we infer that

Fr(V, n) ~- B • C x D, Fr(V' n V", n) ~- B,

Fr(V', n) ~- B x C, Fr(V', n) _~ B x D,

whence the theorem follows.

For the case when V' and V" are disjoint, i.e., when V' c~ V" is the trivial variety, we readily obtain a stronger result.

L E M M A 2.9. Suppose V' and V" are disjoint subvarieties of. a congruence permutable, congruence distributive, semisimple variety V. Then all A ~ V ' + V' ,

A' ~ V' and A" ~ V' , A ~- A" • A" iff A ' ~- A / R ' and A" "~ A/R" , where R' and R" are the smallest congruence relations on A such that A / R ' ~ V' and A /R" ~ V".

Proof By Lemma 2.7, R ' and R" are complements of each other in Con(A), and since A is congruence permutable, this implies that A ~- (A/R') x (A/R").

Suppose A - ~ A ' x A ' , and let S ' and S" be the kernels of the induced projections from A onto A' and A". Then A / S '~- A ' ~ V' and A / S ''~- A " ~ V",

whence R ' ~ S ' and R" ___ S". Also S ' and S" are complements of each other in the distributive lattice Con(A), which implies that S ' = R' and S " = R".

T H E O R E M 2.10. Suppose V' and V" are disjoint subvarieties of a congruence permutable, congruence distributive, semisimple variety V. Then, for ever)' non-zero cardinal ~c,

Fr(V' + V", ~:) ~- Fr( V', ~:) • Fr(V", •).

Proof. In the notation of the preceding lemma, Fr(V', ~c)~-Fr(V' + V", ~:)/R' and Fr(V", ~) ~ Fr(V' + V", K)/R".

3. Absolute retracts

A member S of a class K of algebras is said to be an absolute retract in K if, for every embedding f : S ~ A ~ K, there exists a homomorphism g : A --~ S such that

g f is the identity map on S. It will be shown that the variety RA of all relation algebras contains many finite absolute retracts; in fact, the full algebra of relations

on a finite set is an absolute retract in RA. The results obtained in this section therefore apply, in particular, to RA, and to many of its subvarieties. The same will apply to finite-dimensional cylindric algebras.


T H E O R E M 3.1. Suppose V b a congruence extensile variety, and suppose there

exists a non-trivial, k-generated finite absolute retract in V. Then, for every integer n >- k - 1, Fr(V, n + 1) is not embeddable in Fr(V, n).

Proof Suppose S is a non-trivial, k-generated, finite absolute retract in V. Assume that f : Fr(V, n + 1 ) ~ Fr(V, n) is an embedding. We are going to show that f is injective over S, in the sense that every surjective homomorphism

g : Fr(V, n + 1) -~ S factors through f Since V is congruence extensile, there exist S ' e V, a homomorphism

g' : (V, n) ~ S', and an embedding f ' : S ~ S' such that g ' f = f ' g . Since S is an absolute retract, there exists a homomorphism h : S ' ~ S such that hf' is the

identity map on S.

Fr(V , n + 1)f~f~Fr(V, n)

g g

S ) - " ) S h

Figure l

Letting g # = hg', we therefore have g = g # f Clearly, for different homomorphisms g, the homomorphisms g # will be

different. We are going to use this to obtain a contradiction, by showing that the number m of homomorphisms from Fr(V, n) onto S is smaller than the number m'

of homomorphisms from Fr(V, n + 1) onto S. The algebras Fr(V, n) and Fr(V, n + 1) are freely generated by sets X and Y of

order n and n + 1, respectively. For each set G that generates S, let v(X, G) be the number of maps from X onto G, and define v(Y, G) similarly. Then m is the sum of the numbers v(X, G), and m' is the sum of the numbers v(Y, G). We have v(X, G) <- v(Y, G) for all G, with strict inequality holding whenever 2 < IG[ < n + 1. Since S is non-trivial and k-generated, with k -< n + 1, the conclusion follows.

In connection with Theorem 3.1, we note that the condition n > k - 1 is, very probably, necessary by the following. For every natural number k, Tardos [1988] constructed a variety V such that Fr(V,k) is embeddable in Fr(V, l) but Fr(V, n + l) is not embeddable into Fr(V, n) for any n > k. The definition of V is as follows: the language of V contains a binary operation symbol and for every i > k a unary operation symbol f . The defining equations of V are all the equations

of the following kind:

f z = z where z contains at most i distinct variable symbols.

410 H. ANDRt~KA, B. JONSSON A N D I. NI~'METI ALGEBRA UNIV

This variety is not congruence extensile, though. It would be interesting to find a variety V with the above properties, such that in addition V is congruence extensile and has a finite absolute retract.

By an essential extension of an algebra S we mean an extension A of S with the property that, for every non-trivial congruence relation R on A, the restriction of R to S is also non-trivial. The following simple observation is well known.

LEMMA 3.2. For every member S o f a variety V of algebras, the Jollowing

conditions are equivalent:

(i) S is an absolute retract in V.

(ii) For ever), extension A of S in V, there exists a congruence relation R on A

such that each R-block contains precisely one member of S.

(iii) S has no proper essential extension in V.

The next two results establish a connection between absolute retracts in a variety V and members S of Si(V) that are maximal, in the sense that S has no proper extension in Si(V).

LEMMA 3.3. Suppose V is a variety and S ~ Si(V). Consider the jbllowing

conditions:

(i) S is maximal in Si(V). (ii) S is an absolute retract in Si(V).

(iii) S is an absolute retract in V.

Then (i) =~ (iii) =~ (ii). I f V is semisimple, then all three conditions are equivalent.

Proof. Every essential extension of a subdirectly irreducible algebra is subdirectly irreducible. Hence, if (i) holds, then S has no proper essential extension in V, and (iii) therefore holds by Lemma 3.2. Obviously (iii) implies (ii). Assume now that V is semisimple. If (i) fails, then S has a proper extension A in Si(V). From the semisimplicity of V it follows that A is simple, so there cannot exist a retraction of A onto S. Hence (ii) fails.

EXAMPLE. The implication (ii) =~ (i) is not true in general. As an example, take the variety V of all algebras (A, + , 0, i , f ) such that (A, + , 0 , 1) is a bounded semilattice and x <-f(x) for all x e A. Every non-trivial member of V can be retracted onto its 2-element subalgebra by sending all the non-zero members to 1. The 2-element algebra is therefore an absolute retract in V. But this algebra is not a maximal member of Si(V), because the 3-element chain 0 < a < 1, with f (a ) = 1, is also a member.


T H E O R E M 3.4. For any discriminator variety V, the following conditions are

equivalent: (i) Some non-trivial finite member of V is an absolute retract in V.

(ii) Some finite member of Si(V) is maximal in Si(V).

Proof. By the preceding lemma, (ii) implies (i). To prove the converse, we show that in a congruence distributive variety, every direct factor of an absolute retract is an absolute retract. In fact, suppose A = B • C is an absolute retract, and suppose B' is an extension of B. Then A ' = B' • C is an extension of A. Hence

there exists a congruence relation R on A' having A as a transversal. From the fact that A' is congruence distributive it follows that R is a direct product of congruence

relations R' on B' and R" on C. That is,

((x, y), (u, v)) e R iff (x, u) E R' and (y, v) e R".

It is easy to check that B is a transversal for R'. Thus B is an absolute retract

in V. A member S of a variety V is said to be splitting in V if there is a largest

subvariety V* of V such that S ~ V*, and V* is then called the conjugate variety of

S.

LEMMA 3.5. I f V is a discriminator variety of a finite similarity type, then every finite, subdirectly irreducible member of V is splitting in V.

Proof. Suppose S E Si(V) is finite, and let M be the class of all A ~ HSi(V) such that S is not embeddable in A. Then M is a universal class. Hence, by Theorem 1.I.IV, M = HSi(V*) for some subvariety V* of V. Clearly V* is the largest

subvariety of V with S r V*, and S is therefore splitting.

In Blok and Pigozzi [1982], Corollary 3.2, it is proved that, more generally, every variety in which principal congruences are equationally definable has the

property that every finite, subdirectly irreducible algebra is splitting.

T H E O R E M 3.6. Suppose V is a discriminator variety of a finite similarity type, and S is a finite maximal member of Si(V). I f A is a finitely generated member of V, then A ~ S m • (A/Q), where m is the number of non-equivalent homomorphisms from A onto S, and Q is the smallest congruence relation on A such that A/Q is in the conjugate variety of S in V.

Proof By Corollary 2.5, the number m of non-equivalent homomorphisms from A onto S is equal to the number of co-atoms R in Con(A) with A/R ~ S. Therefore,

412 H. ANDRI~KA, B. JONSSON A N D 1. NI~METI ALGEBRA UNIV

if Q' is the intersection of all these congruence relations R, then A/Q' ~- Sm. To complete the proof, it therefore suffices to show that Q and Q' are complementary factor congruences.

To prove that Q c~Q '= 0, the trivial congruence relation, we note that every co-atom R in Con(A) is above either Q or Q': I f A / R ~-S, then R is above Q', but in the alternative case R is above Q, because the variety generated by A/R does not contain S. If A/R ~-S, then R is not above Q, and therefore R + Q = L the universal congruence relation. Thus Q' is the intersection of finitely many relations

R w i t h R + Q = L w h e n c e Q ' + Q = L

THEOR EM 3.7. Suppose V is a discriminator variety of a finite similarity type, and S is a finite maximal member of Si(V). Then for any positive integer n,

Fr(V, n) ~- S ~r ") • Fr(V*, n),

where V* & the conjugate variety of S in V.

Proof By the preceding theorem.

4. Ordered discriminator varieties

Most of the familiar discriminator varieties V consist of algebras with a natural partial ordering, an ordering that can be defined uniformly for all members of V by a finite set of equations:

x < y iff s,(x, y) = ti(x, y) f o r a l l i ~ I .

Such an ordering necessarily respects the operations of forming direct products,

subalgebras, and homomorphic images. More precisely:

If A = I-] (Aj , j ~ J) and x, y ~ A, then x < y in A iff xj < yj in Aj for all j ~ J. I f A i s a s u b a l g e b r a o f B a n d x , y ~ A , t h e n x < y i n A i f f x < y i n B. If f : A ~ B is a homomorphism and x , y ~ A , then x < y in A implies

f (x ) < f ( y ) in B.

We refer to a variety V with a partial ordering having these properties as an ordered variety. (Actually, the second property is not used in this paper.) If the similarity type of V has a constant 0 such that, for every A e V, 0 A is the smallest member of A, then we say that 0 is a zero element for V.


It is perhaps worth noting that we are speaking here of a variety together with

a partial ordering, not just of a variety that possesses a partial ordering. In any

variety, the algebras have a trivial partial ordering, with x < y i f fx = y . In a

discriminator variety whose type has a constant O, a partial order can be (equation-

ally) defined in such a way that, in a simple algebra,

x - < y i f f x = 0 or x = y ,

and if there are two constants 0 and 1, this can be changed to

x < y iff x =O or x = y or y = l.

All the specific varieties under consideration here consist of Boolean algebras with

additional operations, and it is the Boolean inclusion relation that is of primary interest to us.

Suppose ,4 is a poset (a partially ordered set). For a, b e A, we write a -< b to

indicate that a is covered by b. We say that A is atomic if, for all a, b e A with

a < b, there exists c e ,4 with a -< c -< b.

An algebra A is called residually finite if ,4 is (isomorphic to) a subdirect product of finite algebras.

T H E O R E M 4.1. Suppose V is an ordered discriminator variety of a finite

similarity type. l f A is a finitely generated member of V, then the following statements hold:

(i) I r A is residually finite, then A is atomic. (ii) I f A has a smallest element, and i f Con(,4) has k or more co-atoms R such

that A / R is finite, then A has at least k atoms.

Proof (i). Given a < b in A, we need to find c e A with a < c < b. Since A is residually finite, there exists R ~ Con(A) with AIR finite and a/R • b/R. By Lemma

2.1, R is compact, whence by Theorem l . l . I I I , R is a factor congruence. Let R ' be

the complement of R in Con(A). Then the mapping x ~ f ( x ) = (x/R, x /R ' ) is an isomorphism from ,4 onto (A/R) x (A/R') . In the finite poset ,4/R, a /R < b/R, and

hence a i r ~ C < b/R for some C. Let c be the member of A with f (c) = (C, a/R').

Then c is between a and b, and c covers a as required.

(ii) Suppose A has a smallest element 0, and suppose Ri, 0 < i < k, are pairwise distinct co-atoms of Con(A), with A/Ri finite. Let R = R0 n RI c~ . . . c~ Rk _ 1. Then the algebra A / R is isomorphic to the direct product of the algebras A/Ri, and A / R is isomorphic to a direct factor of A. From this (ii) follows.

414 H. ANDREKA, B. JONSSON AND 1. NEMET1 ALGEBRA UNIV.

T H E O R E M 4.2. Suppose V is an ordered discriminator variety of a finite

similarity type, having a zero element, and suppose there exists a non-trivial finite algebra in V. I f F is V-freely generated by a set X, then every member of X is above

some atom of F.

Proof. Choose S ~ V finite and non-trivial. Given x e X, choose a homomorphism h : F--* S with h(x) 4:0 s. Then h maps F onto a non-trivial subalgebra of S.

The kernel R of h is a factor congruence, and we argue as in the proof of the

preceding theorem to obtain an element c e F with 0 r < C < X.

5. Nonassociative relation algebras

In the next several sections we investigate varieties of relation algebras and, more

generally, of semiassociative relation algebras. In this section we recall some basic

facts about these concepts. Many of these results are valid, more generally, for

nonassociative relation algebras. It is convenient to start with the notion of a conjugate, introduced in J6nsson

and Tarski [1951], Definition 1.11. By a conjugate of a function f : A ~ B, where A

and B are Boolean algebras, we mean a function g : B ~ A such that, for all x e A

and y E B

f ( x )y = 0 iff xg(y) = O.

It is obvious that if g is a conjugate o f f , then f is a conjugate of g. Some other basic

facts are as follows: I f f has a conjugate, then f preserves all existing joins: x = ~ xi implies

f ( x ) = ~ f (x i ) . The conjugate g of f, if it exists, is unique, and is given by the formula

g(Y) = 1-I (x ~ A : f ( x - ) y = 0).

The relationship be tweenfand its conjugate g can be expressed as a pair of identities:

f ( xg (y ) ) < f ( x )y , g(yf(x) ') < g ( y ) x - .

By a nonassociative relation algebra we mean a Boolean algebra A with

additional operations ; , 1' and ~ of ranks 2, 0 and 1, respectively, such that (1) x ; l ' = l ' ; x = x f o r a l l x e A .


(2) For all a e A, the maps

x ~ a ; x and yw-~a~;y

are conjugates of each other, and likewise the maps

x w-~ x ; a and y ~-+ y ; a ~.

If, in addition, (3) (x; 1); 1 = x ; 1 for all x e A,

then A is called a semiassociative relation algebra, and if (4) x; (y; z) = (x; y); z for all x, y, z e A,

then A is called a relation algebra.

The three classes consisting of all nonassociative relation algebras, of all semiassociative relation algebras, and of all relation algebras, will be written NA, SA and RA, respectively. Obviously the three classes are varieties, and RA ~ SA _ NA.

The primary model for the notion of a relation algebra is the algebra Re(U) of all binary relations on a set U. More generally, if V is an equivalence relation on a set U, then the subrelations of V form a relation algebra ~(V) . By a representation of a relation algebra A, we mean an embedding A ~ ~(V) , with V an equivalence relation. The class RRA of all representable relation algebras is a variety. This important result is due to A. Tarski, a proof can be found in J6nsson [1982].

Some important subvarieties of RRA rate special mention. For any positive integer k, the algebra R e ( k ) , - i.e. Re(U), where IUI = k , - generates a variety RA(k), while the algebras R e ( U ) with U infinite all generate the same variety, which will be written RA(~) .

Our primary interest here is in the variety RA and its subvarieties. The varieties NA and SA were introduced and investigated by R. Maddux, who has shown that a large part of the arithmetic of relation algebras extends to these larger classes. In particular, SA turns out to be a discriminator variety, which makes many of our results applicable to it. Most of the results listed in the next two theorems can be found in Maddux [1982] and [1987].

It is convenient to have names for some special elements that will be needed here

and later. Suppose A ~ NA. The element 1' is referred to as the identity element of A, and its complement, 0' = 1' , as the diversity element. An element a e A is said to be

symmetric if aU = a, transitive if a; a < a,

416 H. ANDRI~KA, B. JONSSON AND 1. NE.METI ALGEBRA UNIV.

an equivalence e lement if a = a ~ = a; a,

a left ideal e lement if a = 1 ;a ,

a r ight ideal e lement if a = a; !,

an ideal e lement if a = 1; a = a ; l,

a func t iona l e lement if a~ ; a <- 1'.

We use the c o n v e n t i o n tha t Boo lean mul t ip l ica t ion binds s t ronger than ; , e.g., x; yz

means x; (yz) and no t (x; y)z.

T H E O R E M 5.1. The following statements hoM & every nonassociative relation

algebra.

(i) (ii)

(iii)

(iv) (v)

(vi)

(vii)

(viii)

(ix) (x)

(xi) (xii)

(xiii)

(xiv) (xv)

(xvi)

0 ~ = 0 , 1 ~ = 1 , 1 ' ~ = 1 ' , 0 ' ~ = 0 ' .

(x + y) ~ = x ~ + y~ , (xy) ~ = x~yU, x ~ = x ~

(x; y)~ = y~; x ~, x ~ = x.

0 ; x = x ; 0 = 0 , 1 ; 1 = 1 .

X ~ I f the join u = ~, x i exists, then the joins Y" i , E xi ; Y, E Y; xi also exist,

and are equal to u ~, u; y and y; u respectively.

x~=y~(y:x ;y<_O')=~(y:y;x <-03. (x; y)z = 0 iff (x~; z)y = O, iff (z; y ~ ) x = O.

x ; y < - z i f f x - < ( z - ; y ~ ) - , i f f y - < ( x ~ ; z - ) .

(x; y)z < x; y(x~; z), (x; y)z < x(z; y ~ ) ; z.

x ; y = 0 i f f (x~ ' ; l )y = 0 , i f f x ( 1 ; y ~) = 0 .

I f x < 1', then x is an equivalence element.

I f x <- 1', then x; yz = (x; y) (x; z) and yz; x = (y ; x)(z; x).

I f x <- 1', then x; yz = (x; y)z and yz; x = y(z; x).

I f x < 1", then x; y = ( x ; l )y and y; x = y ( 1 ; x ) .

I f x, y < 1', then x; y = xy.

x = (x; x ~ ) l ' ; x = x; (x~; x) l ' .

T H E O R E M 5.2. Suppose A ~ NA, and let L L and R be the sets consisting,

respectively, o f the ideal elements, the left ideal elements, and the right ideal elements

o f A. Then the following statements hold.

(i) L L and R are closed under the Boolean operations +, �9 and .

(ii) The operation ~ maps L and R bijectively onto each other, while a ~ = a for

a E I .

(iii) For all a 6 L, b ~ R and x 6 A,

(1; x)a = 1 ;xa , b(x; l) = b x ; 1.

(iv) I f a ~ 1, then a is an equivalence element and a(x; y) = ax; a y j o r all x, y ~ A.


If A is a Boolean algebra, then the mapping that sends each congruence relation

R on A into the block O/R of R is an isomorphism from Con(A) onto the lattice J (A) of all ideals of A. If A is a Boolean algebra with additional operations, then the same mapping sends Con(A) into a sublattice of J ( A ) . We call an ideal O/R with R ~ Con(A) a congruence ideal, and we call an element a ~ A a congruence element if the principal ideal Aa = {x ~ A : x < a} is a congruence ideal. The second part of

the next theorem follows from Maddux [1987], Theorem 28.

T H E O R E M 5.3. Suppose A ~ N A . Then an ideal I o f A is a congruence ideal i f f

x; 1, 1; x ~ I whenever x ~ L

and an element a ~ A is a congruence element i f f a is an ideal element.

In a Boolean algebra A, a principal ideal Aa can be made into a Boolean algebra: The join and the meet operations in Aa are the restrictions to Aa of the

corresponding operations in A, the top and bottom elements are of course a and 0, and the complement in Aa of an element x < a is the relative complement a x - . We call Aa a relative subalgebra of A. The mapping x ~ ax is a homomorphism from

A onto Aa.

If A is a Boolean algebra with additional operations, then Aa can also be made into a Boolean algebra with additional operations: I f f is an operation on A, then

the corresponding operation fa on Aa is defined by letting

L ( ~ ) = af(g)

for a sequence ~ of elements of Aa. We again refer to Aa as a relative subalgebra

of A. The construction of relative subalgebras is primarily of interest when the

operations f are operators, i.e., when they preserve finite joins in each argument. Even then, many properties of A are not preserved by the construction. E.g., none of the classes NA, SA, RA or RRA is closed under relativization. However, for special elements a there are positive results. The first part of Theorem 5.4(ii) is

contained in Maddux [1987] Theorem 28.

T H E O R E M 5.4. (i) I f K is one o f the classes NA, SA, RA and RRA, then f o r

every A ~ K and any equivalence element a ~ A we have Aa ~ K.

(ii) For any A E N A and any ideal element a E A, the mapping x ~ ax is a

homomorphism f r o m A onto Aa, and the mapping x ~ (ax, a - x ) is an isomorphism

f r o m A onto Aa • A a - .

418 H. ANDREKA, B. JONSSON AND 1. NI~METI ALGEBRA UNIV

The not ion of a p roduc t o f a finite sequence 2 = (xo, x ~ , . . . , xn ~ ) o f elements

in a semiassociative relation algebra is defined by induction on n. I f ~c is the null

sequence (n = 0), the unique product is 1', and for n = 1 the unique produc t is x0. For n > 1, a is a p roduc t o f ~ iff, for some k < n - 1, a is a p roduc t o f the sequence

(Xo~XI~. . - ,X i I~Xi;Xi+lnXt+2~.. .~Xn-1).

Equivalently, for n > 1, a is a product o f 5c iff a = b; c, where b is a p roduc t o f a

proper initial segment o f 2, and c is a p roduc t o f the complemen ta ry final segment.

We say that k is associative if it has a unique product .

Par t (i) of the next result is Theorem 25 in M a d d u x [1987]; par ts (ii) and (iii)

are immediate consequences. Corol lary 5.6 is par t o f Theorem 29(iv) in Maddux

[1987].

T H E O R E M 5.5. For A ~ SA, each o f the following conditions implies that the

sequence ~ ~ A" is associative.

(i) x, = 1 for some i < n .

(ii) For some i < n, x, is either a left ideal element or a right ideal element.

(iii) 0 is a product o f Y~.

Proo f o f (iii). I f 0 and a are products o f 2, then a; 1 = 0; 1 = 0, hence a = 0. This result is impor t an t because it permits us to omit parentheses in many

calculations. In part icular, the condit ions

x; (y ; z) = 0, (x; y); z = 0

are equivalent.

C O R O L L A R Y 5.6. A nontrivial algebra A ~ SA is subdirectly irreducible iff

1 ; x ; l = l whenever O vL x 6 A.

C O R O L L A R Y 5.7. SA is a discriminator variety.

Proof. The term

t ( x , y , z ) = x ( 1 ; ( x O y ) ; l ) + z ( l ; ( x ~ ) y ) ;1)

is a discr iminator term for Si(SA), where x O y is the symmetr ic difference of x and

V.


We note that NA is not a discr iminator variety, e.g. because there is a

subdirectly irreducible but not simple algebra in NA, see M a d d u x [1978]. However ,

NA is still congruence extensile because Boolean algebras with opera tors are

congruence extensile in general.

T H E O R E M 5.8. The following statements hoM in every simple algebra A E SA.

(i) 0' = 0 or 0'; 0' = l ' or 0'; 0' = l.

(ii) I f a; 1;b = 0 , then a = 0 or b = 0 . (iii) I f a ~ ; l ; a < 1', a; 1 ;a ~ < 1' a n d a > 0 , then a is an atom.

(iv) I f a and b are right ideal elements, then

( ivl) a; b = a unless b = 0 . (iv2) a~; b ~ = b ~ unless a = O.

(iv3) a~; b = 1; ab; 1.

(iv4) a; b ~ = ab ~

Proo f (i) This is p roved in J6nsson and Tarski [1952] for relat ion algebras.

With some care, the p r o o f carries over to the semiassociative case.

(ii) This is contained in Theorem 29 in M a d d u x [1987].

(iii) Fo r A ~ RA this is L e m m a 7.3 in J6nsson [1982]. The p r o o f applies, more

generally, for A E SA. (iv) I f b 5 0 , then a; b = a; 1; b; 1 = a; 1 = a. By symmetry , a~; b ~ = b ~, unless

a = 0. By T h e o r e m 5.1(x), a~; b = 0 iff (a; 1)b = 0, which in the present case means

ab = 0 . I f ab ~ 0 , and hence also a~; b r 0, then a~; b = 1; a ~ ; b ; 1 = 1, and also 1; ab; 1 = 1. Thus in either case a~; b = 1; ab; 1. Finally, a; b u < (a; 1)(1; b ~) =

ab ~, while by Theo rem 5.1(ix), ab ~ = (a; 1)b ~ -< a; (a~; b ~) < a; (1; b ~) = a; b ~.

Thus a; b ~ = ab.

Theorem 5.8(iii) has a consequence that is o f fundamenta l impor tance to us.

T H E O R E M 5.9. For every positive integer k, Re(k) is an absolute retract in SA.

P r o o f The a toms in Re(k) satisfy the condit ions in Theorem 5.8(iii). Hence every

embedding Re(k) ~ A ~ Si(SA) takes a toms into a toms, and is therefore an iso- m o r p h i s m by k being finite. N o w par t (i) =~ (iii) o f L e m m a 3.3 finishes the proof .

We will see later in Theo rem 6.6, that Re(k) for k 4:1 is not an absolute retract in NA.

6. Subvarieties of SA

We begin by exploring some of the consequences of Theo rem 5.8(i).

420 H. ANDRI~KA, B. JONSSON AND I. NI~METI ALGEBRA UNIV

THEOREM 6.1. Let V~, V2 and V 3 be the subvarieties of SA defined by the identities 0' = 0, 0'; 0' = 1' and 0'; 0' = 1, respectively. Then

(i) The varieties Ve are pairwise disjoint, and their join is SA. (ii) Every algebra A e SA has, up to isomorphism, a unique representation

A ~ A I • A2 • A3 with A, e V , for i = 1,2, 3. (iii) For every positive integer n,

Fr(SA, n) ~- Fr(V~, n) x Fr(V2, n) x Fr(V3, n).

(iv) For i = 1, 2, 3, Ve has a unique m&imal nontriv&l subvariety Me, with Si(Mi) = {Me}, where Me consists of just the constants O, 1, 0', !'.

Proof By Theorem 5.8(i) and Lemma 2.9.

Actually Mj = V1 = RA(1). The sole member of Si(V~) is therefore the algebra MI of order two, with 0 = 0' and 1 = 1'. From any Boolean algebra A we obtain a relation algebra by defining 1 '= 1, x ; y = x y and x ~ = x , and these relation algebras are precisely the members of V1. The members of V~ are referred to as discrete relation algebras.

The algebras M2 and M3 are obviously completely determined by the conditions that 0'; 0 ' = 1' and that 0 ' ; 0 ' = I, respectively. Up to isomorphism, M2 is the minimal subalgebra of Re(2), and M3 is the minimal subalgebra of Re(k) with k > 3 .

Obviously V~ is a subvariety of RA. Less obvious is the fact that V2 _c RA. In other words, the condition 0'; 0 ' < 1' implies the associative law. This was first proved by R. Maddux and, independently, by P. Jipsen. A third proof, given below, was then obtained by the authors.

THEOREM 6.2. In the notation of the preceding theorem, V2 = RA(2).

Proof We shall prove that, in an arbitrary SA, the condition 0'; 0' <- 1' implies the associative law. From this the conclusion follows by J6nsson [1982], Corollary 6.4. The proof is based on two arithmetic facts that may be of some independent interest, and are therefore stated as iemmas.

LEMMA A. In any SA, the equation

a; (b; e) = (a;b); c

holds whenever one of the elements a, b and c is below 1".


Proo f o f L e m m a A. First suppose a -< 1'. By Theorem 5.1(xiv),

a; (b; c) = (a; 1)(b; c), ( a ;b ) ; c = (a; 1)b; c.

Since (a; l)b; c is below both a; 1 and b; c, it follows that (a; b); c < a; (b; c). To

prove the opposite inclusion we use Theorem 5.1(ix). This yields

a; (b; c) = (a; 1)(b; c) < b(a; 1; r c < b(a; 1); c = (a; b); c.

The case c < 1' follows by symmetry. Finally suppose b < 1'. Then for all x,

((a; b); c)x = 0 iff (a; b)(x; c ~) = 0

iff ((x; c~); b)a = 0

iff (x; (c~; b))a = 0

iff (a; (b; c))x = O,

whence (a; b); c = a; (b; c). Here we have used the fact that b ~ = b and (x; c~); b = x; (c~; b), since b < 1'.

L E M M A B. In an arbitrary SA, the conditions a; b < 1' and a~; c <- 1" jointly

imply that (a; b); c = ab~c.

Proof. We begin by showing that

(a; b); c -<- a. (1)

By Theorems 5.1(xiv) and 5.5,

(a; b); c = (a; (b; 1))c.

Our claim ( I ) is therefore equivalent to each of the condit ions

(a; (b; l))ca - = 0, (a~; ca - ) (b ; 1) = 0.

Thus it suffices to show that a~; c a - = 0. But we have in fact

a U;ca - < ( a ~ ; c ) ( a ~ ; a ) < 1 ' 0 ' = 0 .

422 H. ANDREKA, B. JONSSON AND 1. NI~MET1 ALGEBRA UNIV

Here we have used the fact that (a; l ' ) a = 0, hence ( a ' ; a - ) 1' = 0 or, equivalently,

a ' ; a < 0'. F r o m (1) and the fact that a; b <- 1' we infer that (a; b); c <- ac, and hence

(a; b); c = (a; b); ((a; b); c) < (a; b); ac,

so that

(a; b); c = (a; b); ac. (2)

We now use Theorem 5.1(ix) to compute

a; b = (a; b ) l ' = a ( l ' ; b~); b(a~; 1') = abe; a~b,

and then apply (2), with a and b replaced by ab ~ and aUb, to infer that

(a; b); c = (abe; a~b); c = (abe; a~b); ab~c <- able .

On the other hand, by Theorem 5.1(xvi),

ab~c < (abUc; a~bc~); ab~c <- (a; b); c.

We now return to the p r o o f of the theorem. Assuming that 0'; 0' < 1', we want

to show that the associative law

a; (b; c) = (a; b); c

holds. We may assume that each of the three elements a, b and c is below either 1' or 0'. By L e m m a A, the law holds if one of the three elements is below 1', In the

remaining case, when a, b, c < 0', we have (a; b); c =ab~c, and by symmet ry

a; (b; c) = ab~c.

T H E O R E M 6.3. In the notation of Theorem 6.1,

F r ( M ~ , n ) ~ - M 2" and Fr(Mi, n)~-M~" f o r i = 2 , 3

for any positive integer n.

Proof. By Theorem 2.6.


T H E O R E M 6.4. For every positive integer n,

Fr(RA(2), n) ~ M 4" • Re(2) 1/2(16n -- 4 n ) ,

where M 2 is as in Theorem 6.1.

Proo f We need to show that

/~(Re(2), n) = �89 16" - 4").

The algebra Re(2) has order 16 and is generated by any one of its elements except the four constants. The number of mappings from an n-element set onto a generating set is therefore 1 6 " - 4 " . Since the automorphism group of Re(2) has order 2, the conclusion follows.

The variety M 3 has many covers. Each of these, except for M 1 + M 3 and

M2 + M3, is generated by a single simple algebra S, containing M3 as a subalgebra. The order of S is therefore at least 8. We consider only the case when ISI = 8.

T H E O R E M 6.5. Suppose V = Var(S) with S ~ Si(SA) o f order 8. Then, for any

positive integer n,

Fr(V, n) ~- M 4" • S u~s'"),

where M3 is as in Theorem 6.1, and #(S, n) is either �89 - 4") or 8" - 4" depending

on whether or not S has a proper automorphism.

Proof. By Theorem 6.2, S must satisfy the identity 0'; 0' = 1, and the minimal subalgebra of S is therefore M3. The identity element 1' must be an atom, for if l ' were the join of disjoint non-zero elements u and v, then u; 0' and v; 0' would be disjoint non-zero elements below 0', and the order of S would therefore be at least 16. Thus 0" is the join of two atoms.

The minimal subalgebra of S is the only proper subalgebra, and S is therefore generated by any one of its elements except the four constants. The number of mappings from an n-element set onto a generating set for S is therefore 8" - 4". One of the three atoms, the identity element, is left fixed by every automorphism, and the order of the automorphism group of S is therefore either 1 or 2. From this the conclusion follows.

T H E O R E M 6.6. Suppose V is a subvariety o f SA, and n is a positive integer. If,

for some positive integer k, Re(k) ~ V, then Fr(V, n + 1) is not embeddable in Fr(V, n).

424 H. ANDRI~KA, B. JONSSON A N D 1. NI~METI ALGEBRA UNIV

Proof. By Theorem 5.9, Re(k) is an absolute retract in SA, and hence also in V.

By J6nsson [1982], Lemma 7.6, Re(k) is one-generated. Hence the theorem follows

by Theorem 3.1.

Theorem 3.1 does not require V to be a discriminator variety, it only requires

that V be congruence extensile. The theorem therefore applies to NA and its

subvarieties. However, as we shall see, Si(NA) has only one finite member that is an

absolute retract in NA.

T H E O R E M 6.7. Re(1) is, up to isomorphism, the only finite member of Si(NA)

that is an absolute retract in NA.

Proof. I f Re(l) can be embedded in the member A of Si(NA), then A must be

discrete, for the condition 0' = 0 holds in Re(1), and therefore holds in A. Since the

only discrete member of Si(NA) is Re(1), it follows that Re(1) is a maximal member

of Si(NA), and by Lemma 3.3 is therefore an absolute retract in NA.

Assuming now that S is a nontrivial finite member of NA, and that S is not

discrete, we are going to embed S in another finite member T of NA in such a way

that T cannot be retracted onto S. Embed S in a finite Boolean algebra T in such a way that at least one atom v

of S that is below 0' is split in T; i.e. v is the join of two or more atoms in T.

Subjecting the embedding to one additional condition to be specified below, we are

going to turn T into a nonassociative relation algebra having S as a subalgebra.

Let U be the set of all atoms of S, and X the set of all atoms of T. For u ~ U

let F(u) be the set of all x ~ X that are below u, and for x ~ X let f(x) be the unique

member of U that is above x. We want the mapping x ~ x ~ to be an involution of

X that sends F(u) into F(u ~). We therefore require the sets F(u) and F(u ~) to have

the same cardinality. For x -< 1', we set x ~ = x. We next define a ternary relation R on X which is to be thought of as the

relation x < y ; z . We define R to be the set of all ( x , y , z ) ~ X 3 such that

f(x) <f(y);f(z) and such that one of the following conditions holds.

( i ) x _< 0', y _< 0', z < 0'. (2) x < 0 ' , y < 0 ' , z < l ' a n d x = y . (3) x < O ' , y < l ' , z < O ' andx=z . (4) x < l ' , y - < 0 ' , z < 0 ' a n d y = z ~.

(5) x < l ' , y _ < l ' , z < - l ' a n d x = y = z .

Finally, for a, b ~ T we define

a ; b = ~ ( x e X : ( x , y , z ) ~ R for s o m e y , z e X w i t h y < a a n d z <b) ,

a~---~(x~:a>_x~X) ,


and we take the distinguished element 1' of T to be the corresponding element of S. It is tedious but not hard to show that T ~ NA, and that S is a subalgebra of T. (For more detail on this kind of construction see Andr~ka, Maddux, N+meti [1988].)

We claim that, since the atom v in S splits in T, T cannot be retracted onto S. Indeed, such a retraction would send some atom y of T that is below v into 0, and it is not hard to show that y; v ~ is equal to v; v ~, a non-zero element of S. Thus

S is not an absolute retract in NA.

T H E O R E M 6.8. Suppose V is a subvariety of NA and n is a positive integer. I f Re(1) ~ V, then Fr(V, n + 1) is not embeddable in Fr(V, n).

Proof. By Theorems 3.1 and 6.7.

T H E O R E M 6.9. Suppose V is a nontrivial subvariety of SA, X is a nonempty finite set, and n = [X I. Then the following statements hold.

(i) Every member of X is above some atom of Fr(V, X). (ii) I f V is generated by its finite members, then Fr(V, X) is atomic.

(iii) I f RRA ~ V, then Fr(V, X) has infinitely many atoms. (iv) I f Re(1) ~ V, then Fr(V, X) has exactly 2" atoms that are ideal elements; if

Re(l) ~ V, then Fr(V, X) has no such atoms.

Proof The first three statements follow directly from Theorems 4.2, 4.1(i) and 4. l(ii), respectively.

To prove (iv), observe first that if Fr(V, X) has an atom that is an ideal element, then Fr(V, X) has a direct factor of order two. Hence Re(l) must be a member of V, since it is the only algebra in SA of order two. Assume now that Re(l) ~ V, and let V* be the conjugate variety of Re(l) in V. By Theorems 3.7 and 6.1,

Fr(V, X) ~ Re(1) z" • Fr(V*, X).

Since Fr(V*, X) has no atoms that are ideal elements, it follows that Fr(V, X) has exactly 2" such atoms.

For V = SA, RA, RRA, Fr(V, n) is not atomic. In fact, it is proved in N6meti [1986] that, for every subvariety V of SA with R A ( ~ ) ___ V, Fr(V, n) is not atomic. The proof goes via showing that first-order logic can be built up in SA and then using G6del's incompleteness theorem. N6meti has also shown that Fr(RA, n) has a nontrivial direct factor that is atomless.

426 H. ANDRI~KA, B. JONSSON A N D 1. NI~MET1 ALGEBRA UNIV.

7. Pairing algebras

All of the results presented in this section and the next one, except for Theorem 7.5 and Corollary 8.5, are variants of results in Tarski and Givant [1987], Section

8.4. However, since their arguments are rather inaccessible, being a side product of a profound and involved metamathematical theory, we are going to give independent proofs.

In these two sections we concentrate on giving an algebraic proof of the fact that all finitely generated members of a particular variety V are one-generated, because from this fact it will follow that Fr(V, co) is embeddable into Fr(V, 1).

Tarski and Givant refer to a pair of functional elements p and q in a relation algebra as conjugate quasiprojections if p~; q = 1. We shall refer to such elements p and q more briefly as paired projections. If A is a subalgebra of :~(V), where V is an equivalence relation on some set U, then two members p and q of A are paired

projections iff p and q are partial functions with the property that for all (x, y) e V there exists z E U such that x = p(z) and y = q(z). In particular, if each block X of V has the property that, for all x, y

x, y ~ X iff (x, y) E X,

and if we let

p = {((x, y), x) : (x, y) e v},

q = {((x, y), y) : (x, y) e V},

then p and q are paired projections in ~ (V) ; we refer to them as standard

projections. We define a pairing algebra to be a relation algebra A with two distinguished

elements p and q, such that p and q are paired projections and the condition 0'; 0' = 1 holds in A. This last condition is added in order to exclude the case when A is a discrete relation algebra and p = q = 1. PA denotes the class of all pairing algebras. We call A a standard pairing algebra if A is a subalgebra of ~(V) , for some equivalence relation V, and p and q are standard projections.

By Tarski and Givant [ 1987], p. 242, every relation algebra that possesses paired projections is representable. An elegant proof of this important result can be found in Maddux [1978a]. Thus every pairing algebra is isomorphic to an algebra of subrelations of an equivalence relation V on a set U, with p and q partial functions on U. The condition 0'; 0" = 1 guarantees that each block of V is infinite.


Our objective is to prove that every finitely generated pairing algebra is

one-generated. In a standard pairing algebra, the relation

R [] S = {((x, y), (u, v)) : (x, u) ~ R and (y, v) e S}

can be expressed as

R [] S = ( P ; R ; P ~ ) n ( Q ; S ; Q ~ ) ,

and if R and S are nonempty, then R and S can be recovered from R [] S by noting that

R = PU; (R [] S); P, S = Q~; (R [] S); Q.

To avoid the exceptional case, we use instead of R [] S the relation T = R [] E w S [] E - , where E is the identity relation, observing that

R = P ~ ; ( T c ~ ( Q ; Q ~ ) ) ; P , S = P ~ ; ( T n ( Q ; E - ; Q ~ ) ) ; P .

R S X . . . . . ~ U X . . . . . ~ ld

,T T, , l T, T T (x, y) ~ (u, v) (x, y) ~ (u, v)

y ~ t ' y ~ v

Figure 2

These ideas must now be transferred into an abstract setting. For items 7.1-7.4 compare Tarski and Givant [1~)87] 4.1(vii), and 8.4(xii), (xiii).

LEMMA 7.1. I f p and q are paired projections in a relation algebra, then

p ~ ; p = l " and q ~ ; q = l ' .

Proof We have p~; q = l, hence by Theorem 5.1(ix),

1' = (p~; q) l ' <p~ ; (p; l ')q <-p~;p.

Similarly, 1' < q~; q.

428 H. ANDRI~KA~ B. JONSSON A N D 1. NI~MET1 ALGEBRA UNIV.

DEFINITION 7.2. Suppose A is a pairing algebra. For x, y ~ A we define

x [] y = (p; x;p~) (q;y ; q~).

LEMMA 7.3. Suppose A is a pairing algebra and x, y ~ A. Let

z = x [] l ' + y [] 0'. Then

x = P~; z(q; q~); P, Y = p~; z(q; O'; q~); p.

Proof For u, v 6 A let

f ,(u) = p ~ ; u(q; v; q~); p.

Using the representability of A, it is easy to prove that, for all u, v, w e A,

f , (u [] w) = u(1; vw; 1).

From this the conclusion readily follows.

THEOREM 7.4. Every .finitely generated pairing algebra is one-generated.

Proof By the preceding lemma, if a pairing algebra is generated by {x, y}, then it is also generated by {z}, where z = x [] l ' + y [] 0'. From this the theorem

follows.

THEOREM 7.5. Every countable member of PA is embeddable in a one-generated member. In particular, Fr(PA, ~o) is embeddable in Fr(PA, 1).

ProoJl Consider the following three properties of a variety V of algebras: (1) Every finitely generated member of V is one-generated.

(2) Fr(V, ~) is embeddable in Fr(V, 1). (3) Every countable member of V is embeddable in a one-generated member. We are going to show that ( i ) implies (2). It is easy to see that if V is

congruence extensile, then (2) implies (3), and reference to the preceding theorem will therefore complete the proof.

Let F,o be the V-free algebra on the set X~ --- {Xo, Xl . . . . }, and for n = 1, 2 . . . . let F, be the subalgebra of F~j generated by the set X, = {Xo, x~ . . . . . x,, i }. Then F, is V-freely generated by Xn. By (1), F, is generated by a single element a,. Hence there exists a surjective homomorphism f , : F n - - ~ Fn+ 1 taking x, into x, for i < n - I and xn ~ into an + ~. Let gn =f , f~ 1 . �9 . f t , and for n > 0 pick b, ~ FI such


that gn+ ~(bn) = xn. Then gk(bn) = x, for k > n + 1. Let A be the subalgebra of Fl

generated by the set {bo, b~ . . . . }. Then there exists a surjective homomorphism h : A ~ Fo taking b, into x, for all n e ~o. From the freeness of F~ it follows that

h is an isomorphism.

8. The variety R A ( ~ )

Every algebra in R A ( ~ ) is a subreduct of a pairing algebra, and the results from the preceding section therefore yield information about the variety RA (~) . It is easy to show that every finitely generated member A of R A ( ~ ) is embeddable in a three-generated member. We will improve this by showing that A can in fact be embedded in a one-generated member of RA(oo). This is Tarski and Givant [1987] Theorem 8.4(xiii). We will then use this fact to show that the subvariety R A ( ~ ) of RA is such that Fr(RA(ov), o~) is embeddable into Fr(RA(ov), 1).

LEMMA 8.1 (Tarski and Givant [1987] 8.4(xiii)). A relation algebra A belongs to R A ( ~ ) iff A is isomorphic to a subreduet of a pairing algebra.

Proof This follows from the fact that relation algebras are congruence extensile, whence HS(K) = SH(K) for K _ RA, and that the class of all relation algebras that are subreducts of pairing algebras is closed under H and P.

In the proofs of the next two lemmas we make use of the fact that the algebra A under consideration is representable, assuming that A is in fact a subalgebra of ~ ( V) for some equivalence relation V.

LEMMA 8.2 (compare Tarski and Givant [1987] 4.7(ix), (x)). Suppose A e R A ( D ) , and for x e A define

v(x) = x(x; 03 ,

P(x ) = v(x; v(x)),

Q(x) = xZ(x2P(x ) - ; 0') - .

f f p and q are paired projections, then so are P(p + q) and Q(p + q).

Proof The relation v(x) is a function; in fact, it is the restriction of x to the largest domain on which x is single valued. Hence P(x) is a function. The relation Q(x) is also a function, for otherwise there would exist (w, u), (w, v )e Q(x) with u :~ v, yielding (w, u), (w, v) r xZP(x)-; 0% (w, u), (w, v) e P(x) (Fig. 3a), contradict- ing the fact that P(x) is a function.

430 H. A N D R E K A , B. J O N S S O N A N D 1. N I ~ M E T I A L G E B R A U N I V

14' W

~ / 0 ' ~ 2 P / ~ q P / ~ P + q v v'

u -- ~ v p d ~ i q p()q [v(p+q) bl U bl R

a b c

Figure 3

We now show that for e = p + q the elements P(e) and Q(e) are paired

projections. This means that, for (u, v) ~ V, we must find an element w such that

(w, u)~ P(e) and (w, v ) ~ Q(e). Choose y , z , w as indicated in Fig. 3b. Then

(y, u) ~ v(e), hence (w, u) ~ e; v(e). Suppose (w, u ' ) ~ e; v(e). Then for some

y ' , (w, y ' ) ~ p + q and (y ' , u') ~ v(p + q) (Fig. 3c). Then y ' = y or y ' = z. In either

case, ( y ' , u) ~ p + q, so that ( y ' , u ') cannot belong to v(p + q) unless u ' = u. Thus

(w, u) ~ P(e). It remains to show that (w, v )~ Q(e). It is clear that (w, v ) ~ e 2, and that if

(w, v ') ~ e 2, then v ' ~ {u, v}. For v ' = v we therefore have (w, v') r e2p(e) ; 0". Consequent ly (w, v) r e2p(e); 0', which implies that (w, v) ~ Q(e).

L E M M A 8.3 (Tarski and Givan t [1987] 7.1(i)). Suppose A e R A ( ~ ) . I f p and q are paired projections in A, then so are (p ; q)0' and qO'.

Proof. Clearly (p; q)0' and q0' are functions. Given (u, v) e V, we need to find

an element w with (w, u) ~ (p ; q)0' and (w, v) ~ q0'. In other words, we need to find

an element w distinct f rom u and v such that (w, u) ~ p; q and (w, v) e q. Choose

three elements u o, u~, u2 in the same block of V as u. For i = 0, 1, 2, choose first vi and then w, as indicated in Fig. 4. F rom the fact that p is a function it follows that the elements w, are pairwise distinct, whence w, ~ u, v for some i. This is our desired

element w.

P P Wt ) /)l ~ /dr

U U

Figure 4

T H E O R E M 8.4 (Tarski and Givan t [1987] 8.4(xiii)). I f the algebra A ~ R A ( ~ )

is finitely generated and possesses paired projections, then A is one-generated.

Proof. By the preceding two lemmas, there exists an element e -< 0' such that the subalgebra A ' o f A generated by e possesses paired projections p and q. By Theorem


7.4, there exists an element a such that A is generated by {a, p, q}, and hence also by {a, e}. Let b = (a [] a)1' (see Definition 7.21). Then a =p~b; p. Consequently, A is generated by the set {b, e}, and therefore by the single element b + e.

Arguing as in the proof of the Theorem 7.5, we infer from Theorem 8.4 and Lemma 8.1 that the conclusion of Theorem 6.5 fails for V = RA(~) . Hence we cannot omit from the hypothesis of that theorem the assumption that Re(k) e V for some positive integer k.

COROLLARY 8.5. Fr (RA(~) , ~o) is embeddable in Fr(RA(~) , 1).

9. Varieties between RA(~) and SA, and non-free generators of free algebras

This section is devoted to the proof of the following result.

THEOREM 9.1. For any subvariety V of SA, tf R A ( ~ ) ___ V, then the V-free algebra on one generator also contains an element that generates it non-freely.

To prove this theorem, we are going to construct unary terms/~ and v such that

SA g vp(x) = x, Re(U) ~/ I~V(X) = x,

where U is an infinite set. If F is the V-flee algebra on one generator a, then the element b =/~(a) will also generate F, because a = v(b), but b will not generate F freely, because #v(b) 4= b.

The construction of/~ and v will be motivated by describing #(R) and v(R) for R in Re(U). We want /2(R) and v(R) to be equal to R, except for certain special relations R. Such patching is possible provided the special relations can be defined by an open Horn formula, because SA is a discriminator variety.

The special relations R that we consider are relations with the following properties:

R totally orders U. U has a first element (relative to R). Every member of U except the last one (if it exists) has a successor.

For special relations R we take/~(R) to be the total ordering of U obtained by moving the first element to the last place. In defining v(R) we consider two cases: If U has a last element, then v(R) is obtained by moving this element to the first place; otherwise v(R)= R.

432 H. ANDRI~'KA, B. JONSSON AND 1. NI~METI ALGEBRA UNIV.

It is now clear how the special elements should be defined abstractly. Given

A �9 Si(SA), call an element x �9 A a total order if

x ; x < x , x x ~ = ! ', x + x ~ = l .

Define

x ~ = xO'(xO'; xO') .

For a total order x, x" is to be thought o f as the covering relation. We say that x

is special if x is a total order and

1; xO' < 1, xO'; 1 = x ' ; 1.

Since special elements are defined by an open Horn formula, there exist by Theorem

1.1.VI terms # and v such that, for any element x in a simple relation algebra,

I~(X) = x(x 'O' ; 1) + (1; xO') ,

v(x) = x(1; x~0 ') + (x0' ; 1) ,

while for all other x e A,/ t(x) = v(x) = x .

It is clear that

RRA ~ v#(x) = x , R e ( U ) ~ / #v(x ) = x .

This establishes the theorem for V = RRA. To complete the p roo f in the general

case, we must show that SA ~ v # ( x ) = x. This will involve some rather lengthy

calculations.

Fix an algebra A �9 Si(SA) and a special element x e A, and let

p = ( x U 0 ' ; 1 ) , y = x p +p~J, q = ( 1 ; y ~ O ' ) , z = y q + q ~ .

Then y = #(x). The major part o f our argument will be devoted to showing that y

is a special element. Once this has been done, we infer that z = v (y ) , and complete

the p r o o f by showing that z = x.

Since A is subdirectly irreducible, we have 0 ' = 0, 0'; 0 ' = 1' or 0'; 0 ' = I by

Theorem 5.8(i). The case 0 ' = 0 is trivial, for then x = y = z = 1. We therefore

assume that 0'; 0' > 1', which implies that 0'; 1 = 1.

C L A I M 1. 0 < p < x .


P r o o f First note that

p p - p~ p -~

P P P pp~ pp-~ p - p - p - p-p~ p p - ~

p ~ 1 0 p ~ p ~ p - ~ 0 1 p~ p ~

Figure 5

p - + x = x~0 ' ; 1 + x > x ~ O ' + x = 1,

because x + x ~ -- 1. Therefore p < x. Equali ty is excluded, for this would yield

(x~0' ; l ' )x = 0, hence x0' ; x -- 0, and finally x0 ' = x0 ' ; 1 _< x0 ' ; x = 0, contradict ing

the fact that x ~ 1'. Also p r 0, because x is special, and therefore 1; x0 ' < 1.

C L A I M 2. The sixteen pairwise relative products o f the elements p, p , p ~ and

p - ~ are given by the table in Fig. 5.

Proof. The element p is a right ideal element. Hence, by Theorem 5.2, p is also

a right ideal element, while p~ and p - ~ are left ideal elements. F r o m this the entries

in the table follow by Theorem 5.8(iv).

C L A I M 3. y is transitive.

Proof. We have

y ; y = x p ; x p - + x p ; p ~ + p ~ ; x p - + p ~ ; p ~.

F r o m the fact that x and p - are transitive it follows that x p - is transitive. F r o m

this and Claim 2 we infer that y; y < y.

C L A I M 4. y y ~ = 1'.

Proof. F r o m Claim 1 and the fact that x is special it follows that

y y ~ = x p - x ~ p - ~ + p ~ p < x x ~ = 1'.

Also,

1 ' = l ' x = l ' x p - + l 'p = l ' x p - + l 'p ~ = l 'y ,

so that 1" < y y ~ .

434 H. ANDRI~KA, B. JONSSON A N D 1. NEMETI ALGEBRA UNIV.

C L A I M 5. p 1'; x ' ~ 0.

Proof. Suppose p i ' ; x' = 0. Since x0 ' -< x0'; 1 = x ' ; 1', this implies that p l ' ; x 0 ' = 0. F rom this it follows that ( p l ' ; l )x0 ' = 0. By Theorem 5.2, p is a right ideal element, and therefore p 1 '; 1 = p( 1'; 1) = p. Consequently pxO' = 0, and in fact p 0 ' = 0 , since p < x . Thus 0 ' - < p = x ~ 0 ' ; 1 , l = 0 ' ; l - < ( x ~ 0 ' ; 1 ) ; l = x ~ 0 ' ; 1 by

semiassociativity. This is a contradict ion, since x is special.

C L A I M 6. y~0 ' ; 1 < 1.

Proof. We are going to show that

(1) p l'; (xc; y ~0') = 0.

From this it will follow by Theorem 5.5 that

( (p l ' ) ; xc); (y~0 ' ; l) = 0,

and in view of Claim 5 this implies that y ~0'; 1 < 1. Observe that x'; p <-x0'; p = 0 by Theorem 5.1(vii). Hence (1) reduces to

p 1'; (x'; x~p ~0') = O.

Applying Theorem 5.1 (vii) twice, we can rewrite this first as (p 1'; 1)(x c; x ~p ~0') = 0,

i.e., as p(x'; x~p u0') = 0, and then as

xC(p;xp 0') = 0.

This equat ion will follow if we show that

(2) p ; x p O'<-xO';xO'.

ClearlypO';xp-O'<xO';xO', but we also havepl';xp O'<pl ' ;p- <-p~;p = 0 .

Therefore (2) holds.

C L A I M 7. y ' >-x'p + ( x 0 ' ; 1) - ;p~ .

Proof. Since (x0'; 1) is a right ideal element and p~ is a left ideal element, this

inclusion can be written as

y~>_x~p + ( x 0 ' ; 1) p~.


It therefore suffices to verify the four formulas

(1) x~p <yO'.

(2) (x0'; l ) - p ~ -< y0'.

(3) x'p (y0'; y0') = 0.

(4) p~(y0 ' ;y0 ' ) < x0'; 1.

The first inclusion holds because x~p - < xO'p- < yO'. We have p~ < y, so the second inclusion reduces to showing that (x0'; 1)-p~ < 0', i.e., that p ~ l ' < x0'; 1. But p~ l ' = p l ' , so this can be written p l ' < x0'; 1, and we actually have p < x0'; 1, for p " + x0'; 1 = x~0'; 1 + x0'; 1 = 0'; 1 = 1. Thus (2) holds.

We have

y0'; y0' = xp-0'; xp-0" + xp -0'; p~'0' + p ~0'; xp-0 ' + p ~0'; p~0',

and the equation (3) therefore reduces to the following four equations.

(5) xCp-(xp-O'; xp-O') = o.

(6) xCp-(xp-O';p~O')=o.

(7) x'p-(p~O'; xp-O') = O.

(8) x~p (p~0';p~0')=0.

The equation (5) holds because xC(x0'; x 0 ' ) = 0, equations (6) and (8) because xC( 1;p ~) < x0'( 1; p~) = xO'p ~ < xx~O'= 1'0 '= 0, and equation (7) holds because p~;p - =0.

Finally, (4) is equivalent to the conjunction of the following four inclusions:

(9) p~(xp-0'; xp-0") < x0'; 1.

(10) p~(xp 0';p~0') < x0'; 1.

(11) p~(p~0'; xp-O') < x0', 1.

(12) p~(p~0';p~0") < x0'; 1.

Inclusions (9) and (10) hold because the second factor on the left side is included in the right side, and ( l l ) holds because p ~ ; p - = 0 . Finally, we prove (12) by observing that p~;p~0 ' = 0, since (p; 1)p~0 ' =pp~O'<- 1'0'.

436 H. ANDR.EKA, B. JONSSON AND 1. NI~METI ALGEBRA UNIV.

C L A I M 8. y 0 ' -< yC; 1.

Proof. By Theorem 5.1(ix), (xC; l ) p - < x ' ( p - ; 1); 1 =xCp-; 1. Therefore

xCp-; 1 = ( # ; 1 )p- = (x0' ; 1)p >- xO'p- . Hence by Claim 7, x 'p -O ' -< y ' ; 1. Thus it suffices to show that

(1) p ~ O ' < x ' p ; l + ( x 0 ' ; 1 ) ; p ~ ; l .

Since p ~ is a non-zero left ideal element, we have p ~; 1 = 1, and (1 ) can therefore

be writ ten p~O' < # p - ; ! + (x0' ; 1 ) - or, equivalently,

p~0 ' (x0 ' ; 1) ~ x ' p - ; 1.

Since x'p-; 1 = (x0' ; 1)p , this reduces to

p~0 ' (x0 ' ; 1)p = 0.

This equat ion holds because p~p < 1'.

C L A I M 9. y is a special element.

Proof. By Claim 1, x = xp - + p, while y = xp + p ~. Hence y + y ~ = x + x ~ = 1. T o g e t h e r wi th C l a i m s 3 and 4, this shows that y is a total

ordering. By Claim 6, l ; y 0 ' < 1, and by Claim 8, y0 ' ; 1 <yC; 1. The opposi te

inclusion holds because yC < yO'. Thus y is special.

C L A I M 10. q = p ~ .

Proof By definition, q = (1; x~p ~0" + 1;p0 ' ) --. We claim that q = (1; p0 ' ) ,

i .e., that 1 ; x ~ p - ~ 0 ' < 1 ;p0 ' . In fact, we will show that p ~ < 1 ;p0 ' , or equiva-

lently, that

p ~ + 1 ; p 0 ' = 1.

We have p ~ 1' = p 1', therefore p ~ = 1; p ~ > 1; p 1', so that

p ~ + l ; p 0 ' - > l ; p l ' + 1 ; p 0 ' = l ; p = l ; p ; 1 = 1 ,

since A is simple and p ~ 0 . Thus q = ( 1 ; p 0 ' ) - , p ~ + q = 1. To complete the proof , we need to show that p~ and 1 ;p0 ' are disjoint. By

Theorem 5.1 (ix)

p ~ ( 1 ; p 0 ' ) < 1 ; p 0 ' ( 1 ; p ~) = 1 ;pp~0 ' = 1; 0 = 0.


Thus p~ =(1 ;p0 ' ) =q .

CLAIM 11. z = x.

Proo f We have x = x p = p and by Claim 10 z = y q - + q ' ~ = y p ~ +

p = xp p ~ + p. It therefore suffices to show that xp < p ~, i.e., that xp '~ < p.

This inclusion follows from the fact that xp '~ < x x '~ = 1' and l'p ~ <p. The proof of Theorem 9.1 is now complete.

We make some remarks here on the possibilities of generalizing Theorem 9.1. In J6nsson and Tarski [1961] it is proved that if a variety is generated by finite algebras then any n-element generator set of the n-generated free algebra generates it freely. Therefore, the assumption that RA(oo) ~_ V cannot be omitted from Theorem 9.1.

On the other hand, by the above quoted theorem of J6nsson and Tarski [1961], in Theorem 9.1 SA cannot be replaced with the variety NA of nonassociative relation algebras, because in N6meti [1987] it is proved that NA is generated by its finite members.

10. Cylindric algebras

The basic reference on cylindric algebras is the two volume treatise of Henkin, Monk, Tarski [1971], [1985]. We recall here the definition and a few basic facts.

A cylindric algebra of dimension n, where n is an ordinal, is a Boolean algebra A with unary operators cg and distinguished elements dij, for i , j < n, such that for all i,j , k < n and x, y e A, the following conditions hold:

(I) The c,'s are self-conjugated closure operations commuting with each other, i.e.

(1) X ~ CiX = CtCiX

(2) y ' c i x = O iff

(3) c iox = clcix

X " c i y = 0

(II) The constants d o satisfy the following:

(4) do.d/k <d,k, d~j=dj,, d~,=l

(5) c , d ~ = l , Ckd,j=d~j if k r

(6) d , ~ ' c i x = x if x < d o , i # j .

438 H. ANDRI~.KA, B. JONSSON AND I. Nt~METI ALGEBRA UNIV.

An equivalent form of saying that c~ is self-conjugated is to say that the complement of a closed element is closed. Thus, in the above definition, (2) can be

replaced with

(2)' ci - q x = - c i x .

We note that (6) expresses that the closure operator c, is "discrete", or is the identity, when "relativized" to de, i # j . An equivalent form of (6) is obtained by substituting (y �9 d,j) for x in (6). The class of all cylindric algebras of dimension n

is written CA,. The primary models for the above axioms are twofold: algebras of n-ary

relations, and algebras of formulas in first-order theories. Here we consider only

algebras of the first kind. Let Re,,(U) denote the set of all n-ary relations on U, i.e. if U n denotes the set

of all n-termed sequences of elements of U, then

Re . (U) = {R : R c_ U.}.

Now the usual set theoretic Boolean operations are meaningful in Ren(U). The unary operations ci denote "erasing the i-th column" in a relation and the constants d~ denote the identity relations: if R is an n-ary relation on U and i , j < n then

c , R = {(So . . . . , s , _ , , u , s ,+ l . . . . ) : s e R , u �9 U}

and

d o = {s ~ u ~ L~, = s j } .

R e , ( U ) will denote also the full algebra of n-ary relations on U with the above operations. By a cylindric set algebra of dimension n and with base U we mean a subalgebra of Re,,(U). By a representation of a cylindric algebra A we mean an embedding of A into a subdirect product of cylindric set algebras. This is equivalent to the following. If V is a disjoint union of Cartesian spaces UT, then the set ~ (V) of subrelations of V forms a natural cylindric algebra. Now, a representation of A is an embedding A ~ ~ (V) for some appropriate V. RCA, denotes the class of all representable cylindric algebras of dimension n. Analogously to the relation algebraic case, RCA n is a variety (a theorem due to Tarski, see Henkin Monk-Tarsk i

[1985] 3.1.108, 109 p. 46). As we shall see later, the theories of cylindric and relation algebras show many

similarities, these similarities are even more outstanding in the case of finite dimensional cylindric algebras and relation algebras. Before showing some of these,


we want to discuss, to some extent at least, the connection between relation and

cylindric algebras. We have seen that relation algebras serve as algebras of binary

relations while cylindric algebras serve as algebras of relations with higher ranks. It

is obvious that the cylindric constants d U correspond to the relation algebraic

constant 12 Below we show that the cylindrifications c, are related to composition

and conversion in relation algebras. Since there is no self-evident way of defining

the composition of two n-ary relations with n > 3, instead of considering some

generalizations of composition ";", we define the unary ci's - which are generaliza-

tions of the operations x ~ x; 1, x ~ 1; x - and from these ci then we can build up

many ways of constructing new relations from old ones. Below we show how we

can recapture the usual composition and inversion of binary relations by using the

cylindric operations. Let n > 3 and let U be any set. For a binary relation R on U

let

/~ = {s e U"" (So, s , ) e g } .

Thus the n-ary relation/~ is the "n-ary version of R". For i , j < n, i r define

s ix = ci(dij " x).

i has a natural meaning in Ren(U), e.g. if n = 3 and The new derived operation s t

R ~ Re(U) then

s~R = {(u, v, w) " (u, w) e R} and s ~ {(u, v, w) " (w, v) e R}.

Let R, S ~ Re(U). Then

RlS=c2(s~ "s~R) and R ~=sos,s2R.2 o i -

Clearly,

/ d = do,.

Now we can use the above as a motivation to define the relation algebra reduct RaA of an A ~ CAn with n > 3:

RaA = { a E A ' a = c i a for all i < n , i : ~ 0 , 1 } ,

440 H. ANDRI~KA, B. JONSSON AND 1. NI~METI ALGEBRA UNIV.

and the operations are the Boolean ones together with

x ; y = c 2 ( s ~ �9 s i x ) ,

X~2 2 0 1 S o S I S 2 X ,

1' = d o l .

It is easy to see that R a A is closed under the above operations. Now, R a A ~ RA if A ~ CA, with n >-4 or i fA ~ RCA 3 ( H e n k i n - M o n k - T a r s k i [1985] 5.3.8 on p. 216, due to Henkin and Tarski). In the other direction, we can build a C A 2 from any

relation algebra as follows:

If (A,; ,~, 1') ~ RA then (A, ci , d i j ) i , j < 2 E CA2,

where

C o X = l ; x , C l X = X ; l . d o l = d l o = l ' and d o 0 = d l l = 1.

By Monk [1961] Theorem 4.4, this C A 2 is actually representable (this also

follows from Henk in -Monk-Ta r sk i [1985] 3.2.65). Let A ~ C A , , . The congruence ideals of A are the ideals that are closed under all

the operations c,, and an element a ~ A is a congruence element iff c ia = a for all i < n. (Such elements are said to be zero-dimensional in Henkin, Monk, Tarski

[1971].) In particular, if n is finite then the element C o q . . . c , l(x) is always a congruence element. From this it readily follows that if n is finite then a non-trivial member A of CAn is simple iff c o . . . c n ~ x = l whenever 0 # x ~ A . It is not difficult to show that if n is finite then A is simple iff it is subdirectly irreducible, therefore CA,, is a discriminator variety if n is finite. (E.g. the term t ( x , y , z ) = x . co . . . c , i ( x ~ y ) + z . co . . . c , ~ ( (x O y ) ) is a d i s c r i m i n a t o r t e r m

for Si(CA,).) We note that if n is infinite then CA, is not a discriminator variety,

e.g. because the subdirectly irreducible CAn's do not coincide with the simple CA,'s but CA, is still congruence extensile because its members are Boolean algebras with

operators. The variety CAj is locally finite. In fact, the subdirectly irreducible members are

simply Boolean algebras with a "trivial" closure operator, c ( x ) = 1 for x # 0. Having recalled what we will need here from the basic monograph on algebraic

logic for quantifier logics, we now turn to applying our universal algebraic results

to the variety CA,.


T H E O R E M 10.1. For any positive integer n,

2 n

Fr(CA,, n) ~- 1-I B~ m'") m = l

with v(m, n) = (2"!) /m!(2"-m)! , where for 1 < m < 2", B,. is the unique member of Si(Chl ) of order 2 m.

Proof. This will follow from Theorem 2.6, once we have shown that p(B,,, n) = v(m, n).

To verify this formula, consider a set X of order n. With each map f " X-~ B,, associate the map f* "At ~ P(X) where At is the set of all atoms of B,,, P(X) is the power set of X, and for a e At,

f*(a) = {x ~ X" a -<f(x)}.

Also, for g "At ~ P ( X ) define the map g# X---~B m by letting

g#(x) = ~ ( a e A t " a s g(x)) for x eX.

It is easy to check t h a t f *# = f a n d g#* = g, and t h a t f i s onto a generating set for Bm iff f * is one-to-one. The number of such maps is therefore 2"(2" - 1 ) . . . (2" - m + 1). Since the automorphism group of B,, has order m!, this yields the indicated formula for ~(B,,, n).

The order of Fr(CA1, n), given in Henkin, Monk, Tarski [1971], Cor. 2.5.62, can be obtained from the preceding result.

COROLLARY 10.2. For any positive integer n,

]Fr(CA~, n)] = 22"+2" '.

Proof. By the preceding theorem', the order of Fr(CA1, n) is 2 N, where

2n m �9 2"! N=f ~1 m!(2 " - m ) ! '

We therefore need to show that the above sum is equal to 2" + 2--1. This is a special case of the equation

m .k! k2* ~=,,=l ~" m ! ( k - m ) !

442 H. ANDREKA, B. JONSSON AND 1. NEMETI ALGEBRA UNIV.

or, upon division by k,

k (k - 1)! 2 k - ' = , , = , ~ ( m - 1 ) ! ( k - m ) ! "

This is just the binomial expansion of (1 + 1) k

We now turn to the variety CAn with n < co. Let n < 09 be fixed for a while. Just

as in the relation algebraic case, the variety CA, is a lattice join of finitely many pairwise disjoint varieties: For any 1 < k < n we define the cylindric algebraic term

a~ as follows: if k < n then

ak=Co...Ck , I ] { d ~ " i < j < k } . ( C o . . . c k l - - I { d ~ " i< j<_k} )

a , = c o . . . c , , _ l [ I { d o : i < j < n } .

For any 1 < k < n let V~ be the subvariety of CA, defined by the identity ak = I. In Henkin, Monk, Tarski [1971], for 1 < k < n, nontrivial members of Vk are called

the cylindric algebras of characteristic k, while nontrivial members of V n are said to

have characteristic 0. Then CA, is the lattice join of the pairwise disjoint subvari-

eties V~, V2 . . . . . V,,, and Fr(CA,, m) ~- 1-I~ = 1 Fr(Vk, m). Also, Vk is generated by the algebra Re , (k ) - i.e. Re,(U) with [U] = k - for l < - k < n , hence these are

locally finite subvarieties of RCAn. From any Boolean algebra A we obtain a CA,

by defining

cix = x and do = l

for all x e B and i , j < n. These algebras are called discrete cylindric algebras, and

the members of V~ are exactly the discrete CA~'s. Also, all the Vk'S (for 1 -< k < n) have a minimal subvariety Mk generated by one finite subdirectly irreducible

algebra Mk. Thus our Theorem 2.6 can be applied to describe these free algebras,

too, as we did for relational algebras at the beginning of Section 6. Here we do not

go into more detail. If k and m are positive integers with m > 2, then Rein(k) is easily seen to be

simple. By Henkin, Monk, Tarski [1971] p. 253, Rem(k ) is one-generated, and by Comer [1984] Theorem 3.8(1) it is an absolute retract in C A m if k < m . It is not

difficult to see that Rein(k) for any finite k, m is maximal in Si(RCAm), hence

Rein(k) is an absolute retract in RCA m. (Also, by splitting atoms in CAm's, with a similar argument to the one in the proof of Theorem 6.6, one can see that among the finite subdirectly irreducible algebras, these are the only absolute retracts in CAm and in R C A m respectively. By a similar method one can show that there are


no nondiscrete finite absolute retracts in CA1 or in the broader variety of closure

a l g e b r a s - here we call a closure algebra discrete if the closure operator is the

identity mapping on the algebra.) I f in the definition of a cylindric algebra of dimension m we omit the require-

ments that the cylindrifications commute, then we obtain the definition of the class

NCA,, of noncommutat ive cylindric algebras of dimension m. The class NCA" is analogous to the class NA of nonassociative relation algebras. We note that an

intermediate class, somewhat analogous to SA, has also been investigated by R.

Thompson, see Thompson [1987]. It is not difficult to see that finite discrete cylindric algebras are absolute retracts in NCAm, for any ordinal m. The argument

for this is basically the same as the first part of the proof of Theorem 6.6.

Finally, NCAm is congruence extensile, because it consists of Boolean algebras with

operators.

We note that if m is infinite then Re,,(k) for k > 1 is neither finite, nor an

absolute retract in CA,.. Still, our Theorem 3.1 can be applied to CA,. with m

infinite, as the next theorem shows.

T H E O R E M 10.3. Suppose m is an ordinal, V is a subvariety ofNCAm, n and k are positive integers and Rein(k) ~ V. Then each of conditions (i)-(iii) below implies that Fr(V, n + 1) is not embeddable into Fr(V, n).

(i) k = 1 (ii) k < m and V _ C A m (iii) V _ RCA,,.

Proof. First assume that m is finite. The cases m = 0, 1 are trivial, since in these

cases V is locally finite, and we therefore assume that m > 2. In this case Rein(k) is

a one-generated, finite, absolute retract in V, and since V is congruence extensile, the conclusion follows by Theorem 3.1.

We are going to reduce the case when m is infinite to the finite case, using the following simple observations.

Observation 1. Suppose r ' is a subtype of a similarity type 3, V is a variety of type 3, K is the class of all r ' -reducts of algebras in V, and V' = Vat(K). I f X is a

non-empty set, then there exists an embedding of Fr(V', X) into the z '-reduct of Fr(V, X) taking each member of X into itself.

Observation 2. For any ordinals p and q with p <- q, and for any cardinal k,

Rep(k) is isomorphic to a subreduct of Req(k). Also, if A ~ NCAq, A e CAq, or

A ~ RCAq, then the p-dimensional reduct of A is in NCAp, in CAp, or in RCAp, respectively.

Assume now that m is infinite, and suppose there exists an embedding f : Fr(V, X) ~ Fr(V, Y), where X and Y are sets of order n + 1 and n, respectively. Each of the elements f (x) with x s X can be expressed in terms of elements of Y,

444 H~ ANDRI~KA, B, JONSSON A N D 1. NI~METI AI_GEBRA UNIV.

using finitely many operations ci, and finitely many elements diz, and we may assume that all the indices i,j, I involved are finite, and therefore smaller than some fixed positive integer p. Let K be the class of all algebras in NCAp that are reducts of algebras in V, and let V' = Vat(K). By Observation 1, there exist an isomorphism g from Fr(V', X ) onto a subreduct G of Fr(V, X) and an isomorphism h from Fr(V', Y) onto a subreduct H of Fr(V, Y), such that g(x) = x for all x c X and

h(y) = y for all y ~ Y. The map f sends X into H, and therefore sends G into H. Consequently h ~fg is an embedding of Fr(V', X) into Fr(V', Y). However, we know that no such embedding exists, since by Observation 2, Rep(k)~ V'. This

contradiction completes the proof.

We note that if m is infinite then Rein(k) ~ V for some finite k if and only if V has a member with positive characteristic. This is so because by N6meti [ 1987a] if m

is infinite then Rein(k) generates a minimal variety. Many results in Henkin, Monk, Tarski [1971] Chapter 2.5 follow from our

Theorems 4.1, 4.2. Let n, m be positive integers. Then Fr(V, n) is atomic if V ~_ CA,, is generated by finite algebras (this is 2.5.7(i), (ii) in Henkin, Monk, Tarski [1971]), Fr(CA,,, n) has infinitely many atoms (Henkin, Monk, Tarski [1971] 2.5.9), and moreover if V c_ CA,, is any variety such that all representable finite cylindric algebras are contained in V then Fr(V, n) contains infinitely many atoms. Also, the

following generalization of Henkin, Monk, Tarski [1971] 2.5.1 l (for finite dimen-

sions) is a corollary of our results: (*) Let n, m be positive integers and assume that V ~_ CA,, contains a discrete

C A m. Then Fr(V, n) contains exactly 2" zero-dimensional atoms. On the other hand, we note that it is proved in N6meti [1987] that Fr(V, n) is

not atomic if m > 3 is any ordinal, V c CA,. and Re,,(U) ~ V for some infinite U.

The cylindric algebraic counterpart of Theorem 9.1, i.e. that F r ( C A m, l) can be non-freely generated by a single element, is true for every ordinal m -> 3. This is the solution of Problem 2.7 in Henkin, Monk, Tarski [1971]. For m -> 4 this is implied

by Theorem 9.1, as follows.

THEOREM 10.4. Let m > 3 be any ordinal, and let V be a subvariety of CA,, such that Rein(U) ~ V .]'or some infinite U. Then the V-free algebra on one generator

contains an element that generates it non-freely.

Proof In the proof of Theorem 9.1, we gave two relation algebraic terms g and v, such that RA r v/~(x) = x while Re(U) 1~ t~v(x) = x for some U. Let Ft', v' be the corresponding CA3-terms as defined in Henkin, Monk, Tarski [1971] section 5.3 (i.e. we replace x; y by c2(s~x " s~ etc.). Then from SA ~ vlt(x) = x we conclude that CA4 ~ v'#'(x) = x, therefore CA,, ~ v'l~'(x) = x for all m > 4. Similarly, from


Re(U) ~ ~tv(x)= x we conclude that Rem(U ) JZl~'V'(X ) = x . This settles the case

m > 4 . The analog of Theorem 9.1 for CA 3 is also true, and this is proved in N~meti

[1986]. (For m = 3 we note that SA ~ Vl4X) = x does not imply CA 3 ~ v'l~'(x) = x

because in the proof of SA ~ v~(x) = x we used the Peircean law (i.e. Theorem

5. l(vii)) several times and in CA 3 the corresponding law does not hold.) For m < 3,

the analog of Theorem 9.1 is not true for CA,.. This is proved in Henkin, Monk,

Tarski [1971] 2.5.23.

11. Open problems

Recall that PA denotes the variety of all pairing algebras as introduced in

Section 7.

PROBLEM 1. Does Fr(PA, m) ~ Fr(PA, n) imply m = n?

By Corollary 8.5 and the proof of Theorem 7.5, every countable member of

R A ( ~ ) is embeddable into a one-generated R A ( ~ ) . By the next lemma, there are finite relation algebras that are not embeddable into one-generated relation alge-

bras.

PROBLEM 2. Does there exist a simple relation algebra that is not embeddable

into a one-generated relation algebra?

L E M M A 11.1. Suppose V is a congruence extensile variety, and suppose S is a

non-trivial f in i te absolute retract in V. Then f o r any m > ]S[, S m is not embeddable in

any one-generated member o f V.

P r o o f Let V, S and m be as in the statement. Assume that A ~ V and that q : S m ~ A is an embedding. We will show that A is not one-generated. Since V is congruence extensile, and S is an absolute retract in V, as we have seen earlier,

every onto homomorphism k : S m -~ S factors through g, i.e. there is a homomor-

phism k # : A --* S such that k = k ~g. Let k~ . . . . , k m be the projections from S m

onto S. Then kl . . . . . km are different homomorphisms from S m into S, because S

is non-trivial, therefore the corresponding homomorphisms k~ . . . . . km ~ are also

different homomorphisms from A into S. Let a e A be arbitrary and let B _ A be the subalgebra of A generated by a. By [S[ < m, there are 1 -< i < j < m such that k,~(a) = k f ( a ) , therefore k~ and k~ agree on B. Since k~ and k~ are different,

.I

this shows that B r A.

446 H, ANDRI~KA, B. JONSSON A N D 1. NI~METI ALGEBRA UNIV.

By using L e m m a 3.3, one can show tha t the subdirect ly i r reducible absolu te

re t racts in R R A are exactly Re(k) with k finite.

P R O B L E M 3. Are there non- represen tab le subdirect ly i r reducible absolu te

re t racts in RA or in SA?

We note that , by spl i t t ing a toms as in the p r o o f o f Theorem 6.6, one can show

that there are no non- represen tab le finite absolu te retracts in CAn, for any ord ina l

n. (Cf. 3.2.53 and 3.2.69 in Henkin , M o n k , Tarsk i [1985].)

In the p r o o f o f Theo rem 6.6, we split a toms in o rde r to show that there are few

absolu te retracts . Therefore answering the fol lowing quest ion might help in solving

Prob lem 3. In J6nsson [1982] L e m m a 7.3, it is p roved that if A ~ R A is simple, and

p c A is a non-zero e lement such that p; l ; p ~ -< l ' , p ~ ; l ; p -< 1', t h e n p is an a tom,

see Theo rem 5.8(iii) herein. The next ques t ion asks if the given cond i t ion is

necessary or not.

P R O B L E M 4. Let A e R A and p E A. Assume that for any inclusion

f : A ,--, B ~ Si (RA), f ( p ) is an a tom in B. Does this imply that p ; 1 ;p ~ -< 1'?

P R O B L E M 5. Is F r ( N A , n) a tomic for finite n? Is F r ( N C A m, n) a tomic for finite

m > 2 , n?

REFERENCES

ANDREKA, H., MADDUX, R. and NI~METI, I. [1988] Splitting in relation algebras. Proc. Amer. Math. Soc., in print.

BERMAN, J. and BLOK, W. J. [ 1987] Fraser Horn and Apple properties. Trans. Amer. Math. Soc. 302. BLOK, W. and PIGOZZl, D. [1982] On the structure of varieties with equationally definable principal

congruences I. Algebra Universalis 15, 195 227. COMER, S. [1984] Galois-theory of cylindric algebras and its applications. Trans. Amer. Math. Soc. 286,

771 - 785. HENKIN, L., MONK, J. D. and TARSKI, A. [1971] Cylindric algebra,;. Part I. North-Holland. HENK1N, L., MONK, J. D. and TARSKI, &. [1985] Cylindric algebras. Part II. North-Holland. JONSSON, B. [1982] Varieties of relation algebras. Algebra Universalis 15, 273 298. J6NSSON, B. and TARSKI, A. [1951] Boolean algebras with operators. Part I. Amer. J. Math. 73,

891 939. J6NSSON, B. and TARSKI, A. [1952] Boolean algebras with operators. Part If. Amer. J. Math. 74,

127- 162. JONSSON, B. and TARSKI, A. [1961] On two properties" of free algebras. Math. Scand..9, 95 -101. MADDUX, R. [1978] Topics in relation algebras. Ph.D. Thesis, University of California, Berkeley. MADDUX, R. [1978a] Sufficient conditions.for representability of relation algebras. Algebra Universalis 8,

162 t72. MADDUX, R. [1982] Some varieties containing relation algebras. Trans. Amer. Math. Soc 272, 501 526. MADDUX, R. [1987] Pair-dense relation algebras. Trans. Amer. Math. Soc., accepted. MONK, J. D. [1961] Studies in cylindric algebra. Doctoral dissertation, University of California,

Berkeley. vi + 83pp.


NI~METI, 1. [1986] Free algebras and decidability in algebraic logic. Dissertation for D.Sc. with Hung. Academy of Sciences, Budapest. Submitted.

NI~METI, I. [1987] Decidability of relation algebras with weakened associativity. Proc. Amer. Math. Soc. 100, 340-344.

N[~METI, I. [1987a] On varieties of cylindric algebras with applications to logic. Annals. of Pure and Applied Logic 36, 235-277.

TARDOS, G. [1988] Personal communication. TARSKI, A. and GIVANT, S. [1987] A formalization of set theory without variables. AMS Colloquium

Publications 41, Providence, Rhode Island. THOMPSON, R. J. [1988] Noncommutative cylindric algebras and relativization of cylindric algebras.

Bulletin of Section of Logic, Lodz, Poland, 17, 75-81. WERNER, H. [1978] Discriminator algebras. Akademie Verlag, Berlin.

Department of Mathematics Vanderblit University Nashville, TN 37325 USA

Mathematical Institute of the Hungarian Academy of Science

Budapest, PF. 127, H-1364 Hungary

free algebras in discriminator varieties

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