algebras with transitive automorphism groups

Algebra Universalis, 31 (1994) 589-598 0002-5240/94/040589 10501.50+0.20/0 �9 1994 Birkh~iuser Verlag, Basel

Algebras with transitive automorphism groups

LASZL6 SZABO

O. Introduction

As a rule a finite algebra with "large" automorphism group is functionally complete. The first general result was found by B. Cs~ikfiny [2], who proved that almost every nontrivial homogeneous algebra (i.e. an algebra whose automorphism group is the full symmetric group) is functionally complete; up to equivalence there are six exceptions. Csfikfiny's theorem was first extended to algebras with triply transitive automorphism groups [8] and later to algebras with doubly transitive automorphism groups [3]; the exceptions are the affine spaces over finite fields. The most general reuslt in this direction is in [4], where the structure of functionally incomplete algebras with primitive automorphism groups is completely described.

In [9] we investigated finite idempotent algebras with transitive automorphism groups. We proved that if an at least three element finite idempotent algebra with transitive automorphism group is simple and has no compatible binary central relation then it is either functionally complete or affine. Moreover, if an at least three element finite idempotent algebra with transitive automorphism group is simple and has a nontrivial semi-projection or a majority function among its term functions then it is functionally complete. The main tools in proving the above results were Rosenberg's Completeness Theorem [5], [6] and functionally completeness criterion for surjective algebras [7].

In the meantime A. Szendrei proved several rather deep results for finite simple surjective algebras [10], [11], [12]. Using them we managed to extend our earlier results for finite idempotent algebras with transitive automorphism groups to arbitrary finite algebras with transitive automorphism groups. Moreover, we ob- tained several additional conditions for a simple finite algebra with transitive automorphism group, which ensure that the algebra is either functionally complete or affine. The aim of this paper is to give and prove these results.

Presented by I. Rosenberg. Received March 23, 1992; accepted in final form January 24, 1993. Research partially supported by Hungarian National Foundation for Scientific Research grant no.

1903.

589

590 LASZL6 SZAB6 ALGEBRA UNIV.

1. Preliminaries

Let A be a nonempty set. Fo r any positive integer n let O~ ) denote the set o f all n-ary opera t ions on A (i.e. m a p s A " ~ A) and let OA = U~=I O~ ). An opera t ion

f rom OA is nontrivial if it is not a projection. By a clone we mean a subset o f OA which is closed under superposi t ions and contains all projections. A ternary

opera t ion f on A is a majority function if for all x, y e A we have f ( x , x, y ) =

f ( x , y, x) = f ( y , x, x) = x ; f i s a Mal'tsev function i f f ( x , y, y) = f ( y , y, x) = x for all x, y e A. An n-ary opera t ion t o n A is said to be an i-th semi-projection (n > 3,

1 < i < n) if for all Xl , . �9 �9 x , e A we have t(xl, � 9 xn) = xi whenever at least two elements a m o n g x I . . . . , x , are equal. An n-ary opera t ion f is conservative if for all

x l , . � 9 x , e A we have f ( x l , �9 � 9 x , ) E {x 1 . . . . , x n } ,

For a nonempty set N, let SN and CN denote the full symmetr ic group on N and the set o f all (unary) cons tant opera t ions on N, respectively. A pe rmuta t ion group

G c_ SN is transitive if for any x, y e N there exists a pe rmuta t ion r~ s G such that

xn = y ; G is said to act primitively on N if (N; G) is a simple algebra and IG[ > 1

( i f IN[ = 2). Clearly, primit ivi ty implies transitivity. A subset F _ OA as well as the algebra (A, F ) is primal or complete if the clone

generated by F (i.e. the set o f all term functions of (A, F)) is equal to Oa ; F as well

as the algebra (A, F) is functionally complete if the clone generated by F w Ca (i.e. the set o f all algebraic functions of (A, F)) is equal to O A. An algebra is surjective if its fundamenta l opera t ions are all surjective.

Let n, h > 1. An n-ary opera t ion f s O~ ) is said to preserve the h-ary relation p _c A h if p is a subalgebra of the h-th direct power of the algebra ( A ; f ) . Then the

set o f opera t ions preserving p forms a clone, which is denoted by Pol p. We say tha t

a relation p is a compatible relation of the algebra (A, F ) if F _ Pol p. A binary relation is called nontrivial if it is distinct f rom the identity relat ion o~ and f rom the full relation A 2.

An h-ary relation p on A is called central if p ~ A h and there exists a n o n e m p t y

proper subset C of A such that

(a) ( a l , . . . , ah) e p whenever at least one a i e C (1 <- i < h); (b) p is totally symmetric, i.e. (a~ . . . . . ah) ~ p implies ( a l ~ , . . . , ah~)~ p for

every pe rmuta t ion rc of the indices 1 , . . . , h;

(c) p is totally reflexive, i.e. (al . . . . . ah) e p if a i = a s for some i r (1 < i , j < h ) .

The largest set C satisfying (a) is called the center of p. Let h > 3. A family T = {O1 . . . . . O,~ } (m > 1) of equivalence relations on A is

called h-regular if each O i (1 <- i < m) has exactly h blocks and O T = Oi c~. - -c~ Om

Vol. 31, 1994 Algebras with transitive automorphism groups 591

has exactly h m blocks (i.e. the intersection Am= I Bi of arbitrary blocks Bi of O~

(i -- 1 . . . . . m) is nonempty). The relation determined by T is

2 r = {(al . . . . , ah) ~ Ah l al . . . . , a h are not pairwise incongruent

modulo Oi for all i (1 < i < m)}.

Note that h-regular relations are both totally reflexive and totally symmetric. Let C = ( C , F ) be a unary algebra and let m ,n > 1. Consider the sets

M = { 1 , . . . , m } and N = {1 . . . . . n}. For given mappings a : M ~ M , p : M ~ N

and g~ . . . . . gm e O~ ~ let us define an n-ary operation h~[g~ . . . . , gin] on C m as follows: For xi = (x ] , . . . , x m) ~ C m, i = 1 . . . . , n set

h , [ g l , . g m ] ( X l , X , ) - - 1~ mo . . . . . . , - - ( g l ( X l u ) , . . . , g m ( X m u ) ) .

Now the m' th matrix power of C, denoted by C tin1, is the algebra with universe C m

and with all the functions h ~ [ g l , . . . , gin] as fundamental operations.

An algebra A is semi-affine with respect to an elementary abelian group ~,, if A an A has common base set A and the quaternary relation of the form

{(x,y,z, t) ~ .A 4 I x - - y § z = t}

is a compatible relation of A; if in addition x - y + z is a term function of A then it is said to be affine with respect to A.

Now we formulate three theorems of A. Szendrei which are the main tools in proving our results:

T H E O R E M A (A. Szendrei [11]). For a finite simple surjective algebra A one o f

the following conditions holds:

(1) A is functionally complete;

(2) A is affine with respect to an elementary Abelian p-group (p is prime);

(3) A is isomorphic to a reduct o f (N; SN) Emj for a nonsingleton finite set N and f o r some integer m > 1;

(4) A has a compatible binary central relation;

(5) A has a compatible bounded partial order.

T H E O R E M B (,~. Szendrei [10]). Let A be a simple surject&e algebra such that

A is isomorphic to a reduct o f ( N ; Su)~mlfor some finite set N (IN[ > 2) and for some

m > 1. I f m is chosen minimal with respect to the existence o f such an isomorphism,

then A is isomorphic to an algebra term equivalent to (N; G ) H for some permutation

group G on N which acts primitively on N i f IN] > 1.

592 LASZL6 SZABO ALGEBRA UNIV,

T H E O R E M C (A. Szendrei [12]). Let A be a finite simple surjective algebra that

is semi-affine with respect to an elementary Abelian p-group A. Then A is either affine

with respect to A or it is isomorphic to a reduet o f (N; Su)Eml for some prime element

set N and for some m > 1.

2. Lemmas

We need the next two lemmas from [9]:

L E M M A 2.1. I f an at least three element finite algebra with transitive automor-

phism group has a compatible bounded partial order then it has a nontrivial compatible

binary reflexive and symmetric relation.

L E M M A 2.2. I f an at least three element finite algebra has a nontrivial compatible binary reflexive and symmetric relation then it has either a nontrivial congruence

relation, or a compatible at least binary central relation, or a compatible relation

determined by an h-regular family o f equivalence relations.

The next lemma was formulated in [11] and originates from [7].

L E M M A 2.3. Let A be a finite surjective algebra. I f for some h > 3, A has a

compatible h-regular relation 2 T with ITI = m and 6) T -- o~, then A is isomorphic to a

reduct o f ( N ; Su) Eml for some h-element set N.

L E M M A 2.4. Consider the matrix power (N; G)~ml for some m >- 1,for some finite set N with INI > 2 and for some permutation group G acting on N. I f the automorphism

group o f (N; G) E'~ is transitive then m = 1. I f in addition G acts primitively on N then

INI is prime and G is generated by a cyclic permutation of order INI.

Proof. Clearly, if the automorphism group of an algebra is transitive then its term functions are all surjective. Moreover, if m > 1 then the matrix power (N; G) Era1 has

unary term functions which are not surjective. Hence m = 1. Consider the algebra (N; G), and suppose that G acts primitively on N. Let a, b ~ N be two distinct elements and let ~, z be automorphisms of (N; G) such that a~ = b and bt = a. Then for any f ~ G fixing a and g ~ G fixing b we have g ( a ) = g ( b t ) - -

g(b)t = bt = a and f ( b ) = f ( a ~ ) = f ( a ) t = at = b showing that the stabilizer subgroups of a and b coincide. Since the stabilizer subgroups of two distinct elements cannot coincide in a primitive permutation group of composite order (see [ 13; Prop. 8.6]), it follows that IGI= IUl = p for some prime number p. This completes the proof.


L E M M A 2.5. Every at least two element finite simple non-unary semi-affine algebra with transitive automorphism group is term equivalent either to the algebra

(Kn;x - y +z , {rx + ( 1 -r)ylr E K~ •

or to the algebra

(K"; x - y +z , {rx + ( 1 - r ) y [r ~Kn• {x + a [a EKn})

where n >- 1, K is a finite field, Kn • n is the n x n matrix ring over K.

Proof. Let A be an at least two element finite simple semi-affine algebra with transitive automorphism group. Then A is surjective, and therefore, by Theorem C, it is either affine or is isomorphic to a reduct of (N; SN) ~ml for some prime element set N and for some m --> 1. If A is isomorphic to a reduct of (N; S u ) H then, by Theorem B and Lemma 2.4, we have that A is a unary algebra, contrary to our assumption. Hence A is affine.

D. M. Clark and P. H. Krauss proved in [1] that every at least two element finite simple affine algebra is term equivalent either to the algebra

(K";x - y +z , {rx + ( 1 - - r ) y ] r ~K~• ekx)

or to the algebra

( K " ; x - y + z, {rx + ( 1 - - r ) y I r ~ K . • ekx, {x + a [a eK"})

where n -> 1, 0 <- k -< n, K is a finite field, K. • ~ is the n • n matrix ring over K, and ek ~ K~ • is the diagonal matrix

( ~ 0 ' i ) (~ l , , ,

" . %

, . ~

with the first k entries in the diagonal equal to 1. Since the term functions of A are all surjective and, clearly, etx is surjective if and only if l = n, we have that k = n. Thus ekx = e ,x is the unary trivial operation. It is well-known and easy to show that the automorphism groups of the algebras given in the lemma contain the permutation group {x + a [ a ~ K ~} which is transitive. This completes the proof.

594 L~,SZL6 SZAB0 ALGEBRA UNIV.

3. Results and proofs

THEOREM 3.1. For an at least three element -finite simple algebra A with

transitive automorphism group one o f the following conditions holds:

(a) A is functionally complete;

(b) A is term equivalent either to (Kn; x - y + z, {rx + (1 - r)y : r ~ K, • ~ }) or

to ( K ~ ; x - y + z , { r x + ( 1 - r ) y : r ~ K ~ • where n > 1,

K is a -finite f ield and Kn • ~ is the n x n matrix ring over K;

(c) A is term equivalent to (A ; a) fo r a prime element set A and for a cyclic

permutation a o f order IA [; (d) A has a compatible binary central relation.

Proo f Let A be a simple at least three element finite algebra with transitive automorphism group. The transitivity of the automorphism group implies that A is a surjective algebra. Therefore, by Theorem A, we have that A satisfies one of the conditions (1)-(5). Cases (1) and (4) coincide with cases (a) and (d), respectively. In case (2) Lemma 2.5 implies (b), in case (3) Theorem B and Lemma 2.4 yield (c).

Finally suppose that A has condition (5). Since A is simple, taking into consideration Lemmas 2.1 and 2.2, we obtain that A has a compatible at least binary central relation or a compatible relation determined by an h-regular family T. It is known that if a surjective algebra has a compatible at least binary central relation then it has also a compatible binary central relation (see e.g. [7]). Therefore, the first case implies (d). It is also known that if a surjective operation preserves a relation determined by an h-regular family T, then it preserves the equivalence relation Or (see e.g. [7]). Therefore, in the second case Or is a congruence relation of the simple algebra A. Consequently, Or = o. Then Lemma 2.3, Theorem B and Lemma 2.4 imply (c). This completes the proof.

We do not know if case (d) can in fact occur. So we formulate the following

PROBLEM. Is every at least three element -finite non-unary algebra with transi-

tive automorphism group either functionally complete or equivalent to one o f the affine

algebras given in condition (b)?

The next results show that in several cases the answer is positive.

COROLLARY 3.2. For an at least three element finite simple non-unary algebra

A with transitive automorphism group that has no proper subalgebras one o f the

conditions (a) or (b) holds.


P r o o f Since A is surjective, and it is known (see e.g. [7]) that the center of a compatible central relation of a surjective algebra is a (proper) subalgebra, our statement follows from Theorem 3.1.

T H E O R E M 3.3. For an at least three e lement f i n i t e s imple algebra with transit ive

au tomorph i sm group that has a nontr iv ial at least ternary term func t ion t sa t i s fy ing the

ident i ty t ( x , y . . . . . y ) = x one o f the condit ions (a) or (b) holds.

P r o o f Let A = (A, F) be an at least three element simple finite algebra with transitive automorphism group and let t be a nontrivial at least ternary term function of A satisfying the identity t (x , y . . . . , y ) = x . Since A is non-unary, using Theorem 3.1, we have to show only that A has no compatible binary central relations. Suppose that p is a compatible binary central relation of A with center C and let c ~ C.

We call a subset I ___ A an ideal if t(al . . . . , an) E I whenever al ~ L Since an intersection of ideals is an ideal again, we may speak about an ideal generated by a subset of A. For any a ~ A denote by I (a) the ideal generated by {a}. Clearly, if I is an ideal and n ~ Aut A then In is again an ideal, and I (a )n = I (an) . Because of the transitivity of Aut A, the cardinalities of the 1-generated ideals are equal, and greater than one since t is not the first projection. So the 1-generated ideals form an Aut A-invariant partition of A. Denote by 0 the corresponding equivalence relation. Then 0 is distinct from the identity relation and Aut A __ Pol 0. We show that 0 _c p. Let a, b ~ A with I(a) = I(b) , and consider the subset I a = {x I (x, a) ~ p }. Then I a is an ideal. Indeed, if xl E Ia and x2 . . . . . x n c A are arbitrary elements, then (xl, a), (x2, c) . . . . , (x,, c) e p implies that ( t (x l . . . . , x,), a) = (t(Xl, x 2 , . . . , x,), t(a, c . . . . , c)) E p, i.e. t(Xl . . . . , xn) E Ia. Now, since Ia is an ideal with a e /~ , we have b ~ I (b) = I(a) ~_ Ia and (b, a) e p. Hence 0 ___ p.

Consider the subalgebra o- of A 2 generated by 0. Then 0 c a _ p and F u Aut A ~ Pol a, i.e. a is a nontrivial compatible binary reflexive and symmetric relation of the algebra ~i, = (A; F u Aut A). Therefore ~, is not functionally complete. Now, by the transitivity of Aut A, the algebra i has neither compatible bounded partial orders nor compatible binary central relations. Therefore, taking into consideration Theorem A, we have that ~, is either affine or is isomorphic to a reduct of (N; SN)E~ for a nonsingleton finite set N and for some integer m > 1. If.~ is affine then A is semi-affine. From this, by Lemma 2.5, we have condition (b). If .~ is isomorphic to a reduct of (N; S N ) tin1 for a nonsingleton finite set N and for some integer m > 1, then Theorem B and Lemma 2.4 imply that ,~ and A is a unary algebra, a contradiction. This completes the proof.

COROLLARY 3.4. For an at least three e lement f i n i t e s imple algebra with

transi t ive au tomorph i sm group that has a M a l ' t s e v f unc t ion among its term func t i ons

one o f the condit ions (a) or (b) holds.

596 LASZL6 SZABO ALGEBRA UN1V.

COROLLARY 3.5. I f an at least three element finite simple algebra with transitive automorphism group has a nontrivial semi-projection among its term functions, then it is functionally complete.

Proof It is well-known (see e.g. [4]) that a nontrivial semi-projection cannot preserve a quaternary relation given in the definition of semi-affine algebras. Therefore our statement follows from Theorem 3.3.

T H E O R E M 3.6. I f an at least three element finite simple algebra with transitive automorphism group has a majority term function, then it is functionally complete.

Proof Let A = (A, F) be an at least three element simple finite algebra With transitive automorphism group, and let d be a majority term function of A. Using Theorem 3.1, we have to show only that A is not affine and has no compatible binary central relations.

It is well-known (see e.g. [4]) that a majority function cannot preserve a quaternary relation given in the definition of semi-affine algebras. Hence A is not semi-affine.

Now suppose that p is a compatible binary central relation of A with center C, and let c ~ C. We call a subset I ~ A an ideal iff d(x, y, z) ~ I whenever at least two of the arguments belong to L The set A and the one-element subsets are obviously ideals. Moreover, for any a E A the set I~ = {x [ (x, a) ~ p} is also an ideal. Indeed, if for example x, y ~ Ia and z ~ A is arbitrary element, then (x, a), (y, a), (z, z) ~ p implies that (d(x, y, z), a) = (d(x, y, z), d(a, a, z)) E p, i.e. d(x, y, z) ~ Ia. Clearly, if I is an ideal and zc ~ Aut A then I~ is again an ideal.

Define a binary relation 0 by setting (a, b) ~ 0 if and only if there is a minimal ideal (i.e. an ideal properly containing one-element ideals only) containing a and b. Then 0 is distinct from the identity relation and Aut A ~ Pol 0. We show that 0 _ p. Indeed, let (a, b) ~ 0. If a = b then (a, b) E p, too. If a vL b then put u = d(a, b, c) (c is a central element of p) and let I be the minimal ideal with a, b E L Now a = d(a, b, a), b = d(a, b, b) ~ I u. Since a and b are distinct, u is distinct from one of them, say u # b. By definition u E L We have u, b ~ I n Ib, SO by minimality of L it follows that I ~_ Ib, implying that (a, b) ~ p. Hence 0 ~ p.

Consider the subalgebra ~ of A 2 generated by 0. Then 0 _ ~ a _ p and F ~ Aut A _ Pol o-, i.e. cr is a nontrivial compatible binary reflexive and symmetric relation of the algebra ,~ = (A; F u Aut A). Therefore .~ is not functionally complete. By the transitivity of Aut A, the algebra ~ has neither compatible bounded partial orders nor compatible binary central relations. Therefore since A is not semi-affine, by Theorem A, we have that ~, is isomorphic to a reduct of (N; SN) Eml for a nonsingleton finite set N and for some integer m > 1. Taking into consideration Theorem B and Lemma 2.4, from this it follows that ~, is a unary algebra, a contradiction. This completes the proof.


T H E O R E M 3.7. I f an at least three element finite simple algebra with transitive automorphism group has a nontrivial conservative operation among its term functions then it is functionally complete.

Proof. Let A = (A, F) be an at least three element simple finite algebra with transitive automorphism group and let t be a nontrivial conservative term function of A. It is easy to check that a nontrivial conservative operation cannot be semi-affine. Therefore A is not affine. Moreover, it is well-known that every clone generated by a nontrivial idempotent operation contains either a majority function or a Mal'tsev function or a nontrivial semi-projection or a nontriviat binary idempotent operation (see e.g. [3]). Therefore the clone [t] generated by t contains an operation of one of the types listed above. In the first three cases our statement follows from Theorem 3.6, Corollary 3.4 and Corollary 3.5. Now let f ~ [t] be a nontrivial binary operation. Clearly, f is also conservative. Denote f ( x , y) simply by xy. Observe that f has the identities (xy)y = x ( x y ) = xy. Using Theorem 3.1, we have to show only that A has no compatible binary central relations. Suppose that p is a compatible binary central relation of A with center C.

We call a subset I _ A a right (left) ideal iff xy ~ I whenever x ~ I (y ~ I). Since an intersection of right (left) ideals is a right (left) ideal again, we may speak about a right (left) ideal generated by a subset of A. For any a e A denote by R(a) (L(a)) the fight (left) ideal generated by {a}. Clearly, if I is a right (left) ideal and rce Aut A then In is again a right (left) ideal, and R(a)n = R(ar 0 (L(a)n = L(an)) for any a e A. Because of the transitivity of Aut A the cardinalities of the 1-generated right (left) ideals are equal, and greater than one s incef i s not a projection. So the 1-generated right (left) ideals form an Aut A-invariant partition of A. First we show that A is either a 1-generated fight ideal or a 1-generated left ideal.

Suppose that R(a) is a proper subset of A for some a e A (and then, clearly, for every a s A). If n = IR(a)l and k is the number of pairwise different 1-generated right ideals, then we have IAI = kn. Now let a e A be an arbitrary element and consider the 1-generated left ideal L(a). I f x ~ A\R(a) , then from xa ~ R(x) and a r R(x) it follows that xa = x showing that x ~ L(a) and (A \R(a ) )w {a} ___ L(a). Therefore [L(a)I > (k - 1)n + 1 -> kn /2+ 1. Since IL(a)t is a divisor of IAI =kn, this implies that [L(a)l = [A t and L(a)= A. I f L(a) is a proper subset of A, then an analogous argument shows that R(a) = A.

Now we can suppose that A = R(a) for some a e A (and then, clearly, for every a e A). Let x, y e A be arbitrary elements and c e C. Since R(x) = A, there are elements x 2 , . . . ,xn such that ( . . . ( (xx2)x3) . . . )x , =y . Then from (x,x) , (c, x2), . �9 �9 (e, x , ) e p we have (xc, y) = ( ( . . . ( (xc)e) . . . )c , ( . . . ((xx2)x3). . .)x, ,) ~ p and xc ~ C showing that C is a left ideal. Denote by 0 the equivalence relation whose blocks are the 1-generated left ideals. Then 0 is distinct from the identity relation and Aut A ~ Pol 0.

598 LfilSZLO SZAB6 ALGEBRA UNIV.

We show that 0 ___p. Let a ,b ~ A with L ( a ) = L (b ) . If a e C or b e C then (a,b) e p . Suppose a , b r and let c ~ C . Since b s L ( a ) there are elements x2 . . . . . x , such that xn(. . . ( x 3 ( x 2 a ) ) . . .) = b. Moreover, ca = a since ca = c implies that a ~ L ( a ) = L ( c a ) = L ( c ) c C, a contradiction. Then from (c, x n ) , . . . , (c, x2), (a, a) e p we have (a,b) = ( c a , b) = ( c ( . . . ( c ( c a ) ) . . . ) , x , ( . . . ( x 3 ( x 2 a ) ) . . . ) ) ~ p.

Hence 0 _ p. This fact, as we have seen in the proof of Theorem 3.6, leads to a contradiction, which completes the proof.

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[2] CSAKANV, B., Homogeneous algebras are functionally complete, Algebra Universalis 11 (1980), 149-158.

[3] P~.LFY, P. P., SZAB6, L. and SZENDREI, ~., Algebras with doubly transitive automorphism groups, in: Finite Algebra and Multiple-Valued Logic (Proc. Conf. Szeged, 1979), Colloq. Math. Soc. J. Bolyai, vol. 28, North-Holland, Amsterdam, 1981, 521-535.

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[5] ROSENBERG, I. G., Ober die funktionale Vollstdndigkeit in den mehrwertigen Logiken (Struktur der Funktionen yon mehreren Ver/inderlichen auf endlichen Mengen), Rozpravy (2eskoslovenske Akad. V6d 1Rada Mat. Pfirod. V6d 80 (1970), 9-93.

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[8] SZAB6, L. and SZENDRE1, A,, Almost all algebras with triply transitive automorphism groups are functionally complete, Acta Sci. Math. (Szeged) 41 (1979), 391-402.

[9] SZAB6, L., Idempotent algebras with transitive automorphism groups, Acta Sci. Math. (Szeged) 56 (1992), 215-222.

[10] SZENDREI, /~, Simple surjective algebras having no proper sualgebras, J. Austral. Math. Soc. (Series A) 48 (1990), 434-454.

[11] SZENDREI, A., Theprimal algebra characterization theorem revisited, Algebra universalis, to appear. [12] SZENDREI, A., Completeness criterion for semi-affine algebras, Proceedings of the 22nd Interna-

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Bolyai Institute Szeged, Hungary

algebras with transitive automorphism groups

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