Periodica Mathematics Hungarica Vol. 32 (3), (1996), pp. 229-236

FURTHER REFINEMENTS OF THE PRIMAL ALGEBRA CHARACTERIZATION THEOREM

LAsz~6 SZAB~* (Szeged)

Abstract

The compatible relations and the automorphisms of finite algebras are investi- gated. Moreover, stronger versions of Rosenberg’s Primal Algebra Characterization Theorem and of A. Szendrei’s Primal Algebra Characterization Theorem are given.

Introduction

In [6], [2] and [3] th e author and his co-workers described the nontrivial finite functionally incomplete algebras with 3-transitive, 2-transitive and primitive auto- morphism groups, and later in [7], [8] and [9] th e author described the finite simple functionally incomplete algebras with transitive automorphism groups. The main tools in the proofs were Rosenberg’s Primal Algebra Characterization Theorem [4],

[5] and its stronger versions given by A. Szendrei in (lo], [ll] and [13]. In the proofs it turned out that if an algebra has a compatible relation of certain types listed in the Primal Algebra Characterization Theorem then it has also a compati- ble relation preserved by every automorphism. The aim of the present paper is to investigate the compatible relations and the automorphisms of finite algebras and to give stronger versions of Rosenberg’s theorem (Theorem 3.1) and of Szendrei’s theorem (Theorem 3.3).

1. Notions and notations

Let A be a nonempty set. By a clone we mean a set of finitary operations on A which is closed under superposition and contains all projections. If A = (A; F)

*Research partially supported by Hungarian National Foundation for Scientific Re- search grant no. 1903.

Mathematics subject ctassificution numbers, 1991. Primary 08A40. Key words and phrases. Primal algebra, compatible relation, automorphism.

0031~5303/96/S5.00 @ AkadCmioi Kiado’, Budapest

Akoddmiai Kiadb, Budapest Kluwer Academic Publishers, Dordrecht

230 SZAB6

is an algebra then CloA and Clo, A denotes the set of all term operations and of all n-ary term operations of A (n 1 l), respectively. For a unary algebra C and m > 1, the m’th matrix power of C will be denoted by Cf”] (for the definition see e.g., [ll]). The automorphism group of A is denoted by Aut A.

For a finitary relation p on A the set of operations preserving p forms a clone, which is denoted by Polp. A binary relation p is called nontrivial if it is distinct from the identity relation w and from the full relation A’; the converse of p is the relation p-l = UY,X>: (5, Y> E ~1. F or an h-ary relation p and a permutation ?r on A put p” = {(UlT, . . . , ahr): (al,. . . , ah) E p}.

The readers are supposed to be familiar with Rosenberg’s Primal Algebra Characterization Theorem [5], and with the notions of primal, quasiprimal, semi- affine and affine algebras (for the definitions see e.g., [ll]).

Following D. Hobby and R. McKenzie [I] we call an algebra A = (A; F) Abelian if it satisfies the so-called term condition: for all n 2 k > 1, for every n-ary term operation f of A, and for arbitrary U, V E Ak, 7i, 6 E Anbk,

f(21, ii) = f(ti, 6) * f(V, ii) = f(C, b).

Furthermore, A is strongly Abelian if it satisfies the strong term condition: for - - all n L Ic 2 1, for every n-ary term operation f of A, and for arbitrary U, v E

Ak, a,b,cE An-k,

f(ii, ii) = f(C, 6) * f(ii, E) = f(5, E).

It is known that strongly Abelian and semi-affine algebras are Abelian, affine alge- bras are Abelian but not strongly Abelian, and matrix powers of unary algebras are strongly Abelian.

2. Preliminary results and lemmas

We need a characterization of Abelian and strongly Abelian algebras in terms of compatible relations. (Concerning strongly Abelian algebras cf. [14; Lemma 3.51.)

LEMMA 2.1. An algebra A = (A; F) is Abelian if and only if A has a 4-ary compatible relation p such that

(a) (a,~, b, b) E p and (a, b, a, b) E p for all a, b E A, and (b) for any elements a, b, t, y E A, with (a, b, x, y) E p, we have a = b if and only

ifx=y. Moreover, an algebra A = (A; F) is strongly Abelian if and only if A has a 4-ay compatible relation u such that

(c) (a, b, a, b) E o and (a, b, c, c) E u for all a, b, c E A, and (d) for any elements a, x, y E A, (a, a, x, y) E u implies x = y.

PROOF. First consider the subalgebra p generated by the quadruples (a, a, b, b) and (a, b, or, b) with a, b E A. We have

P = {(f(& 4, f(% $1, f(C, df@, Q):

REFINEMENTS OF THE PRIMAL ALGEBRA CHARACTERIZATION THEOREM 231

7121, O<h<n, ti,GgAk, 7i,b~A”-~, f~Clo,A}.

Clearly p is the least 4-ary compatible relation of A satisfying (a). Moreover, A is Abelian if and only if p has property (b).

Second, consider the subalgebra u generated by the quadruples (a, b, a, b) and (0, b, c, c) with a, b, c E A. We have

fJ = {(K% q, f(% b), f(% El, f(C, E)):

nil, O<k<n, uL,ijcAk, G,b,c~An-~, f~Clo,A}.

Clearly p is the least Gary compatible relation of A satisfying (c). Moreover, A is strongly Abelian if and only if u has property (d). This completes the proof. H

COROLLARY 2.2. If A = (A;F) is an Abelian (strongly Abelian) algebra then the algebra (A; F U Aut A) is also Abelian (strongly Abelian).

PROOF. Let A = (A; F) b e an Abelian (strongly Abelian) algebra. By Lemma 2.1, there exists a compatible 4-ary relation p (u) of A satisfying the properties (a) and (b) ((c) and (d)) of Lemma 2.1. Consider the relation

7= n{pY ?r E Aut A} (T = n{u”: a E Aut A})

It is easy to check that r is a compatible 4-ary relation of A satisfying the properties (a) and (b) ((c) and (d)). M oreover, it is preserved by every automorphism of A. Therefore, by Lemma 2.1, we have that (A; FUAut A) is Abelian (strongly Abelian). n

The following lemma was proved in [73.

LEMMA 2.3. If a 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 regular relation.

LEMMA 2.4. If a finite algebra has a nontrivial compatible at least ternary totally reflexive and totally symmetric relation then it has either a compatible at least ternary central relation preserved by every automorphism or a compatible regular relation preserved by every automorphism.

PROOF. Let A = (A; F) be a finite algebra having a nontrivial compatible at least ternary totally reflexive and totally symmetric relation p. Then it is easy to see that the relation

u= n{pY 7r E Aut A}

is a nontrivial compatible at least ternary totally reflexive and totally symmetric relation of A preserved by every automorphism. Hence F UAut A c Pol u. Consider the algebra (A; Pola). Since u is an at least ternary totally reflexive relation, Polo contains all operations taking at most two values. Therefore, it is easy to show that the algebra (A; Polu) has no nontrivial binary compatible relations and cannot be semi-affine. Prom this, by Rosenberg’s Primal Algebra Characterization Theorem [5], we have that (A; Pol u) (and thus (A; F U Aut A)) has either a compatible at

232 SZAB6

least ternary central relation or a compatible regular relation. This completes the proof. n

By the basics of tame congruence theory [l], a finite simple algebra has type 1, 2, 3, 4 or 5. Moreover, it is of type 1 if and only if it is strongly Abelian, and it is of type 2 if and only if it is Abelian but not strongly Abelian.

h3MMA 2.5. Let A be a finite simple algebra of type 2, 3, 4 or 5.

(1) if A has a nontrivial compatible binary reflexive and symmetric relation p then it also has a nontrivial compatible binary rejlexive and symmetric relation u such that u s p and u is preserved by every automorphism of A. Moreover, A has either a compatible at least binary central relation preserved by every au- tomorphism or a compatible regular relation preserved by every automorphism ofA.

(2) If A has a nontrivial compatible bounded partial orderp then it has also a com- patible bounded partial order u such that p C CT and for every automorphism ~ofAwehaveeithera”=oora*=a-i.

PROOF. Let A = (A; F) b e a finite simple algebra of type 2, 3,4 or 5. Define UA to be the set of all sets of the form f(A) where f is a unary polynomia1 operation of A with If(A)1 > 1, and define MA to be the set of minimal members of UA.

First suppose that A has a compatible, binary, reflexive and symmetric rela- tion p and consider the relation 7 = w U U{X’: X E MA}. By [l; Lemma 5.241, we have r C p. Furthermore, it is easy to check that if X E MA and r E Aut A then XT E MA. Therefore every automorphism of A preserves r. Let u be the compati- ble relation of A generated by 7. (Notice that, by [l; Lemma 5.241, if A is of type 2 or 3 then u = r.) Then r c u E p and it is easy to see that u is preserved by every automorphism of A. Applying Lemma 2.3 for the simple algebra (A; F U Aut A), we have that it has either a compatible at least binary central relation or a compatible regular relation. This completes the proof of the first statement.

Second suppose that A has a compatible bounded partial order p. Then, by [l; Lemma 5.261, A’ is of type 4 or 5 and there are four compatible partial orders

Co,(‘l,Eo,<~ OfAsuch that C Et (i=O,l), Cl =CO’, it1 =Ei’, tofltl =w, and either cc G p c [O or (‘1 E p s &. We can suppose that CO s p E to. It follows that to is a bounded partial order. If r E Aut A then tt is also a compatible partial order and by [l; Lemma 2.261, we have cc C (0 or 50” c <i = [il. Since A is a finite algebra, this implies that 60” = 60 or <c = to’. Hence u = & satisfies the condition of the second statement. This completes the proof. n

The next lemma is a direct consequence of Corollary 1.6 and Lemma 3.3 in

LEMMA 2.6. Let A be a reduct of the m-th matrix power of a finite vnary algebra U (m 2 1). If A is simple and has no compatible regular relation, then U is a 2-element algebra and A is isomorphic to an algebra term equivalent to (U’)l”‘l for an appropriate reduct U’ of U.

REFINEMENTS OF THE PRIMAL ALGEBRA CHARACTERIZATION THEOREM 233

Finally we need the following lemma which is a weaker version of Theorem 2.1 in 1121.

LEMMA 2.7. For arbitrary finite simple algebra A that is semi-afine with respect to an elementary Abelian p-group (p prime) one of the following conditions hold:

(1) A is afine with respect to an elementary Abelian p-group (p prime); (2) A is isomorphic to an algebra which is a reduct of a matrix power of a unary

algebra; (3) A has a compatible regular relation.

From Lemma 2.6 and 2.7 we obtain

COROLLARY 2.8. Let A be a finite simple algebra that is semi-a&e with respect to an elementary Abelian p-group (p prime). If A has no compatible regular relation then it is either afine or is isomorphic to an algebra term equivalent to a matrix power of a 2-element unary algebra.

3. Results

The next theorem is a refinement of Rosenberg’s Primal Algebra Characteri- zation Theorem [4], [5].

THEOREM 3.1. For a finite non-primal algebra A = (A; F) one of the follow- ing conditions holds:

(X1.1) A has a compatible bounded partial order p such that for every z E Aut A we have either p” = p or p” = p-‘;

(3.1.2) A has a compatible binary relation {(a, ox): a E A} where rr is a permu- tation of A with [Al/p cycles of the same length p (p prime);

(3.1.3) A is semi-afine with respect to an elementary Abelian p-group (p prime);

(3.1.4) A has a nontrivial congruence relation; (3.1.5) A has a compatible central relation preserved by every automorphism of

A if it is at least binary; (3.1.6) A has a compatible regular relation preserved by every automorphism

ofA.

PROOF. Let A be a finite non-primal algebra. If A is not simple then we have (3.1.4). If A has a compatible unary central relation then we have (3.1.5). If A is Abelian then, by Lemma 2.1, the algebra A = (A; F U Aut A) is also Abelian. Applying Rosenberg’s Primal Algebra Characterization Theorem for A, we have that one of the conditions (3.1.1), (3.1.2), (3.1.3), (3.1.4), (3.1.5) and (3.1.6) holds.

Now suppose that A is a simple non-Abelian algebra having no compatible unary central relation and no compatible binary relation {(a, as): a E A} where z is a permutation of A with IAl/p cycles of the same length p (p prime). Then A is of type 3, 4 or 5. A cannot be semi-afhne since semi-affine algebras are Abelian.

234 SZAB6

Therefore, again by Rosenberg’s Primal Algebra Characterization Theorem, A has either a compatible bounded partial order or a compatible at least binary central relation or a compatible regular relation.

If A has a compatible bounded partial order then, by Lemma 2.5(2), we have (3.1.1). If A has a compatible at least ternary central relation or a compatible regular relation then our statement follows from Lemma 2.4. Finally, if A has a compatible binary central relation then, by Lemma 2.5(l), we have (3.1.5) or (3.1.6). This completes the proof. n

In [ll] and [13] A. Szendrei characterized the finite simple algebras having no proper subalgebras.

THEOREM 3.2 (A. Szendrei [ll], [13]). F or a finite simple algebra A having no proper subalgebra one of the following conditions holds:

(3.2.1) A is quasiprimal; (3.2.2) A is afine with respect to an elementary Abelian p-group (p is prime); (3.2.3) A is isomorphic to an algebra term equivalent to U[“l for some 2-ele-

men2 unary algebra U and for some integer m 2 2; (3.2.4) A has a compatible at least binary central relation; (3.2.5) A has a compatible regular relation. (3.2.6) A has a compatible bounded partial order. .

The next theorem is a stronger version of Theorem 3.2.

THEOREM 3.3. For a finite simple algebra A having no proper subalgebra one of the following conditions holds:

(3.3.1) A is quasiprimal; (3.3.2) A is afine with respect to an elementary Abelian p-group (p is prime); (3.3.3) A is isomorphic to an algebra term equivalent to TJml for some 2-ele-

ment unary algebra U and for some integer m 1 2; (3.3.4) A has a compatible at least binary central relation preserved by every

automorphism of A; (3.3.5) A has a compatible regular relation preserved by every automorphism

ofA. (3.3.6) A has a compatible bounded partial orderp such that for every x E Aut A

we have either p” = p or pT = p-l.

PROOF. Let A = (A; F) be a finite simple algebra that has no proper subalge- bra. First suppose that A is not Abelian, i.e., it is of type 3, 4 or 5. Apply Theorem 3.2. The first three conditions are the same in Theorem 3.2 and Theorem 3.3. If A has a compatible at least ternary central relation or a compatible regular relation then, by Lemma 2.4, we have (3.3.4) or (3.3.5). If A has a compatible binary central relation or a compatible bounded partial order then, by Lemma 2.5, we have (3.3.4) or (3.3.5) or (3.3.6).

Now suppose that A is Abelian. If A has a compatible regular relation then, by Lemma 2.4, we have (3.3.4) or (3.3.5). From now on suppose that A has no compatible regular relation. Consider the algebra A = (A; F U Aut A) which, by

REFINEMENTS OF THE PRIMAL ALGEBRA CHARACTERIZATION THEOREM 235

Corollary 2.2, is also Abelian. Apply Theorem 3.2 for A. Since A is Abelian it cannot be quasiprimal. Moreover, by our assumption on A, the algebra A has no compatible regular relation. If A has a compatible at least binary central relation or a compatible bounded partial order then we have (3.3.4) or (3.3.6). If A is affine then, by Corollary 2.8, we have (3.3.2) or (3.3.3). If A is equivalent to a matrix power of a P-element unary algebra then, taking into consideration Lemma 2.6, we have (3.3.3). This completes the proof. H

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(Received: January 30, 1995) (In final form: August 29, 1995)

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